IN MEMORIAM
FLORIAN CAJORl
HIGH SCHOOL ALGEBRA
BY
J. H. TANNER, Ph.D.
PROFESSOR OF MATHEMATICS IN CORNELL UNIVERSITY
NEW YORK .:• CINCINNATI •:• CHICAGO
AMERICAN BOOK COMPANY
The Modern Mathematical Series,
lucien augustus wait,
(^Senior Professor of Mathematics in Cornell University,)
GENERAL EDITOR.
This series includes the following works :
ANALYTIC GEOMETRY. By J. H. Tanner and Joseph Allen.
DIFFERENTIAL CALCULUS. By James McMahon and Virgil Snyder.
INTEGRAL CALCULUS. By D. A. Murray.
DIFFERENTIAL AND INTEGRAL CALCULUS. By Virgil Snyder and J.
Hutchinson.
HIGH SCHOOL ALGEBRA. By J. H. Tanner.
ELEMENTARY ALGEBRA. By J. H. Tanner.
ELEMENTARY GEOMETRY. By James McMahon.
The High School Algebra and the Elementary Algebra cover substantially the
same ground : each of them is designed to meet college entrance requirements in
elementary algebra ; the High School Algebra, however, presents the briefer and
simpler treatment of the two.
COi'VKlGHT, 190T, BY
J. H. TANNER.
TANNBB'8 HIGH 8CH. ALG.
W. P. I.
-3+
PREFACE
In the preparation of this book the author's aim has been :
(1) To make the transition from arithmetic to algebra as easy
and natural as possible, and to arouse the pupil's interest by
showing him early some of the advantages of algebra over
arithmetic.
(2) To present the several topics in the order of their sim-
plicity, giving definitions only where they are needed, and insur-
ing clearness of comprehension by an abundance of concrete
illustrations and inductive questions.
(3) To provide a large, well-chosen, and carefully graded set of
exercises, the solution of which will help not only to fix in the
pupil's mind the principles involved, but also further to unfold
those principles.
(4) To omit non-essentials, and yet provide a book that fully
meets the entrance requirements in elementary algebra of any
college or university in this country.
Among other features of this book to which attention is in-
vited are: (1) the careful statement of definitions and principles;
(2) the emphasis laid upon translating formulas and equations
into verbal language, and vice versa; (3) the inclusion of many
formulas from physics which the pupils are asked to solve for
the various letters which they contain; and (4) the extensive
cross-references, as well as the many " hints " and " suggestions "
found among the exercises and problems, all calculated to throw
sidelights upon the work.
On the request of several prominent mathematics teachers the
author has put an elementary chapter on quadratic equations
(Chap. XII) before the chapters on radicals, imaginaries, and the
theory of exponents. This arrangement is made possible by the
treatment of radicals of the second order given in § 121 (only such
iv PREFACE
radicals are met with, in Chap. XII), and it has important peda-
gogical as well as practical advantages over the more usual
arrangement. Those teachers, however, who prefer the usual
order, may omit § 121 altogether, and take Chaps. XIV and XV,
except §§ 162, 163, and 170, before taking Chap. XII.
In order to avoid unnecessary repetition, the work on graphs
has nearly all been collected into a single chapter. This arrange-
ment has made it possible to give this topic a somewhat more
adequate treatment than is usual in a book of this kind, and to
do so without giving it more than its rightful amount of space.
By this arrangement also those schools w^hich do not take graphs
in their first year's work will find their algebra work unin-
terrupted, while appropriately .placed footnotes indicate the
connection in which the parts of this chapter may be most advan-
tageously read by those who wish to include graphs.
Scattered through the book are a few articles (marked with
a *) which, should be taken in connection with the review work
when time permits ; the omission of these articles, however, does
not anywhere break the continuity of the work. For the benefit
of the brightest pupils in a class there have been inserted here
and there references to the author's Elementary Algebra, where
the topics concerned are discussed somewhat more fully ; a few
copies of the Elementary Algebra placed in the school library can
in this way be made to serve a very useful purpose.
It is with great pleasure that the author acknowledges his in-
debtedness to the many experienced teachers of Algebra in High
Schools and Academies in all quarters of our country whose sug-
gestions to him have added so much of value to this book. Special
acknowledgments are due to Prof. J. M. McPherron of the Los
Angeles High Schools for reviewing the manuscript before it
went to press, and to Miss Cora Strong of the State Normal
School of Greensboro, N.C., who secured a leave of absence from
her school so that she might give her entire time to assisting in
the work on this book ; to her mainly belongs the credit for the
excellent exercises which are found in the following pages.
CONTENTS
[See Index in back of book for particular topics.]
CHAPTER PAGE
I. Introduction
I. Literal Numbers . 1
II. Elementary Operations 8
11. Positive and Negative Numbers 16
III. Addition and Subtraction — Parentheses'
I. Addition 27
II. Subtraction 32
III. Parentheses 35
IV. Multiplication and Division
I. Multiplication . . . . . . . .38
11. Division 45
V. Equations and Problems
Review Exercise — Chapters I-V 67
VI. Type Forms in Multiplication — Factoring
I. Some Type Forms in Multiplication ... 71
11. Factoring 77
VII. Highest Common Factor — Lowest Common Multiple
I. Highest Common Factor 98
II. Lowest Common Multiple 105
VIII. Algebraic Fractions 109
IX. Simple Equations 125
X. Simultaneous Simple Equations
I. Two Unknown Numbers 144
11. Three or More Unknown Numbers .... 156
Review Exercise — Chapters VI-X . . . . .166
V
VI
CONTENTS
C^KVT^^ PAGE
XI. Involution and Evolution
I. Involution 170
II. Evolution 176
XII. Quadratic Equations (Elementary)
I. Equations in One Unknown Number . . ' . 190
II. Simultaneous Equations involving Quadratics . 206
XIII. Graphic Representation of Equations . . . 219
XIV. Irrational Numbers — Radicals 234
XV. Imaginary Numbers 257
XVI. Theory of Exponents (Zero, Negative, and Fractional
Exponents) 265
XVII. Quadratic Equations (Supplementary to Chapter XII) 277
Review Exercise — Chapters XI-XVII .... 284
XVIII. Inequalities 287
XIX. Ratio, Proportion, and Variation
I. Ratio 293
II. Proportion 295
III. Variation . 301
XX. Series — The Progressions
I. Arithmetical Progression 307
II. Geometric Progression 313
XXI. Mathematical Induction — Binomial Theorem . 321
XXII. Logarithms 329
Table of Common Logarithms 341
INDEX 343
HIGH SCHOOL ALGEBRA
• CHAPTER I
INTRODUCTION
I. LITERAL NUMBERS
1. Algebra. In the following pages, we shall continue to
use the symbols 0, 1, 2, 3, etc. to represent numbers, and
the signs +, — , x, -^, and =, to denote addition, subtrac-
tion, multiplication, etc.; that is, we shall use all of these
characters just as we used them in arithmetic. We shall
presently see, however, that algebra greatly simplifies the
solution of certain kinds of problems (§ 3), that it introduces
new kinds of number (§§ 13, 146, 164), and that it makes
extensive use of letters to represent numbers.
2. Numbers represented by letters. In arithmetic, num-
bers are almost always expressed by means of the symbols
0, 1, 2, 3, etc., but letters also are sometimes used. For
example, in interest problems p often stands for principal^
r for rate^ t for time^ i for interest^ and a for amount.
In algebra, on the other hand, the use of letters to repre-
sent numbers is very common ; thus, just as in arithmetic
we speak of 4 books, 7 bicycles, 85 pounds, 3 men, etc., so
in algebra we use, not only these expressions, but also such
expressions as a books, n bicycles, x pounds, y men, etc.
Numbers represented by letters are often called literal
numbers.
In the case of literal numbers the operations of addition,
subtraction, etc. may be indicated just as these operations
are indicated with arithmetical numbers. Thus, for example,
1
2 HIGH SCHOOL ALGEBRA [Ch. I
if n represents one number and k another, then n-{-k stands
for their sum, n — k for their difference, n x k for their
product, n-i-k or - for their quotient.
k
EXERCISE I
If a stands for 3, b for 2, and x for 12, find the value of each
of the following expressions :
- 2 a -\- X o 3 abx — ah
3. • o. •
h ab-{-hx
^ a-\-hx ^ X 4:a x
1.
a-\-h.
2.
x — a.
3.
x-i-a.
4.
5b--*
3a a 2b 2
7 ah -{-2 X —10 ^ 10 ^ 4- ^~^^
4a-f 46 a b
11. If s represents 16, what number is represented by 2s?
s 3 s
by Js, or (as usually written) -? by — ?
4 8
12. If a suit of clothes costs 8 times as much as a hat, and
if h stands for the cost of the hat, how may the cost of the suit
be represented ?
13. Does h-\-Sh (i.e., 9h) represent the combined cost of the
hat and suit in Ex, 12 ? Explain your answer.
14. The side of a square is 5 feet long. How long is the
bounding line of this square? How long is the bounding line
if the side is x feet long ?
15. A boy's present age is 15 years ; indicate, without perform-
ing the subtraction, his age 4 years ago. What was his age n
years ago ? What will it be y years hence ?
16. At 5 cents each how many erasers can be bought for 15
cents ? for x cents ? for n dollars ?
17. What number multiplied by 8 gives the product 40? If
8 a; = 40, what is the value of x? If 5y -\-2y = 21, what is the
value of y ?
* 6b means 5 times b ; so too ab means a times b ; and 3 ax means the
product of 3, a, and x.
2-^ INTliODUCTION 3
3. One advantage of literal numbers. The following ex-
amples show how the solution of problems may often be
simplified by using letters to represent numbers.
Prob. 1. A gentleman paid $45 for a suit of clothes and a
hat. If the clothes cost 8 times as much as the hat, what was
the cost of each ?
ARITHMETICAL SOLUTION
The hat cost "a certain sum/' and since the clothes cost 8 times
as much as the hat, therefore the cost of the clothes was 8 times
" that sum," and the cost of the two together was 9 times " that
sum." Hence 9 times "that sum" is $45, and therefore 'Hhat
sum" is $5, and 8 times "that sum" is $40; i.e., the hat cost
$5, and the clothes $40.
This solution may be put into the following more systematic
form, still retaining its arithmetical character.
A certain sum = the cost of the hat ;
then •.• * 8 tim,es that sum = the cost of the clothes,
9 times that sum = the cost of both,
i.e.j 9 times that sum = $45.
that sum = $ 5, the cost of the hat,
and 8 times that sum = $ 40, the cost of the clothes.
ALGEBRAIC SOLUTION
The solution just given becomes much simpler if we let a
single letter, say x, stand for the number of dollars in " a certain
sum " and " that sum " as used above, thus :
Let X = the number of dollars the hat cost.
Then Sx = the number of dollars the clothes cost,
and x + S x = the number of dollars both cost ;
i.e., 9 a; = 45.
a; = 5, and 8 a? = 40 ;
i.e., the hat cost $5, and the clothes cost $40.
N.B. The letter x, above, stands for a number, not for the cost of the hat.
* The symbols •.• and .-. stand for the words "shice" and "therefore,"
respectively.
4 HIGH SCHOOL ALGEBRA [Ch. I
Prob. 2. If a locomotive weighs 3 times as much as a car, and
the difference between their weights is 50 tons, what is the weight
of the locomotive ?
SOLUTION
Let w = the number of tons in the weight of the car.
Then Sw = the number of tons in the weight of the locomotive,
and, since the difference between their weights is 50 tons,
3w — w = 50,
i.e.f 2w = 50,
whence iv = 25,
and 3w = 75;
i.e., the locomotive weighs 75 tons.
Prob. 3. Of three numbers the second is 5 times the first, and
the third 2 times the first ; if the sum of these numbers equals
the third number increased by 42, what are the numbers ?
SOLUTION
Let n = the first of the three numbers.
Then 5n = the second number,
and 2 71= the third number ;
now since the sum of the three equals the third number increased
by 42,
.-. n-{-5n-j-2n = 2n + 4:2,
i.e., Sn = 2n-\- 42,
hence 6 n = 42. [Subtracting 2 n from each of the equal
sums above.]
n = 7, 5n = 35, and 2 n = 14 ;
i.e., the numbers are 7, 35, and 14, respectively.
Remark. Observe that the steps in each of the foregoing solu-
tions are :
1. To let some letter, say x, stand for one of the unknown
numbers (preferably the smallest).
2. To express the other unknown numbers in terms of x.
3. To translate into algebraic language those relations between
the unknown numbers which the problem states in words ; this
3] INTRODUCTION 5
translation gives an equation, and from it the required numbers
are easily found.
Observe also that while the above problems can be solved by
arithmetic, the algebraic solution is much simpler.
EXERCISE 11
Solve the following problems :
4. In a room containing 45 pupils there are twice as many
boys as girls. How many boys are there in the room ?
5. If a horse costs 7 times as much as a saddle, and if the
difference in the cost of the two is $ 90, find the cost of each.
6. A house is worth 5 times as much as the lot on which it
stands, and the two together are valued at $4200. Find the
value of each.
7. If the house and lot of Ex. 6. differ in value by $4200, how
much is each worth ?
8. The double of a certain number taken from 10 times the
same number leaves 72. What is the number ?
9. If n represents a certain number, how may we represent:
(1) the number, plus 4 times itself, plus 5 times itself ?
(2) the sum of the number, its double, and its half ?
What does 5 -a? -f- 7 n — 3 n represent ?
10. A number, plus twice itself, plus 4 times itself, is equal to
56. What is the number ?
11. Divide 98 into three parts such that the second is twice
the first and the third is twice the second.
12. Divide 160 into three parts such that two of them are equal,
while the third is twice either of the others.
13. In a yachting party consisting of 36 persons, the number
of children is 3 times the number of men, and the number of
women is one half that of the men and children combined. How
many women are there in the party ?
14. If I have s nickels, how many cents have I ? How many
cents in s dimes? in s quarters? in the sum of s nickels, s
dimes, and s quarters ?
6 TUGn SCHOOL ALGEUnA [Ch. I
15. A boy found that he had the same number of 5, 10, and 25
cent pieces, and that the total amount of' his money was ^ 3.20.
How many coins of each kind had he ?
16. Alice buys Christmas gifts at 25 cents, 15 cents, and 10
cents — the same number at each price. If she spends $2 in
all, how many gifts does she buy ?
17. If a; stands for a certain number, what would stand for
(1) the double of the number, increased by 7 ?
(2) the difference between 3 times the number and 8 ?
18. How would you represent two numbers whose difference is
4 ? two numbers whose sum is 13 ?
19. Find the number whose double, with 4 added, equals 46.
20. Find two numbers, differing by 7, whose sum is 35. Also
find two numbers whose sum is Q^ and whose difference is 15.
21. William has 8 cents more than his sister Harriet, and
together the two have 80 cents. How much money has each?
If Harriet's money is made up of an equal number of nickels and
one-cent pieces, how many nickels has she ?
22. In a family of seven children each child is 2 years older
than the next younger. If the sum of their ages is 84 years, how
old is the youngest child ?
23. A father's age is now 3 times that of his son ; 5 years hence,
the sum of their ages will be 62 years. Find the present age of
each. (Cf. Ex. 15, p. 2.)
24. Four years ago Isabel was twice as old as Mabel ; the sum
of their present ages is 32 years. How old is each ? •
25. In a business enterprise, the combined capital of A, B, and
C is f 21,000. A's capital is twice B's, and B's is twice C's.
What is the capital of each ?
26. In the triangle MNP, NP'is 2 inches longer
than MN, while PM and JfJVare of equal length.
If the sum of the three sides is S6 inches, find
the length of each.
27. An east-bound and a west-bound train
leave Chicago at the same hour, the first running twice as fast
N P
3] INTRODUCTION 7
as the second ; after one hour they are 90 miles apart. Find the
speed of each.
28. In a fishing party consisting of four boys, two of the boys
caught each the same number of fish, another caught 2 more than
this number, and the fourth, 1 less. If the total number of fish
caught was 29, how many did each catch ?
29. An estate valued at $24,780 is to be divided among a family
consisting of a mother, two sons, and three daughters. If the
daughters are to receive equal shares, each son twice as much as a
daughter, and the mother twice as much as all the children to-
gether, what will be the share of each ? ^ ^
30. ABCD represents the floor of a room.
Find the dimensions of the floor if its bound-
ing line is 48 feet long.
31. A gallon of cream is poured into two
(a;+4) feet
pitchers, one of which holds 7 times as -D G
much as the other. How many gills does each pitcher hold?
32. If i of a number is added to the number, the sum is 120.
What is the number ?
Suggestion. Let 3 x = the number.
33. If ^ of a number is added to twice the number, the sum is
35. What is the number ?
34. Of two numbers, twice the first is 7 times the second, and
their difference is 75. Find the numbers.
Suggestion, Let 1 x = the first number, then 2 x = the second.
35. An estate of f 19,600 was so divided between two heirs
that 5 times what one received was equal to 9 times what the
other received. What was the share of each ?
36. A tree whose height was 150 feet was broken off by the
wind, and it is found that 3 times the length of the part left
standing is the same as 7 times that of the part broken off. How
long is each part ?
37. If two boys together solved 65 problems, and if 8 times the
number solved by the first boy equals 5 times the number solved
by the second boy, how many did each boy solve ?
8 HIGH SCHOOL ALGEBRA [Ch. I
II. ELEMENTARY OPERATIONS
4. Addition. In algebra, as in arithmetic, such an ex-
pression as 7 + 3 is read " 7 plus 3," and means that 3 is to
be added to 7.
To perform this addition we begin at 7 and count 3 for-
ward, obtaining the result 10, which is called the sum of
these two numbers.*
So also if a and b stand for any two numbers whatever,
the expression a-\-b is read "a plus 5," and means that h is
to be added to a.
The result obtained by adding two or more numbers is
called their sum, and the numbers that are to be added are
called the summands.
5. Subtraction. To what number must 3 be added to
obtain the sum 8? If 8 was obtained by adding 3 to some
number (i.e.^ by counting 3 forward}, how may we, starting
with 8, find the number at which the counting began ?
Here, as in arithmetic, the operation of finding this num-
ber is indicated by the expression 8 — 3, which is read " 8
minus 3." We may say that 8 — 3=5 because 5 + 3 = 8.
The process of finding one of two numbers when their
sum and the other number are given, is called subtraction.
It consists, as we have just seen, in counting backward, i.e.,
in undoing the work of addition, which consists in counting
forward.
If a and b stand for any two numbers whatever, the ex-
pression a — b is read " a minus 5," and means that b is to be
subtracted from a.
The result obtained by subtracting one number from an-
other is called their difference (also the remainder). The
number which is to be subtracted is called the subtrahend, and
* If fractions are to be added, we first reduce tliera to a common denomi-
nator and tlien add their numerators ; it is still a counting process.
4-C] INTRODUCTION 9
the one from which the subtraction is to be made is called
the minuend.
6. Inverse operations. Of two operations which neutralize
each other when performed in succession, each is called the
inverse of the other. Thus the operations of addition and
subtraction are each the inverse of the other (cf. Exs.
8-10, below).
EXERCISE- III
Read each of the following expressions, then name its parts :
1. 8 + 12 = 20. 2. 9-7 = 2. 3. 12y-9y = 3y,
4. Since 4 + 9 = 13, therefore 13 - 9= ? 13 -4 = ?
5. In subtraction, what name is used to denote the given sum ?
the given summand? the required summand ? Illustrate, using
Ex. 2 above.
6. Add 4 to 7 by counting. Where do you begin to count ?
In what direction do you count?
7. By counting, subtract 4 from 11. Do you count in the
same direction as in Ex. 6 ?
8. How may you combine the subtrahend* and remainder to
get the minuend ? Why ?
9. How would you test the correctness of an answer in sub-
traction? Illustrate. Could you use subtraction to test the
correctness of a sum ?
10. When is one operation said to be the inverse of another ?
Using the numbers 8 and 6, illustrate the fact that subtraction is
the inverse of addition.
11. If m and n stand for any two given integers whatever, can
you, by counting, find the value of ?7i + n ? of m — n ?
12. What is the value of 5 - 3 ? of5-4? of5-5? of5-6?
of 5 — 8 ? In order that subtraction be possible, how must the
subtrahend compare in size with the minuend ?"*
* With our present (arithmetical) meaning of number such a subtraction
as 5 — 8 is, of course, impossible ; in Chapter II, however, we shall so extend
the meaning of number as to make the subtraction a — b possible even when
b is greater than a.
10 niGH SCHOOL ALGEBRA [Cit. I
7. Multiplication, (i) In arithmetic, multiplication is
usually defined as the process of taking (additively) one
of two numbers, called the multiplicand, as many times as
there are units in the other, called the multiplier. In this
sense, 6x4 (read "6 multiplied by 4") means 6 + 64-6 + 6;
i.e.^ this multiplication may be regarded as an abbreviated
addition.
Strictly speaking, however, the above definition of multi-
plication applies only when the multiplier is an arithmetical
integer : under this definition, for instance, we could not find
such a product as 8 x 51, because we could not take the mul-
tiplicand two thirds of a time any more than we could fire a gun
two thirds of a time.
(ii) A broader definition of multiplication, and one bet-
ter suited to our present purpose, may be stated thus :
Multiplication is the process of performing upon one of
two given numbers (the multiplicand) the same operation
as that which is performed upon unity to get the other given
number (the multiplier) ; the result thus obtained is called
the product of these numbers. The multiplicand and multi-
plier are called factors of the product.
To illustrate, consider again the question of multiplying
8 by 5|-. The multiplier, 5|-, is obtained from unity by tak-
ing the unit 5 times, and J of the unit twice, as summands,
i.e., 5f = 1 + 1 + 1 + 1 + 1 + 1+ J;
and, therefore, by this new definition of multiplication,
8x5f = 8 + 8 + 8 + 8 + 8 + | + f = 40 + J^ = 45f
(iii) Just as 6 X 5 means that 6 is to be multiplied by 5,
so 5 X 3 means that b is to be multiplied by 3. Similarly,
kxn xy means that h is to be multiplied by n^ and that
their product is then to be multiplied by y.
Instead of the oblique cross ( x ), a center point (•) placed
between two numbers (a little above the line to distinguish
it from the decimal point) is frequently used as a sign of
7-8] INTttOhVCTlON . 11
multiplication. And even the center point is usually omitted
if doing so causes no confusion. Thus, 8xri = o-'/i=3n;
so, too, p xr xt=p • r ' t=prt, and 3 x 7 = o • 7. But the
sign (cross or center point) must not be omitted between two
arithmetical numbers. (Why not?)
EXERCISE IV
Kead each of the following expressions, then name its parts :
1. 8x3 = 24. 2. f. 15 = 10. 3. 5«.4 = 20a.
4. What is the value of 5 • 3 ? How is this product obtained
under the old definition of multiplication [§ 7 (i)] ?
5. Using the new definition [§ 7 (ii)], show that 5 • 3 means
5 + 5 + 5. Similarly, explain the meaning of 9 • 4; of 4 • 9.
6. Show that 2| • 8 has the same meaning under the old defi-
nition of multiplication as under the new.
7. To get f from 1, we divide 1 into how many equal parts ?
How many of these parts do we take '? What, then, should be
done to 10 in multiplying it by J ?
As in Ex. 7, find the following products :
8. 16. f. 9. 12. 2f 10. 7.5f.
If a = 2, 6 = 5, and a? = |, find the value of each of the follow-
ing expressions :
11. Sabx. 13. 2bx — ax + 3b. 15. Saax + 4:bx.
12. 5b + 6x-ab. 14. 7abx + 3a-2b. 16. aab — 10 x.
8. Division. Division is the inverse (i.e.^ihe "undoing")
of multiplication. Thus, since 4 x 9 = 36, therefore 36 -=- 9 = 4,
and 36 -4 =9.
The expression 36 -^ 9 = 4 is read " 36 divided by 9 equals
4." Here 36 is called the dividend, 9 the divisor, and 4 the
quotient.
In multiplication we have given two numbers, and are
asked to find their product ; in division we have given the
product (now called the dividend} and one of the factors
HIGH SCH. ALG. — 2
1^ tilGH SCHOOL ALGEBRA [Ch. 1
(now called the divisor')^ and are asked to find the other
factor (now called the quotient).
Hence we may say : division is the process of finding from
two given numbers, called dividend and divisor, respectively,
a third number (called the quotient) such that the divisor
multiplied by the quotient equals the dividend.
E.g., 36 -- 9 = 4, because 4 x 9 = 36.
If 8 and t represent any two numbers whatever, then each
of the expressions, s -?- f , -, s/U and s : t indicates that s is to
be divided by t.
If the divisor is not exactly contained in the dividend,
then, as in arithmetic, the indicated division is called a
fraction.
E.g.^ ^, -— , — , and — ^t_ are called fractions.
6 D n y .
It is to be remarked, however, that literal numbers
may be fractional in form but integral in value, and
vice versa. Thus, -, though fractional in form, has the inte-
gral value 3 if a = 12 and 5 = 4.
9. Powers, exponents, etc. (i) In algebra, as in arithmetic,
such a product as 5 • 5 • 5 is usually written in the abbrevi-
ated form 5^, the small 3 showing the number of times that
5 is used as a factor.
Similarly, 23 z= 2 • 2 • 2, a^^a-a-a, 2^ . 52= 2 • 2 • 2 • 5 • 5,
7i2jt>4 z= n^ ' p^ = n ' 71 • p ' p ' p • p, etc.
The expression k^ is usually called the fourth power of h.
In this expression, 4 is called the exponent and h the hase of
the power.
(ii) Hence the following definitions : A power of a number
is the product arising from using the given number one or
more times as a factor.
An exponent is a number placed (in small symbols) at the
8-9] INTRODUCTION " 13
right and slightly above a given number, to show how many
times the latter is to be used as a factor.
Thus, if X represents any number whatever, and n any
arithmetical integer,* then the expression x^ is called the nth
power of 07, and means the product arising from using x as
a factor n times ; n is the exponent of the power.
Note. Observe that under the above definitions a^ has the same mean-
ing as a ; the exponent 1, therefore, need not be written.
The second and third powers of numbers are, for geometrical reasons,
often called by the special names of square and cube respectively. Thus a^
is called " the second power of a," " the square of a," and also " a squared."
EXERCISE V
Eead each of the following expressions, name its parts, and
test the correctness of the results :
1. 18-6 = 3. 4. ?5^ = 4a. 7. ^ = Z2.
2. 28^14 = 2. 5. 9.^81. k^^4a^
3. 6_|o = 70. 6. 2^.32 = 288. * 32 3 *
Read the following expressions and tell what operations are
indicated in each case; then find the numerical value of each
expression when a = 5, 6 = 2, 7i = l, and ic = 4.
.5 ,^ 3 6^ „ a^-lOW
9. a\ 12. ^^1^. 15.
16 3 a
10- «' + ^'- 13. 8W-10a^. ^^ 75 -aW
11. 7wV. 14. 7i^ + bax\ ' y?-a '
17. Write 7«7-7«7 by means of the exponent notation.
Also a ' a'a\ 5 • 5 • a; • a; • a; ; and 9 • 9 • 9 • 9 • a • a • ?/ • ?/ • 2/.
18. How may we use multiplication to test the correctness of
an example in division ? Why ?
19. The sum of any two integers is integral. Is this true of
their difference ? of their product ? of their quotient ? Illus-
trate your answers.
* We shall later (Chapter XVI) enlarge the scope of such a symbol as x" by
giving it a meaning even when n does not represent an arithmetical integer.
14 HIGH SCHOOL ALGEBRA [Ch. I
20. When the dividend is not exactly divisible by the divisor,
what name is given to the indicated quotient ?
5
21. How are fractions defined in arithmetic ? Is — a frac-
tion under the arithmetical definition ? If not, why not ?
10. The order in which arithmetical operations are to be
performed. What is the value of 2 + 6 • 5 — 8 ^ 2 ? Is it
28, 16, 'or 12? In order that such an expression shall have
the same meaning for all of us, mathematicians have agreed
that, when there is no express statement to the contrary :
(1) A succession of multiplications and divisions shall
mean that these operations are to be performed in the order
in which they occur from left to right.
(2) A succession of additions and subtractions shall mean
that they are to be performed in the order in which they
occur.
E.g., 9 . 8 -- 6 . 2 = 72 -- 6 . 2 = 12 . 2 = 24,
but 9 • 8 -J- 6 • 2 is 710^ equal to 72 -h 12, i.e., to 6.
So, too, 7 + 9-6 + 3 = 16 -6 + 3 = 10 + 3 = 13,
but 7 + 9 — 6 + 3 is 710^ equal to 16 — 9, i.e., to 7.
(3) A succession of the operations of addition, subtrac-
tion, multiplication, and division shall mean that all the
operations of multiplication and division are to be performed
before ani/ of those of addition and subtraction, and in accord
with (1) above. The additions and subtractions are then
to be performed in accord with (2) above.
E.g., 2 + 6- 5-8--2 = 2 + 30 -4 = 28.
Note. While such an expression as 3 • a -^ 2 • a; • ?/ means [(8 a) ^ 2] • a; • y,
the expression Sa -^2xy is usually understood to mean (3 a) ~ (2 xy) ;
i.e., 3 a and 2 xy are here understood to vei>resent products rather than unper-
formed multiplications.
11. Signs of aggregation, (i) Any desired departure from
the order of operations given in § 10 may be indicated by
employing one or more of the so-called signs of aggregation ;
1-11] tNTROTWCrtON 15
among these are the parenthesis ( ), the brace \\, the bracket
[ ], and the vinculum .
(ii) An expression within a parentliesis, brace, or bracket,
or under a vinculum, is to bo regarded as a whole, and is to
be treated as though it were represented by a single symbol.
U.g., (2 + 6) . 5 - 3 - (7 + 8 -^ 2) = 8 • 5 ^ 3 - 11, i.e,,
'2^. So, too, (4 + 6) -^ 2 = 5, while without the parenthesis
its value would be 7.
It may sometimes be useful even to employ one sign of
aggregation within another,
Mg., 72- J47-7(15-10)S =72- ^47-35? =72-12 = 6.
EXERCISE VI
Find the value of each of the following expressions:
1. 20 + 5-3. 4. 12-2x4. 7. 16-2-1-6.
2. 20-5 + 3. 5. 12 -(2x4). 8. 16 - (2 + 6).
3. 20 -(5 + 3). 6. 9.(6-2). 9. 11 • 4 - 6 • 3-2.
10. 28 -(6 + 13) -(10 -2). 13. 42-7x5-5 + 6x2.
11. 32-9 + 6--2 + 1. 14. 12 + 9-3-30-2 + 8.
12. 32- (9 + 6) --(2 + 1). 15. (12-f 9).3-(30-2 + 8).
16. J25 - (10 + 13)S -2 + 31-5 + 4.
17. 16- 9 -4(36 -3-2) +54 -(17 -12-5).
Read each of the following expressions, and tell in what order
the indicated operations are to be performed :
18. ac-^b. 21. (c-by. 24. 6a H-2c-2d ^
19. a(c-^b\ 22. c'-^b'-2d.
20. c-b\ 23. c----(b^-2d). 25. ^ -— .
^ ^ c<i^+[2(c — a)]-
26. If a = 8, b — 3, c = 12, and 2d= 1, find the numerical value
of each of the expressions in Exs. 18-25.
CHAPTER II
POSITIVE AND NEGATIVE NUMBERS
12. Introductory. * Suppose the present reading of a ther-
mometer is 5° above zero, and the temperature is falling ;
what will be the reading when it has fallen 1°? 2°? 3°?
4°? 5°? 6°? 7°? 8°?
* Note to the Teacher. It will stimulate interest in the work, as well
as enlarge the pupil's view, if the teacher will amplify and present to the
class the following considerations (cf. also El. Alg. pp. 18, 19) :
1. Man's earliest idea of number came from counting., and in this way
there arose the number system consisting of the integers 1, 2, 3, 4, •••.
Presently he advanced a step and counted backward as well as forward, and
by groups of things as well as by single things ; this led the way to the opera-
tions of addition, subtraction, multiplication, and division.
2. Having devised this number system and these operations to meet the
needs of his daily life (just as later he invented the clock, the steam engine,
the telephone, etc.), he soon found the system inadequate: 7 h- 3, for ex-
ample, represents no number in the above system. His increasing desire for
exactness, however, as civilization advanced, demanded that division should
be always possible; for this and other reasons he invented fractions, and
included them in his number system .
3. Many of the things with which we are concerned bear a relation of
opposition to each other (assets and liabilities, thermometer readings above
zero and below zero, etc.), and the change from one of these opposites to
the other may be regarded as a subtraction (cf. § 12) ; for this and other
reasons it was found advantageous again to extend the number system, and
thus to make subtraction always possible.
4. The use of literal notation, too, quite apart from the foregoing con-
siderations, demands a number system in which addition, subtraction, etc.,
are always possible. If subtraction, for example, is not always possible,
such an expression as a — 6 may or may not represent a number ; hence,
should a — h occur in the solution of a problem before the relative values of
a and b were known, our work would come to a standstill. It is wiser, there-
fore, to include negatives in- our number system, and let the solution proceed,
giving the necessary interpretation to our results later.
16
12-18] POSITIVE AND NEGATIVE NUMBERS 17
Wliicli one of the arithmetical operations (addition, sub-
traction, etc.) did you use in answering these questions?
Again, if a business man's financial losses are large enough,
they will decrease his capital, not only to zero but through
zero, and bring him into debt.
So, too, if a traveler now in north latitude goes south far
enough, he will pass through zero latitude, and into south
latitude.
Observe that, in each of the statements just made, the
change from a given condition to its opjjosite is essentially
a process of subtraction; it is a subtraction, moreover, in
which we can subtract not only to, but through zero.
In our present number system, we can subtract only to
(not through^ zero ; in order, therefore, to express m the
simplest possible wag the numerical relations between such
opposite things as those given above (gains and losses, lati-
tude north and south, etc.), we must extend our number
system so as to make subtraction through zero possible.
13. The number system extended. The arithmetical inte-
gers arranged in a series increasing by one from left to right,
and therefore decreasing by one from right to left, are
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
Addition is performed by counting toward the right (cf.
§ 4), and subtraction by counting toward the left, in this
series. Moreover, addition is always possible because this
series extends without end toward the right, and subtraction
is arithmetically possible only when the subtrahend is not
greater than the minuend because this series is limited at
the left.
Hence, to make subtraction with arithmetical integers
always possible, it is only necessary to continue the above
series indefinitely toward the left.
Let the result of subtracting 1 from 1 be designated by ;
of subtracting 1 from 0, by — 1 ; of subtracting 1 from — 1,
18 HIGH SCHOOL ALGEBRA [Ch. II
by — 2 ; of subtracting 1 from — 2, by — 3, etc. ; with these
new numbers included, the series becomes
..., -6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6, 7, ...,
whicli extends without end toward the left as well as toward
the right.
Since in this enlarged series each number is less by one
than the next number at its right (and hence greater by
one than the next number at its left), therefore addition and
subtraction with arithmetical integers may, as before, be
performed by counting toward the right and left respectively.
E.g.^ to subtract 8 from 5, ^.e., to find the number which
is 8 less than 5, we begin at 5 and count 8 toward the left,
arriving at — 3 ; hence, 5 — 8 = — 3.
Similarly, 4-6=— 2, 4-9=- 5, 11 -16 =-5, etc. ;
hence, besides indicating a particular place in the enlarged
number series, — 5 also indicates that the subtrahend is 5
greater than the minuend.
Again, to add 7 to —4, i.e.^ to find the number which is
7 greater than — 4, we begin at — 4 and count 7 toward the
right, arriving at 3 ; hence —4 + 7 = 3.
Note. Such an expression as 4 — 9 = — 5 is, of course, not to be under-
stood to mean that 9 actual units of any kind can be subtracted from 4 such
units ; 4 of the 9 units may be immediately subtracted, leaving the other 5
units to be subtracted later if there is anything from which to subtract ; in
this sense the number —5 may be said to indicate a postponed subtraction,
and thus to have a subtractive quality ; hence the appropriateness of attaching
the minus sign to such numbers.
14. Positive and negative numbers. Numbers which are
less than zero * are called negative numbers ; while numbers
greater than zero are, for distinction, called positive numbers.
In writing positive and negative numbers, the latter are
always preceded by the minus sign, while positive numbers
may be written either with or without the plus sign. Thus,
* "Less than zero" in the sense suggested in § 13,
13-15] POSITIVE AND NEGATIVE NUMBEES 19
— 5, —2.3, —48, and — | are negative numbers, while 3,
+ 84, and 1.7 are positive numbers.
Such a number as — 2 is read either as negative two or
as minus two, while + 6 is read as positive six or plus six.
15. Algebraic numbers, etc. Positive and negative num-
bers together (including, of course, fractions as well as inte-
gers) are often called algebraic numbers, wliile positive
numbers alone are called arithmetical numbers.
By the absolute value of a number is meant merely its size
without regard to its quality/ ; thus — 2 and + 2 have the
same absolute value; so also have — 17.25 and + 17.25.
Two numbers which have the same absolute value but
which are of opposite quality, are called opposite numbers ;
such, for example, are 5 and — 5.
Note. The signs + and — as used above are called signs of quality.
We shall continue, however, to use them as signs of operation also ; hence
it is sometimes necessary, in order to avoid possible confusion arising from
this double use, to inclose a number with its quality sign in a parenthesis
(cf. Ex. 20, p. 21).
EXERCISE VII
By counting along the algebraic number series given in § 13 :
1. Add 3 to 8. 5. Subtract 3 from 8.
2. Add 7 to — 3. 6. Subtract 7 from 7.
3. Add 4 to — 4. 7. Subtract 7 from 4.
4. Add 5 to — 12. 8. Subtract 5 from — 3.
9. If temperature above zero is indicated by positive numbers,
how may temperature below zero be indicated ?
10. Interpret the following temperature record taken from a
U. S. Weather Bureau report : Albany, -f- 8° ; Bismarck (S.D.),
- 11° ; Buffalo, - 2° ; Denver, - 5° ; and Galveston, + 34°. .
11. In the above record how much warmer is it at Albany than
at Bismarck ? at Buffalo than at Denver ? at Buffalo than at
Galveston ? Illustrate your answers by diagrams.
20 HIGH SCHOOL ALGEBRA [Ch. II
12. The total value of a man's property (his assets) is $ 15,860,
and his total indebtedness (liabilities) is $1420. What is the net
value of his estate? What is the net value of an estate of which
the assets are a dollars and the liabilities h dollars ?
13. If h exceeds a in Ex. 12, is a — 6 positive or negative ?
How should a — 6 be interpreted in this case ? What is meant
by saying that the net value of an estate is — $ 750 ?
14. If assets are indicated by positive numbers, interpret the
financial conditions indicated by the following numbers : $ 783 ;
- $ 2568 ; - $ 374.20 ; and $ (856-1232).
15. Kegarding longitude west of Greenwich as positive, indi-
cate by a number and sign that a place is : (1) in 24° east longi-
tude; (2) on the meridian of Greenwich; (3) in 10° west longitude;
(4) in 15° east longitude.
16. At 6 A.M. a thermometer records 15° below zero; by noon
it has risen 32°. Indicate by a number and sign the reading at 6
A.M. and at noon. Illustrate by a diagram.
17. If on the line X'OX distances to the ^^ ? ^
left of are called negative, locate the point
on this line whose distance from is : + .3 in. ; — .5 in. ; + .1 in.
18. If positive numbers are used to denote assets, gains in busi-
ness, increase of any kind, temperature above zero, easterly motion,
south latitude, west longitude, distance down stream, etc., what are
the corresponding meanings to be attached to negative numbers ?
19. An ocean steamer is in 12° east longitude. If east longitude
is indicated by positive numbers, and if the vessel moves west-
ward through 7° of longitude per day, indicate by a number and
sign the longitude of the vessel 4 days hence ; 1|- days hence ; 2
days ago. Illustrate by a drawing.
20. What number must be added to — 12 to make the sum 4 ?
Which, then, is the greater, —12 or 4 ? How much greater?
"Which of these numbers has the greater absolute value ?
•
16. Addition of algebraic numbers. As we have already
seen (§§ 4, 13), to add a positive number to any given num-
ber, we begin at the given number and aoxmi forivard.
la-lC] POSITIVE AND NEGATIVE NUMBERS 21
Moreover, since a negative number represents an unper-
formed subtraction (§ 13, Note), therefore adding a negative
number means performing this subtraction. Hence, to add
a negative number to any given number, we begin at the given
number and count hachvard.
E.g.^ to add — 6 to 54 we begin at 54 and count 6 backward;
i.e., 54 + (- 6) = 54 - 6 = 48 (cf. Exs. 19, 20, below).
Such an expression as 11—4—8 + 3, which, by what is
said above, equals 11 + (— 4) + (— 8) + 3, is usually spoken
of as an algebraic sum.
EXERCISE VIII
Perform the following additions, and explain your work :
1. 2. 3. 4. 5. 6.
9
12 -8 9 14 -7
8 8-3 -6 -3
7.
8.
9.
10.
11.
12.
-4
23
-11
-31
7
-9
-12
-15
-26
45
5
— 5
13.
14.
15.'
16.
17.
18.
-3
7
-6
11
-22
-72
4
5
-8
-4
31
^b
-8
-9
— 2
-9
15
-21
19. If a boy weighing 54 lb. were weighed while holding a
toy balloon which pulls upward with a force of 6 lb., what
would be the combined weight? If +54 lb. represents the
weight of the boy, what would represent the iveight of the
balloon ?
20. In Ex. 19 the combined weight of the boy and the balloon
is (+54) + (— 6) lb. ; hence adding the negative number destroys
part of the positive number ; is this true in general for additions
of positive and negative numbers ? Illustrate your answer.
22 HIGH SCHOOL ALGEBRA [Ch. II
21. When does the addition of a negative number to a positive
number destroy the latter wholly ? When only in part ? Illus-
trate your answers by means of assets and liabilities (cf. Ex. 12,
p. 20).
22. Ex. 21 suggests a useful rule for algebraic addition, viz. :
(1) To add two numbers with like signs, find the sum of their
absolute (arithmetical) values, and to this prefix their common
sign.
(2) To add two numbers with unlike signs, find the difference
of their absolute values, and to this prefix the sign of the larger.
Test this rule in the examples on p. 21.
23. A wheelman after riding 37 miles westward from a certain
point rides back 12 miles. If distances to the westward are indi-
cated by positive numbers, show that 37 + (— 12) miles indicates
both his direction and his distance from the starting point.
24. Indicate by a sum of positive and negative numbers what
temperature is now registered by a thermometer which stood at
54° above zero, then rose 2°, later fell 9°, and then rose 2i°.
25. Eind the following indicated algebraic sums : 18 + (— 3) +
(_10) + 2; 42+(-27)4-(-64); _5 + 18 + (-11) + 23.
26. Is algebraic addition sometimes performed by arithmetical
subtraction ? Is it so when the two summands have like signs ?
when they have unlike signs ? Illustrate your answers.
17. Subtraction of algebraic numbers. We already know
how to subtract positive numbers (of. §§ 5, 13); therefore
we now need to consider only the question of subtracting
negative numbers.
Now, since subtraction is the inverse of addition (§ 6)
and since 7 + (-8) = 4 (§ 16), therefore 4 -(-3)= 7; i.e.,
4-(-3) = 4 + 3. Similarly: 11 - (- 5) = 11 + 5; -3-
(-12)= -3 + 12; etc.
Hence, subtracting a negative number from any given
number gives the same result as adding its opposite to the
given number.
10-17] POSITIVE AND NEGATIVE I^ UMBERS 23
EXERCISE IX
Subtract the numbers written below, each from the one above
it, giving the necessary explanation in each case:
1. 2. 3. 4. 5.
12 4 -2 8 -4
8 7 5 -3 -2
6.
7.
8.
9.
10.
-3
11
6
-4
-31
-10
-8
-15
-12
-25
11. The weight of a boy while holding a toy balloon, which
pulls upward with a force of 6 lb., is 48 lb. If we take away
(subtract) the balloon, how much will the boy weigh ? Show that
in this case 48 — (— 6) = 54.
Supply the missing numbers in the following equations :
12. •.•-5h-? = -2, .-. _2-(-5) = ?
13. ... _54-? = -9, ... -9-(-5) = ?
14. |-(_|) = ? 16. 10-3-(-5) = ?
15. _l|-(-5f) = ? 17. 23-(-a) + (-3) = ?
If a = 5, b = — 6, c = — 3, and d = 2, find the value of each of
the following algebraic sums :
18. a-b + c + d. 21. a^^b-(d^-c).
19. a-^c — (b — d). 22. ad — b-{-(a^ — b).
20. a—(b + G-d). 23. 2 a-^ - c — 3d^-\-b.
24. Using positive numbers to represent assets, illustrate the
fact that subtracting a negative from a positive number increases
the latter (cf. Ex. 11).
25. Make up concrete examples (like Ex. 11 or Ex. 24) to illus-
trate Exs. 4, 8, and 3, above.
26. Solve Exs. 1, 2, 5, and 9, above, by counting ; and explain
in each case why you count in one direction rather than in the
opposite direction.
24 HIGH SCHOOL ALGEBRA [Ch. II
27. A rule for subtraction is often stated thus : " Reverse the
sign of the subtrahend and proceed as in addition." Show that
this rule is correct when the subtrahend is a negative number.
28. Mt. Washington is 6290 feet above sea level, Pikes Peak
is 14,083 feet above sea level, and a place near Haarlem, in Holland,
is 16|^ feet below sea level. By subtraction find how much higher
Pikes Peak is than Mt. Washington ; and also how much higher
Mt. Washington is than the place near Haarlem.
29. When is algebraic subtraction equivalent to arithmetical
addition (cf. Ex. 2%, p. 22) ? Illustrate your answer.
30. Write the following algebraic sums so that they shall not
contain minus as a sign of operation, then find the value of each :
36-19-13+2; 14a-26i«+9}a-15a; -6a;-47a;-3a;.
18. Product of algebraic numbers. Rule of signs.* The
product of any two algebraic numbers is tlie result obtained
by performing upon the multiplicand the same operation as
that which is performed upon positive unity to obtain the
multiplier [§ 7 (ii)].
E.g,, since 3 = 1 + 1 + 1,
therefore 8 • 3 = 8 + 8 + 8 = 24,
the product 24 being obtained from 8 just as 3 is obtained from
positive unity.
Similarly, - 8 • 3 = (-8) + (- 8) + (- 8) = - 24. [§ 16
Again, since — 3 = — 1 — 1 — 1, i.e., since — 3 is obtained by
subtracting positive unity three times,
therefore 8 • (- 3) = - 8 - 8 - 8 = - 24,
and _8.(-3) = -(-8)-(-8)-(-8) = 8 + 8 + 8 = 24.
Observe that two of the above products are positive and
two are negative. How do the signs of the factors compare
when the product is positive? when it is negative? How
does the absolute value of the product (cf. § 15) compare
with the absolute values of the factors ?
* Teachers who prefer to give more drill on addition and subtraction at
this point may omit §§ 18 and 19 until after Chapter III has been read.
17-19] POSITIVE AND NEGAriVE NUMBERS 25
Applying the tlefinitioii of a product to any two numbers
whatever, just as we did to 8 and 3,-8 and 3, etc., above,
we see that : (1) if two factors have like signs, their product
is positive; (2) if they have unlike signs, their product is
negative; and (3) the absolute value of the product equals
the product of the absolute values of the factors.
EXERCISE X
Find the following products and explain your work :
1.-5-2. 4. 9 . - 3. 7. 3 . - 4 . - 5 . 2.
2.-5.-2. 5. -7|.-6. 8. 2'-'^x,i.e.,2'-^-x.
3.-12.8. 6.-^.-5. 9. _4«.3.-5.-7.
10. Show that ( - 2)\ i.e., _ 2 . - 2 • - 2 • - 2, is 16. What is
the value of (- 2)' ? of (- 3)^ • (- 4)^ ?
11. What is the sign of (- 7)^ ? Why ? What is its absolute
value ? What is the sign of (- 7)'^ - 2 . 5 ?
12. A succession of multiplications (as in Ex. 7, for instance)
is called a continued product. Can the sign of a continued product
be obtained without actually performing the multiplication ?
How ? What is the sign if there are 5 negative factors ?
13. An odd power of a negative number (i.e., a power whose
exponent is odd) has what sign ? An even power ? Is a power
of a. positive number ever negative ? Explain.
If a = — 4, 6 = — 2, c = 3, rf = — 1, and e = 2, find the value of :
14. abode. 16. ah^d'. 18. If — c^—d^.
15. cHh\ 17. {a + hf. 19. 4.c'-cd^d\
Find the value of (a-\-h) - (x — y) :
20. when a =2, b = —3, x =8, and 2/ = — 5.
21. when a= — 4, b = 6, x = 3, and ?/ = — 1.
22. when a = ^, b = —2a,x= — \, and y = \.
19. Division of algebraic numbers. Division is the inverse
of multiplication (cf. § 8); i.e., it consists in finding one of
26 HIGH SCHOOL ALGEBRA [Ch. 11
two numbers when their product and the other number are
given. Hence the results of § 18 may be used to show how
to divide algebraic numbers.
Thus, since 8-3 = 24, 8.(-3) = -24, _8.3 = -24, and
(_8).(-3)=24,
therefore 24 -- 3 = 8,
_24-(-3)=8,
_24-3=3_8,
24 - (- 3) = - 8.
So, too, whatever the given numbers. Therefore : (1) the
absolute value of the quotient of two algebraic numbers is
the quotient of their absolute values, (2) this quotient is
positive if the dividend and the divisor have like signs, and
(3) it is negative if they have uiilike signs.
EXERCISE XI
Find the value of each of the foljowing indicated quotients :
1. 14-2. 4. -31- (-If). 7. 15 -(-1).
2. 14-- (-2). 5. -24-9. a -365--(-9i).
3. _18^41 6. (-6)2-(-2)3. 9. _63a2^(-9).
10. Of what operation is division the inverse? How, then,
may the correctness of a quotient be tested ? Illustrate.
11. If the dividend is positive, and the divisor negative, what
is the sign of the quotient ? Compare the signs of divisor and
quotient when the dividend is positive ; when it is negative.
Find the value of each of the following expressions :
12. 24-28-(-7)+(-16)-(-4).(-3).
13. _8.(-6)--24-27--(-6)^3.
14. j28-(-7)-2.(-4-2) + 24j-(-2)3.
Verify that = —. -„ :
•^ x-i-y X — y x^ — y^
15. when a = Q, b = 2, a? = 10, and y = 6.
16. when a= — 8, b = 12, x= — 9, and y = 7.
CHAPTER III
ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS
— PARENTHESES
I. ADDITION
20. Algebraic expressions, monomials, etc. Any combina-
tion of letters, or of letters . and numerals, representing a
number is called an algebraic expression.
The terms of an algebraic expression are the parts into
which it is separated by the signs + and — (or, rather, these
parts together with the signs preceding them). Thus, 3 a,
+ m^, and — 5cx are the terms of the expression 3 a + w^ — 5 ex.
An algebraic expression which is not separated into parts
by + or — signs is said to consist of a single term, and is
called a monomial. Thus, 3 a, 5mx^, and —llcV are
monomials.
An expression consisting of two or more terms is called
a polynomial. Thus, S a -{-7 m^— 5 ex is a polynomial.
A polynomial consisting of two terms is usually called a
binomial, and one of three terms, a trinomial. Thus, 2 s—5 xy
is a binomial, while 4 a; — a + 7 ^^^ is a trinomial.
21. Coefficients. Any one of the factors of a term, or the
product of two or more of them, is called the coefficient
(co-factor) of the product of the remaining factors. Thus,
in the term 5 axy^., the coefficient of axy^ is 5, the coefficient
of x'lp' is 5 a, the coefficient of 5 xy^ is a,
A coefficient consisting of numerals only is called a numer-
ical coefficient, while one that contains one or more letters
is called a literal coefficient. Thus, in the term — 3 a^m^ the
HIGH SCH. ALG. — 3 27
28 HIGH SCHOOL ALGEBRA [Ch. Ill
numerical coefficient of ax^m is — 3 ; but — 3 a and — '6 am
are literal coefficients of xhii and x?^ respectively.
Eemakk. The word " coefficient " is usually understood to
mean '' numerical coefficient " and to include the sign preceding
the term. Observe also that ax means the same as lax (cf. § 9,
Note).
22. Positive and negative terms. Like and unlike terms.
A term preceded by the sign + is called a positive term, and
one preceded by the sign — is called a negative term. If
the first term of an algebraic expression is positive, its sign
is usually omitted, but the sigli of a negative term is never
omitted.
Terms which either do not differ at all, or which differ
only in their coefficients, are called like terms, also similar
terms ; terms which differ in other respects are called unlike
terms, also dissimilar terms. Thus, 3 x^y^ — 5 a^y, and | x^y
are called similar terms.
EXERCISE XII
1. Name the coefficient of d^x in each of the following terms :
3 a^x, — 5 d^x, dx. 4 dhx. — | a^x. —^ , — 9 a^x.
2. In Ex. 1 which coefficients are literal and which numerical?
Which terms are positive and which negative ?
3. Do the positive terms in Ex. 1 necessarily represent positive
numbers for all values that may be assigned to the letters in-
volved ? Try a = 3 and a; = - 2.
4. What is the coefficient of x — ym each of the follow-
ing expressions : 13 (a? — y), — a (x — y), ^ m (x — y), and
(4 — a^) (x — y) ? Which of these coefficients are numerical ?
Which literal ?
5. Consult a good dictionary for the derivation of the words
"monomial," "binomial," "trinomial," and " polynomial." Write
three monomials, three binomials, three trinomials, and three
polynomials.
21-23] ADDITION OF ALGEBRAIC EXPRESSIONS 29
6. Distinguish between the meanings of 5 in the expressions
5 X and oc^. What name is given to the 5 in each case ?
7. What are like terms ? By what other name are they known ?
In what respects may they differ and still be like terms ?
8. Are 3 x^y, — 2 x-y, and | x^y similar ? Are 4 ax^ and — 6 bx^
similar ? Are these last two terms similar if 4 a and —6b are
regarded as their coefficients ?
9. Write three sets of like terms, some terms being positive
and some negative, and each set containing at least four terms.
23. Addition of monomials. Since 5 times any given num-
ber plus 2 times that number is 7 (^^e., 5 + 2) times the
given number, therefore 5a + 2a=(5-f2)a=7a, whatever
the number represented by a. Similarly, 8 mx^i/ + 8 mx^i/ —
(3 + S)mx^i/ = 11 mx^i/. Hence,
To add two or more similar monomials, add their coeffieie7it>i
and to this result annex the common literal factors, each with
its proper exponent.
It is usually more convenient to write the terms to be
added under one another, as in arithmetic, thus :
3 XT/ 153 ahnx 18 aks
8 xi/ 74 a!^mx — 7 aks
Wxy ' ? ? (cf. § 16)
If the monomials to be added are dissimilar, they cannot
be united into a single term, but their sum may be indicated
in the usual way ; thus, the sum of 5 a and 2 a:^ is 5 a H- 2 a;^.
Similarly, the sum of 3m and — 6a is 3m + (— 6 a), which
equals 3m — 6 a (cf. § 16).
EXERCISE XIII
Add the following sets of similar termsf and explain your work:
1. 2. 3. 4.
6 n 18 a^ —9 mx 31 abvr
3 n - 10 a^ 5mx - 22 abx"
— 2n — 3 a^ — 6 mx — 6 aba^
30 HIGH SCHOOL ALGEBRA [Ch. Ill
5. State a convenient rule for adding any number of like
terms. Does your rule apply to cases in which some of the terms
are negative ?
Find the algebraic sum of:
6. 4 a?^, — 2 x^y — 5 x^. 10. 12 a^n, a^n, — 4 a% — 9 ahi,
7. 11 ax, ax, — 9 ax. 11. S xz, — 8 xz, — xz, 2 ics;.
8.-4 cs^, — cs^, 8 cs^. 12. — a6^, — 7 a6^, a5% — 5 ah*.
9. — 3 aa;"2/j ''^ cl^Vj ctx^y. 13. — ic^/? ~ ^ ^I/) 12 a;?/, — 3 a;y.
Simplify the following expressions; i e., unite like terms, and
indicate the results where the terms are unlike :
14. 3 bxy^ + (- 4 bxy^) + (- 12 bxij^ + 5 bxy^ + bxy^ + (- bxy^.
15. — 4 mp^ + 13 a'^x + 7 mp^ -f 3 mp^ -f ( — 5 ax^ — 2 acc^ + mp^
16. 25 c^s^^ - 10 6^^ - bH - c^s^ -\-3bH-{- bH + cV - 8 (fs\
17. 7.5a; + fx — a; — i£C + iic + i^x — 3.45 x + li a;.
18. 3d*-5 c/ + 2ic^^-11.5d^-7ic^'^-5d^ + d*-c^.
19. - 6 (a- 62) 4-3 (a- 62) _ («_ ^,2) _ 5 (ct_ 52^ j^^a-b"),
20. 23 a^ + 5 6" - 8 a^^** - 13 6« + 24 a^^" _ 19 a^ + a^ - 6" - a^ft".
21. How many a^'s in 5 a; + 3 a; ? in 10 a? — 2 a; ? in 8 a; — aa; ?
in4aj — aj? ina; — Sa^ + lla;? in ma? -f na; — 3 a; ?
22. How many s^^'s in 8 s^^ + 2 sH ? in 3 sH + as't ? in 3^ s^^
+ 2 ms^^ - s2^ ? in 3 asH + (- 2 fts^^) ?
23. Add 15 a;2, - 2 aa;2, - 7 a^^^ 4 fta^^^ and - 2 Za;^.
[In Exs. 23-25 let «, 6, and ^ belong to the coefficients.]
24. 2 aa;?/) ~ 8 ^2/> ^^^ ^ ^^2/- 25. 5 xz^, — axz^, and 2 bxz^.
24. Addition of polynomials. Any two polynomials, e.g.^
S a^ — 7 xi/ -}- 12 7/^ ai^d 5 a^ 4- 6 2:3/ — 3 «/2, may be added
thus:
3a2_7^y + 12^2
5 a^ _|_ (5 2;^ _ 3 ^2
8a2_ a;y+ 9?/2
28-26] ADbtriON OF ALGEBRAIC EXPRESSIONS 31
This procedure may be stated tliiis : To add two or more
polynomials^ write them under 07ie another so that similar terms
shall stand in the same column, and then add each columii sep-
arately as in § 28.
EXERCISE XIV
Add the following sets of polynomials :
1.
2.
3.
4.
4a-26
6m
4-3^2
ax — 5
y
8|) + 2.s-3^2
2a4-56
4m
-7 7l'
2ax-
1
Wp-bs-^lt^
5.
6.
7.
6 m — 4 ^i
+ 7p
Sa
-40^ + 5?/
- a^a; - 8 6 -h 2 /
- 2m+ n
-5p
6a
-i-5x'- y
— 2 a?x —^'f
-Sm-{-2n
-4.P
-4.a
^2^
8 a-x + 2 6 - 9 .v*"
8. 12 ace — 5 a?^ — 9 2/, — 3 ao? — 6 a^^ + 2 ?/, and ax + x'^' — y.
9. 3 m — 7 71 + 2p, m H- 4 ?? — 6p, and n — 2m +p.
10. a^ — 2 .T + 1, a; — 3 + 8 a^, and 4 a; — 3.
11. 2a-76 + 3(a^-l), 4.b- a-6{x' -I), and 6+ (a;--l).
12. 3(a + &) - 2 c' - 5, 7 - 6{a + &) + 4 c', and c' - (a + h).
13. 2(?7i - n) + 4(|> + 1), 3 a — 5(m — w), and — 7{p + 1) — 9 a.
14. 2f A:-6.5? + 3im, 5fc-6im, and4m + 2Z.
15. s''-^t-^v,2t-^s\^.\i^i^v-^t+2s\
Supply the missing coefficients in the following equations :
16. 12a; — 4a?/ — 5aj-h7a2/= ^x + '^ay.
17. aoi? — 2 xy -\- dxy — CO? = ? x^ -\- ? xy.
18. 6rs-s^ + 5cs3+(2-3c)ns=?rs+?s3.
19. (a2-c)p + (3a4-5c-5)i?=?/>.
20. Add 3 a?2 + 4 a;/ — 2/"* — 7 a;?/ + 2 a:^?/, 10 a^y -f- (5 c - 10) xy-,
(c + 1) a:^ — 3 2/% and xy -\- ^ xy"^ -{■ {2 d — l)x^y.
25. Checking results. If a result (iu addition, for example)
is correct, then it must, of course, remain correct when we
assign any arbitrary values to its letters. This is the basis
32 BIGH SCHOOL ALGEBRA [^'h. Ill
of a very useful test of the correctness of algebraic work
(usually called a '' check ").
Thus, find the sum of 10 a^ - 3 7/ and - 2 a;^ _^ 9 2/.
SOLUTION CHECK
. 10.^-3., = 10- 6=. 4 1 j^^^^^^^
-^^ + ^^ =-2 + 18=26 L,d ,= 2
80^ + 62/ = 8 + 12 = 20 J
And since the sum of 4 and 16, the values of the summands, is
20, which is also the value of the sum, therefore the work is prob-
ably correct.
EXERCISE XV
In Exs. 4-11, p. 31, check your results as above by putting any
convenient arbitrary values for the letters.
II. SUBTRACTION
26. Subtraction of monomials. Since subtraction is the
inverse of addition, therefore (cf. § 23) :
To suhtraet one of tivo similar monomials from the other, sub-
tract the coefficient of the subtrahend from that of the minuend^
and to this remainder annex the common literal factors.
The work may be arranged thus :
73 a2 27n^k mx^f Uax
24 a2 -Sn^k 19 2;^ - 9 ax
49rt2 ~J5M ~1 ~~f~ (cf. § IT)
If the given monomials are dissimilar, the subtraction can,
of course, only be indicated.
A good practical rule for subtraction is : To subtract one
of tivo similar monomials from the other^ reverse the sign of the
subtrahend and proceed as in addition. In order to avoid
confusion in reviewing one's work, it is best, however, not
actually to reverse the written sign but only to conceive it
to be reversed.
25-27] SUBTRACTION OF ALGEBRAIC EXPRESSIONS 38
EXERCISE XVI
Perform the following indicated subtractions :
1. 18-5; -18-5; _18-(-5); 18-(-5); 9-(-9).
2. 3. 4. 5. 6.
7a 16ba^ - 18 m^ -ISr^a^ 26 ^V
4 a -Sbx^ Im^ - Tr^cc^ _ 9 ^y
7. 8. 9. 10.
3 ex" - 6 7nY 6. 8 k'a^y'- - 21 a^m*
11. Show that " changing the sign of the subtrahend and pro-
ceeding as in addition" will give the remainder in each of the
above exercises.
12. From 7 aoc^y take 3 ax-y ; from 5 njf take — 8 7q)^ ; from
4 (a - 2 b^) take - 11 (a - 2 ¥) ; from the sum of 13 2/"V and - 5 y-^'^
take 4 i/^a;^.
13. Indicate the subtraction of b^ from 3 a-; of 4(0^ + ?/^) from
— 6 (c + 2/) ; of 2 a;?/ from the sum of x^ and 2/" ; of — a^ft" from the
sum of 3 a* and — 6^**.
14. How many x^y's in 8 aic-^/ — 2oify? in mxry-{-nx-y—2 cx^y ?
in 7 cx'y-(- 3 x'y)?
15. Supply the missing coefficient in : 125 7nz — 97 mz= ? Z',
c2a2-(-9a2)=?a2; 5 ax^ + ? ao;^ = 2 aa;=^ ; 4:C8-?s = 2bs.
27. Subtraction of polynomials. One polynomial may be
subtracted from another by writing the subtrahend under
the minuend, similar terms under one another, and subtract-
ing term by term, thus :
55_3^2+ Q^l 32:4-5
85 + 5:^2 _ 9^^ _7a;4 4- 4 -2:?:
2 6 - 8 ^2 _^ 15 ab 10ai^'^)-\-2x
34 HIGH SCHOOL ALGEBRA [Ch. Ill
EXERCISE XVII
In the following pairs of expressions subtract the second from
the first, and check your results as in § 25 :
1. 8a-562, 2a + h\ 3. ^x' + x, Zx-2x',
2. 3m2-7, m^-lO. 4. s' + ^t, 2t-5s',
5. a^-2ab + b', -3 a' + 12 ab-12b\
6. x^-\-5oi^y + 7 xif — 2y^, 3 ic/ — ar' — 2 / — 5 a?y.
7. Check your answer to Ex. 6 by letting x = 2 and y = l.
8. From the definition of subtraction show that the minuend
equals the sum of the subtrahend and remainder. What means
of checking the result does this suggest (cf . § 17) ?
9. In each of Exs. 1-6, p. 31, subtract the second expression
from the first.
10. From c^ -\-d subtract c^ — d — 4 A;. Check result in two ways.
11. From 5 x^ + 4 a'b take 8 a^6 — 2 x^ + 5 abx, and check result.
12. Subtract 15 y^ + 10 aV + 4 mV from 34 aV - 10 mV, and
check result.
13. Subtract 15 — 3 a? + 10 aj^ from 12 a^ + 5 j also from — 2x;
also from 0.
14. Subtract 5^ a^-2i + a;-4J a;^ from 7 a;^~2i- a; + a^-4;
also from 0.
15. Subtract the sum of 5 a — 31 6^ + 2 a^ and 26 6^ — 4 a; from
ix^-2a' + 7b\
16. From what must a^b"" + 3 ca; + cZ*" be subtracted if the result
is to be 4 a^b'' — d'-^2cx?
In Exs. 17-20 let a, b, and m belong to the coefficients :
17. From 2ax — Sby-\- mxy take x — 2y-\-^^xy.
18. From (m - 2 6) ?/ + 3\z take 2z+(a- b)y\
19. From Qr -l)xy -\- bY - 3 mx^ take 2 V'xy - 5 mx^ + af.
20. From (a^ - 3 a6 + m^)^ -f (4 a^ - 5 «?> + 2 6*' + 7 m^) ?/ +
2 amz^ subtract {cir - 5 o6 + &^ - m')^ - 3|- a^62^ + (a^ - 2 a6 + m") y.
27-28] PARENTHESES 35
III. PARENTHESES*
28. Parentheses removed and inserted. Such an expres-
sion as 2 a; — (?/ — 3 0) means that i/ — 3 2 is to be subtracted
from 2 X ; hence (§ 27)
2x-(iy-Zz) = 2x-y + ^z,
Similarly, a — (^—h + c— d — e) = a-\-h — c + d-{- e; etc.
These equations (read from left to right) show that a
parenthesis inclosing any number of terms, and preceded by
the minus sign, may be removed provided that the sign of each
term within the parenthesis is reversed.
Again, reading the above equations from right to left
shows that any number of the terms of an expression may
be inclosed within a parenthesis preceded by the minus sign,
provided that the sign of each term so inclosed is reversed.
Remark. A parenthesis preceded by the plus sign may, of
course, be removed or inserted without changing the signs of the
terms inclosed. (Why ?)
EXERCISE XVIII
By means of § 27 show that :
1. 5a — (3a + ^)=5a — 3a — & = 2a — &.
[What is the sign of 3 a in (3 a + 6) ?]
2. ^x — 2y—{—^x-\-y)=^x — 2y-{-^x — y=zlx — Sy.
3. m^ — 3 np 4-p^ = m^ — (3 np —p^).
4. a — 2b-^c — 4.x = c — 4:X-{—a-\-2b).
5. Using § 11, show that 8 - (10 - 7) = 8 - 3 = 5 ; and then
show that the same result may be obtained by using § 27, i.e.,
show that 8-(10-7)=8-10 + 7 = -24-7 = 5.
Simplify each of the following expressions by two methods,
, as in Ex. 5, and compare results :
* " ^Parenthesis " here means any sign of aggregation whatever (cf. § 11).
36 HIGH SCHOOL ALGEBRA [Ch. Ill
6. ll_(3 + 6). 9. 27 + (-5-3)-13-7.
7. ii_(_3 4.6). 10. 27 -(-5 -3) + 13 -7.
a _(8-5) + 10. 11. -(6-4 + 9)+3-(-2 + 7).
12. In Ex. 9 what is the quality sign of 13 ? What does the
minus sign preceding 13 indicate ?
In Exs. 13-19 remove parentheses and unite similar terms :
13. 7x — Sac-\-(x — 2 ac). [Compare § 28, Remark.]
14. 3a — 46 + (6 — 2a). 17. x — y + {x-\-y) — {^x — y).
15. 2f-{-x' + f-xy). 18. a-2/2-(a-3)-(-3/-l).
16. 5a2 + 3?>-(-2a). 19. -(2m-5)-(-6+a^-3m).
In each of the following examples inclose the last two terms
in a parenthesis preceded by — :
20. 2s-3^ + w. 23. aa^ — 4 5x — 3 + 2^.
21. 6 4-5 X- — 3?/. 24. 2 /i — 3 A; — 7 a? — 5.
22. a^-f2 6 + c*. 25. 3 m* — 2 m"ic — 5 ma^" + a;*.
29. Parentheses within parentheses. It often happens
that one parenthesis incloses one or more others. In such
cases the expression within an inner parenthesis forms a
single term of the next outer parenthesis [cf. § 11 (ii)].
These parentheses, too, may be removed as in § 28,
thus :
3 a^ - J 9 m - [ - a^ - (4 s3 _ 5 ^^^) ^ ,,3-| j
= 3 a^ — 9 m -f- [ — a^ — (4 s^ — 5 m) + s^] [Removing brace
= 3 a^ — 9 m — a^ — (4 s^ — 5 m) -+- s^ [Removing bracket
= 3a^ — 9m— a^ — 4s* + 5m + s^ [Removing parenthesis
= 2 a^ — 4 m — 3 s'. [Collecting terms
Let the pupil simplify the above expression by first
removing the innermost parenthesis, then the next inner-
most, and so on, and compare his work with what is here
28-29] PAIiENTHESES 37
EXERCISE XIX
Simplify the following expressions; that is, remove the paren-
theses and combine like terms :
1. s - [t^ -\- (u'^ ~ s)']. 5. (m — 4p) - (a —p + m).
2. s-[e-(ir-s)]. 6. Sx'-\2a-(-x'-{-a)l,
3. 6a-[6-(-2a-|-36)]. 7. _ J2a - (-ic^.^, ^^ j^
4. a^^l-f-(2f-3x')l 8. -J_(-a^ + a)J.
9. mx^— [8?/— (6a; — W.1') — 2aj.
10. _ (60 - 25) -[92 -(18 + 27) I .
11. 3p + 4g + [7i)-2g-(5i)-3-5g)].
12. a — y— \a — (—y — a — 2)1.
13. a;-S3a.--[-(-3a; + 22/)+o?/]-32/:
14. 8a;2^2a;?/-[3x2-?/2-(2a;.?/-.T- + /)]-2/2.
15. 2 a' - 5 [3 6" - (a' - 2 c - 5 a')] _ [7 6" + 5 c - 6« - 2 a'] \ .
16. 8a-25-{-(3c-f?)-[4c-d-(-8aH-2&)]-2d;.
17. 4.-[5y-l3-(2x-2)-4.xl'}- Ja; + 52/-^+3J.
18. In Ex. 2, how many minus signs affect u^? How often,
then, will its sign be reversed by removing the parentheses ?
What will be its sign finally ? Answer the above questions for
- «2 in Ex. 7.
19. By considering the number of minus signs affecting the
respective terms, remove together all parentheses in Ex. 9. Also
in Exs. 11, 12, and 14.
20. In the expression 3 m — 4 a -f- 10 a;^ — 5 ?/ + 3 a^^ — 8 aa;, in-
close the 4th and 5th terms in a parenthesis preceded by the
minus sign ; then inclose this parenthesis, together with the two
preceding terms, in a bracket preceded by the minus sign.*
21. Make the changes asked for in Ex. 20, in the expressions
3m + 4a-10a;2_5^_^3^^2_g^^^ 3m-4a -2a?2-h52/-3a6-,
and — 5 a;'' + 3 ?/'" - 4 a - 14 6c + 8 A;2.
* The value of the expression is, of course, to be left unchanged.
CHAPTER IV
MULTIPLICATION AND DIVISION OF ALGEBRAIC EXPRES-
SIONS
L MULTIPLICATION
30. Law of exponents in multiplication. What is the
meaning of 5^ (cf . § 9) ? of o^ ? of a;" ?
How many times is s used as a factor in the product
8^-8^? Is 8^ ' 8^ equal to 8^ ? Explain why.
How may the exponent of the product ^ - 8"^ he obtained
from the exponents of the factors s^ and 8^ ? Would your
answer remain true if we were to put other exponents in
place of 3 and 2 ?
Is cc^x^a^ equal to x^'^, i.e., to 2^^+2+5? W^hy ? Is x"x^ equal
toaj«+*? Why?
The results of these considerations may be expressed in
symbols thus :
wherein a may stand for any number whatever, but m, w,
and p are positive integers.
Translated into common language, this law of exponents is :
The product of two or more power8 of any number is that
power of the given number who8e exponent i8 the 8um of the
exponents of the factors.
31. Product of two or more monomials. The product of
any two or more monomials may be obtained by a simple
extension of § 30.
J7.^., in the product of 2 ax^ and 3 ab'^x, how many numeri-
cal factors ? What is their prodilct ? How many a\s in the
38
30-31] MULTIPLICATION OF ALGEBRAIC EXPRESSIONS 39
entire product? How many 5's ? How many a;'s ? ^\^rite
down this product, using the exponent notation. What is
its sign ? Why ?
What is the product of 3 aV^ — 2 abaP'^ and 5 «5 ? Is it
— 30 a'^h'^x^ ? Explain in detail, mentioning the sign, the
coefficient, the letters, and their exponents.
These considerations lead to the following rule for obtain-
ing the product of two or more monomials: To the product
of the numerical coefficients of the several monomials, annex the
different letters which these monomials contain, giving to each
letter an exponent equal to the sum of the exponents of that
letter in the several monomials.
EXERCISE XX
Find the following indicated products, and explain your re-
sults, especially the signs and exponents:
1. 2. 3. 4. 5.
3a2 3a2 6-3^ -Smh -7aV
5a^ -5a^ -2.3^ -3ms^ -2 2^
6. 7. 8. 9. 10.
-5-23 mV 2 a;" -Sa^^" _4a;2y
8.2^ -ba^x -QxP -Sab^ -9x^y^
11. 7 s2 . _ 3 as . - 2 a\ 14. - 2 a"6 • 6f ab^'- 1 2| a'^bK
12. 3 mx^ • 2 m^ • — 7 am. 15. 7.5 mv'^ - — 4| am^ • —3 a"^.
13. _ 2i s? . - s^ . 3f a^^. 16. - 12 A;^/^ . _ aA:Z" • 2^ a"A:^
17. 2 (a -6)2. - 5 (a -b) ' 3 x'y ' -2^ x (a-b) .2.5(a-6)Y.
18. Define and illustrate the meaning of exponent, power,
base (of. § 9). May the base be negative ? May it be a frac-
tion ? May the exponent be fractional or negative ?
19. If 7z represents a negative number, is n^ positive or nega-
tive (cf. Ex. 13, p. 25)? How does 3^ compare with (-3)^?
2^ with (-2)^?
20. What is the meaning of 2/"~^ ? In this expression may n
be less than 2 ? What is the product of 4 a^ and —Ta""-^?
40 HIGH SCHOOL ALGEBRA [Ch. IV
21. Determine, by inspection, the sign of the result in each of
the following products when a = — 2, 6 = 3 : (a — 6)^ ; (a — 6)^ ;
(a + hy ; {att^Y ; (a^b^y ; (a^ — by. State your reason in each case.
32. Product of a monomial and a polynomial. From the
definition of a product [cf. § 7 (ii), § 18],
5 . (2 + 9) = 5 . 2 + 5 . 9,
since 2 + 9 is obtained by taking positive unity 2 times,
then 9 times, and adding the two results.
Similarly, a{m -\- x— y) = q^i -\- ax — ay^
whatever the numbers represented by a, m, x, and y.
Hence we may say : To find the product of a polynomial and
a monomial multiply each term of the polynomial by the mono-
mial and add the partial products.
The actual work may conveniently be arranged thus :
Check
3 a%- 4x2 + 11 2/2 = 10
-2xy = - 2
- 6 aH^y + 8 x^y - 22 xy^ = - 20
EXERCISE XXI
when a = 1, X = 1,
and y = 1
1. How is 2 -\-a — x obtained from + 1 ? How, then, may
2/(2 + a — ic) be obtained from y ?
2.
3.
4.
Multiply
3x-5y
a-4a6 + 362
2 m — Sn^ — mn
by
2x
-56
— 4 mn
Find the following products, and check results (cf. § 25) :
5. (2a^-4:y')'7xy. 9. -Sx^y^(2 x"" -4:X^y).
6. (4:ax-5xy)(-2x^). 10. 5 a\a^-~4 a^ -2 a-\-7),
7. (~2s^-Sst)(-9st% 11. -7ax'(Sax^-3a'x-5ax).
8. {-6u'-\-nv)(-4:v'x). 12. - 8 a^^" (3 - 4 a"* + 12 6").
13. -12xy{2}x^-5lx-4:).
14. 2iabc(1.5a'-{-7.5ab'-6b').
31-33] MULTIPLICATION OF ALGEBEAIC EXPRESSIONS 41
15. {^x'z-^x^^-4.xz''-^xz + ll){-^xz).
16. [a^+(a + iy4-(a-l)ic + l].2aa:2_
17. («/ - 2 xy -.15 a;y + 4 ary - 7 a?y (- cc^^/^-s).
18. 2 (%2-3 i;) [(^2 - 3 i;)^ - 13 a; (?^2_3 '^^+2 a;2(^2_ 3 ^,)_l].
19. 3a[7ar^-4(2a;2 + aaj) + «(2a-3a; + l)].
20. Multiply da—bh-\-c—x — y by — 1, and show that the
result agrees with § 28.
21. By what nust x — ly — 2az be multiplied, to obtain the
product 12 ah: — 14 a^y — 4 ah ?
22. Find a monomial and a polynomial whose product is
6 ax" -10 a'x - 14 aV 4- 8 aV.
23. Are the values of m and n in Exs. 9, 12, and 18, limited in
any way ? If so, how ?
33. Product of two polynomials. Since m + ^ is obtained
by taking positive unity m times, then n times, and adding
the two results, therefore (of. § 32)
(a -\- h ■{- c) ' (m -\- n) = (a -\- h -\- c)m + (a + 5 + c)n
= am -\-hm+ cm -\- an -{- hn -\- en.
Similarly for any polynomials whatever ; i.e.^ the 'product of
two polynomials is obtained by multiplying each term of the
multiplicand by each term of the multiplier^ and adding the
partial products.
If any two or more terms of a product are similar, they
should, of course, be united.
Such a multiplication and its check may be arranged thus :
Check
a2 + 2 a& - 62 = + 2,
a+ h =+2.
(a2+2a&-&2).a= a^ + 2a%- aW'
(a'^ + 2a&-62).6=: ggft + 2 a&2 _ &8
a3 + 3a-26 _f. aW- - 63 = + 4,
when a — \
and 6 = 1
Remark. The product of three or more polynomials may be
obtained by multiplying the product of the first two by the third,
this product by the fourth, and so on [cf. § 10 (1)].
42 BIGB SCHOOL ALGEBRA [Ch. IV
EXERCISE XXII
Perform the following indicated multiplications :
1. {x+2d)'(x-a). 9. (x^-xy + y^)'(x-y).
.2. {Sa^-5x)'(2a''-{-3x). 10. (a^ -xy -i-y^) - (x + y).
3. (7-2m3).(3-5m=^). 11. (0^-2 ay -\-y^) • {a- y).
4. (a'A-ab-\-b')'{a-b). 12. (5 s^ - 2 i^^^ . (5 ^3 _^ 2 ^^^^
3. (2s-Sf)-(Ss-4.t). 13. (7ey2-2c/«).(7ey2 + 2^).
6. {x-{-a)'(x-{-a), 14. (-4/g- 6 r) • (3^)-^?^)^
I.e., (ic + ay. 15. (m^ — w^ + p^) • (5 m — 2 np).
7. (3m2-10)l 16. (2a;-32/ + 0-4)2.
8. {Gx'-Sayy. 17. (a^- 3 a6 + 62_2 c)^.
18. (3m2 — 2mn) • (5m + 3w^) = ? Check your result by
letting m = 1 and n — 1. If, in the product, the exponent of m
should be wrong, would this check reveal the error ? Explain.
Would the error be revealed if m were taken equal to 2 ?
Multiply (and check your work) :
19. m* — 2 m^ — 6 m^ + m — 1 by 3 m^ + m — 2.
20. 2ii? — 1xy + Sx^ — 4.x-\-2y-\-lhjxy — Zx — 2y.
21. a2-62_c2_2a6-25c+2acby a-26 + c.
22. I.^x^-2xy-2.3y^-^x-2.^yhy3ly-l^x.
23. x"" + y"" hy X — y '^ hj y? -\- y'^ ', by a?" — i/**.
24. ic"" — 3 a;'"~^y + 3/"* — 3 x?/'"-^ by a?^ — 2 a;?/ + /.
25. 2x+(3n — l)?/by (n 4-l)a^ — (3n + l)y.
26. r2-(2ri-«2) by4?2_4 (^2r^-r2)-l.
34. Degree and arrangement of integral expressions. In
multiplications with polynomials, and elsewhere, it is often
advantageous to arrange the terms of a polynomial in a par-
ticular order; such arrangements will now be explained.
-^ A term is said to be integral if it contains no letters in its
denominator ; it is integral in a particular one of its letters if
that letter does not appear in its denominator. A polynomial
83-35] MULTIPLICATION OF ALGEBRAIC EXPRESSIONS 43
is integral, or integral in a particular letter, if each of its
terms is so.
U.g., 3 ax^ + ^^ — ^f ^ is integral in 6, m, x, and v ;
a 3
it is fractional in a; its first and last terms are altogether
integral, while its second term is integral only in 5, m, and ^.
e^ By the degree of an integral term in any letter (or letters)
is meant the number of times this term contains the given
letter (or letters) as a factor. Thus, 7 a^x^ is of the second
degree in x, and of the fifth degree in a and x together.
^The degree of an integral polynomial is the same as the
degree of its highest term. Thus, Sa^ — 5 a^jp^y — 2 bx^y^ is
of degree 4 in a;, 3 in ?/, and 5 in a; and 7/ together.
A polynomial is said to be arranged according to ascending
powers of some one of its letters if the exponents of that
letter, in going from term to term toward the right, increase ;
and that letter is then called the letter of arrangement. If
the exponents of the letter of arrangement decrease from
term to term toward the right, the expression is said to be
arranged according to descending powers of that letter.
Thus, 2x^ — 5 ax^y — 7 Pxy^ 4- 3 m^y^ is arranged according
to descending powers of x, and ascending powers of y,
35. Multiplication in which the polynomials are arranged.
If each of two polynomials is arranged according to powers
of some letter which is contained in each, then their product
will arrange itself according to powers of that letter, and the
actual multiplication will take on an orderly appearance.
E. g. , Ito 6nd the1>roduct«l 7 ic - 2 ic^ + 5 + x^ by 3 ic + 4 a;2 - 2, arrange the
tft)rk<to«B:
Check
a-3- 2x2+ 7x + 5 =11/
4 x2 + 3 x - 2 =5,
4 a;5 - 8 x* + 28 x3 + 20 x2
3x* - 6x3 + 21x2 + 15 X
- 2 x3 + 4 x2 - 14 X - 10
4x6 _5a4 + 20x3 + 45x2 + X- 10 =55,
HIGH SCH. ALG. — 4
when X = 1
44 niGIl SCHOOL ATMEBJIA [Ch. TV
EXERCISE XXIII
1. Is 1^ a^oc^ integral or fractional ? In what letters is ' :;^
integral ? ' In what letters is it fractional ?
2. May an integral term have a fractional coefficient ? Illus-
trate. Write a term integral in m and n and fractional in d.
3. When is a polynomial fractional in a particular letter^?
Write a binomial fractional in a and b ; a trinomial integral in a
and b, but fractional in c.
4. In Ex. 11 below,
(1) Of what degree is each term of the multiplicand ? each
term of the multiplier ?
(2) Of what degree in a? is the first term of the multiplicand ?
Of what degree in y ? Answer the same questions for the other
terms of the multiplicand.
5. How is the degree of an integral polynomial determined ?
Give three illustrations from the exercises on p. 42.
6. Name the degree as regards x of each polynomial in Exs.
11 and 12 below.
7. Arrange the expression
3ay^f + xf>-S xY — 6 T^y^ + x^y — Sf-{- 5 y^
according to ascending powers of y. How is it then arranged
with reference to aj? Of what degree is this expression? What
is the degree of 3 x^y^ ?
Multiply :
8. 6x^-2 + 5x + Si^hj x^-\-5-x.
[In Exs. 8-16 arrange both multipUer and multiplicand according to some
letter contained in each, and observe that the product has then a correspond-
ing arrangement.]
9. 2a + a3-a2_iby 4-a2-fa.
10. Sa-x-4.ax^-{-x''-a^hj a^-ax-^a^.
11. 3xy'^-f-Sx^y-^x''hy-2xy-\-a^-^y\
12. x-y'^ — xy^ -f y^ — x^y-\-x^ by x^-\-xy — y^.
13. 4.hh-hr^-h^-\-27^\)yh-2r.
35-37] DIVISION OF ALGEBRAIC EXPRESISIONS 45
14. Q>y^ + ^ xY + 2x'-3o?y-xfhy y'' + Zx'-2xy.
15. af — 5 aj^2/^ —^f — &xy^-\- 15 ^y^ + 2 x^y hj 'd y"^ -{- x^ — 2 xy.
16. 4.f-lQ>s^t^-^&^-\-^sH + ^st^hj^s'-f-^sH.
17. a"+i - 3 a"+- + a "+^ - 2 «"+'* by 2 a"-i + 3 a"-^ - 4 a'*-=^.
18. x^'y^ + i»"+^?/ + 0^**+^ — £c"+^?/2 by or^ + 2/^ — ic?/^ — x^y.
19. In Ex. 8, of what degree is the multiplicand ? the multi-
plier? the product? The term of highest degree in the product
is the product of what two terms ?
20. Of what degree is the multiplicand in Ex. 11 ? the multi-
plier? What, then, should be the degree of the product? Should
all the terms of the product be of the same degree ? Why ?
II. DIVISION
36. Law of exponents in division. Since division is the
inverse of multiplication (§ 19), therefore the results of § 30
may be employed to find the law of exponents in division.
Thus : since a^ - a^— a^ therefore a^ -j- a^ = ? Is o^ -r- oc^
equal to x^^ i.e., to a:;^^? Why?
. Write the following indicated quotients and explain your
answer in each case : aJ -^a^\ 8^ -i- s^; 2^" -^ 2^ ; x^ -i- x.
How is the exponent of the product of two powers of any
given number obtained (cf . § 30) ? How, then, should the
exponent of the quotient be obtained ?
If m and n are positive integers, m greater than /t, and x
any number whatever, then (cf. § 9, also Exs. 18, 20, p. 39)
the above results may be expressed in symbols thus :
j,m _j^ w-re __ j,m — n
This equation states the law of exponents in division ;
translate this law into common language (cf. § 30).
37. Division of monomials. Since the quotient multiplied
by the divisor always equals the dividend (§8), therefore
12 a^-f- 3 a:^ = ? that is, what is the number which, when
multiplied by 3 x'^., gives 12 a^ as product (gL § 31) ?
46 HIGH SCHOOL ALGEBliA [Ch. IV
Similarly: 8 a^rr^ ^ 4 aV = ? Why? 24mV-^8mY = ?
Why? -18 a^^"-^ 6^362=? Why? Sml^xy^ -i- (^-^.m^xy) ^^^
How is the sign of the quotient determined ? the coeffi-
cient? the exponents? How may § 8 be used 'to test the
correctness of the quotient ? From the above write a rule
for dividing one monomial by another, mentioning the sign,
coefficient, letters, and exponents of the quotient (cf. § 31).
EXERCISE XXIV
Perform the following divisions ; check results by § 8 :
1. Q>a^^2a. 4. - 1^ a^h' ^ (S ah\
2. 15aV-f;3ax2. 5. 10 c«dV -- 5 c^de.
3. 12 mV -T-4:X^. 6. — 45 m^n^ -^ ( — ^ m^n).
^ -48aV ^^ -lmh\ ^^ 3.1 (xyz)'
12 aV ' ' --i-mV' ' -.Sa^yV'
g * 15 7ty ^2 -<33mny ^^ - 12 g^^c^^
-mfgY\ 13 _J/^. ,„ 2^
,»»+3
25 //p2 -T\fh' 6x^
10. — :r—rr' 14. 18. .
— 7 a^6 4^ m"?^'^ 2 a;"
19. If two monomials have like signs, what is the sign of their
product ? of their quotient ? How do we find the exponent of
any given letter in the quotient of two monomials ?
In Exs. 20-25, multiply the first monomial by the second;
also divide the second monomial by the first :
20. - 16 «^ 51 m''. 23. I c (m + n), - 10 c' (771 + 71)1
21. 42 (p + qy, — 14 (p + qy. 24. 8 ax^Y'"+% — 2 ^^^+2^2 (m+»)^
22. 5afy% 15 a^^+y. 25. 13 (.t-:^)^, -26(x-zy.
38. Division of a polynomial by a monomial. Since (§ 32)
a (m + 2; — ^) = «m + ax — ay,
therefore (am + t«:r — ay} -i- a = m + x — y,
whatever the numbers represented by a, w, a: and y. Hence,
;}7-;]8] niVTSlON OF ALGKliUAIC EXPRESSIONS 47
To divide a polynomial hy a monomial^ divide each term of
the polynomial hy the monomial^ and add the quotients so
obtained.
E.g., (15 aV - 10 hx^y + cV) -^ 5 x^ =^ S a^x - 2 bx^y + i c\
EXERCISE XXV
Perform the following indicated divisions :
, 4a»-12a^ ^ 9 mhi^ + 12 mn^ - 30 m^n*
D.
4 a^ — 3 mn^
-2Axy + lSiK^y\ ^ -18a^-81a; + 9a^
xy
c2
14 7-2-
-3^0^
2l7-2s^ + 8
-39>
26 a^m'
7 r
^-52 aV-
«%®
1 a'W-
13 a-m^
-4a«6^ + 6
a^«Z>^
^ 3 m - 2 n + 11 a; ^
9mV + 12myi2-30mV ^
3m7i2 ' • -2a26«
11. How may any polynomial whatever be divided by a mono-
mial ? How are the signs of the several quotient terms deter-
mined ? their coefficients ? their letters ? their exponents ?
Divide [and check the work in each case (cf . Ex. 10, p. 2Q>)'] :
12. 07-1-4 ax^ — 3 m^x — 6 a7nx by — x.
13. a^6%i3 - 4 a36V« + 12 ci'ft'ic by 4 rt^^s^
14. 1 7-^5 + J cr^^s^ — I r^s^ by 2 rs ; also by f rs.
15. a"* — 2 a^'^^ - 5 a"*+2 + 9 a"»+^ by a"* ; also by a\
16. 2«+* — 3 z""^ -h 4 aV — 2;^ by — 1 2:1
17. - 10 (h - 1)*-' - 6 (/i - 1)^A: + 15 (7i - 1)^A:2 by - 5 (h - 1).
18. x(x-\- yy —^(x-\- yf + x'^ (x +yy by — a? (aj + ?/)l
19. 2 (s - ^)- - s^ (s - O'^+i - 5 (s - ^)-+3 by i (s - O'"-^
Separate each of the following expressions into two factors,
one of which is a^ :
20. c^a^-hd^^. 22. -Sx^y-{-5x^z-7x^.
21. aV — a^x^ + aj2. 23. —x^ + Q eV -f — - •
4
48 HIGH SCHOOL ALGEBRA [Ch. IV
In Exs. 24-26, group the like powers of y (cf. Exs. 20-23) :
24. ty^ -f c?/' - ry' -~3sy^-{- y\
25. ay* --2by — S cy^ — my* + dy^ — 9 y.
26. (a + l)/-(«-l).v' + /-3 2/^-(3a4-4)2/3 + a/.
39. Division of a polynomial by a polynomial. Since, by
§ 35, the product of (4:a^ + 2>x-2^ and (jx^ -2x^+1 x^b)
is 4 a:^ — 5 a^ + 20 a;^ + 45 a:2 + a; — 10, therefore, with this last
expression as dividend, and a^— 2a:^+7a7 + 5as divisor, the
quotient must be 4:x^-\- S x — 2; i.e.,
( 4 rcs _ 5 ^4 + 2 :z^ + 4 5 2^ + a; - 1 ) -f- (2^3 _ 2 2^ + 7 rr + 5)
= ^x^ + Sx-2,
The process of obtaining this quotient from the given
dividend and divisor will now be explained.
Since the dividend is the product of the divisor by the
quotient, therefore the highest term in the dividend is the
product of the highest term in the divisor multiplied by
the highest term in the quotient (cf . Ex. 19, p. 45) ; and
therefore if 4 x^, the highest term in the dividend, is divided
by a^, the highest term in the divisor, the result, 4a^, will be
the highest term in the quotient.
Moreover, since the dividend is the algebraic sum of the
several products obtained by multiplying the divisor by each
term of the quotient, therefore if 4 a;^ — 8 ic* + 28 a^ 4- 20 a^,
the product of the divisor by the highest term of the
quotient, is subtracted from the dividend, the remainder,
viz., 3a::* — 8a^ + 25a:2^^_;1^0^ ^[n }yQ ^\^q g^^ Qf ^\^q prod-
ucts obtained when the divisor is multiplied by each of the
other terms of the quotient except this one.
For the reason given above, if 3 a;*, the highest term of this
remainder, is divided by a^, the highest term of the divisor,
the result, 3 x, is the next highest term of the quotient.
By continuing this process all the terms of the quotient
may be found. The work may be arranged as follows :
J9] DIVISION OF ALGEBRAIC EXPRESSIONS 49
DIVIDEND
x3-2a:2+7x+5
4x2 + 3x-2
(x3-2x2 + 7x+6).4x2= 4 x5-8a;H 28x3+20x2
3x4- 8x3 + 25x2+ x-10 — -'
(a;3_2a;2+7x + 5).3x = Sx^- 6x3+21x2+ 15 x Quotient
- 2x3+ 4x2-14 X- 10
(x3-2x2+7x+5) .(-2)= - 2x3+ 4x2-14 x- 10
Check
When X = 1, dividend = 55, divisor = 11, and quotient = 5, as it should.
Even if it is not known beforehand that the dividend is
the product of two polynomials, the process of division may
still be applied as above. This process may be formulated
thus :
(1) Arrange both dividend and divisor according to the
descending powers of some one of the letters involved iii each,
and place the divisor at the right of the dividend.
(2) Divide the first term of the dividend by the first term of
the divisor y and write the result as the first term of the quotient.
(3) Multiply the entire divisor by this first quotient term^
and subtract the result from the dividend,
(4) Treat the remainder as a new dividend^ arranging as
before^ and repeat this process until a zero remainder is reached^
or until the remainder is of lower degree in the letter of arrange-
ment than the divisor.
EXERCISE XXVI
Divide (and check your results by § 25) :
1. ar^+7a; + 12by ic + 3. 5. IS x +6x^-i-6 by 3x-{-2.
2. ay^-x-20hj x-5. 6. 8 + 3 a^- 14 a; by 2-3a;.
3. b'-6b-16hjb + 2. 7. 10a^+lla2_8byl-2al
4. s2-14s + 49by s-7. 8. Sx^ -4.a^-7 hj - sc^-l.
9. c3 + 6c24.12c+8by c+2.
10. 2a^-{-llx''-\-19x-\-10hy2x''-i-7xi-5.
U. 75 m- + m^ - 15 m' -125 by 25-\-m'- 10 ml
50 HIGB SCHOOL ALGEBRA [Ch. IV
12. p*-\-4:p^-\-6p'' + 5p + 2hjp'+p + l.
13. 2ic^-f6a^ — 4a; — 5a^ + lbya^ — a; + l.
14. 3a^+3a^ + 3 + 3a + a^ + 5a3by 1 + a.
[Here, as in arithmetical "long division," labor may be saved by "bring-
ing down " at any stage of the vs^ork only so much of the remainder as is
needed for the next step.]
15. Divide 6 a^o^ - 4 a^x-4:ac(f + a"^ -\- x'^ hj a^ -\- x^ - 2 ax.
SOLUTION
X* -iax^ + 6 a^x^ - 4 a% + a* bc^ - 2 ax + a^
x^ — 2 ax^ + (i^x^ ^•■^ — 2 ax -{■ a^
— 2 ax^ + 5 a'^ic-^ — 4 a^x
-2ax^ + 4 a^x^ - 2 a^x
a-x- — 2 a^x + a*
a-x^ -2a^x-\- g^
Note. To make the explanation of § 39 apply when two or more letters
are involved, replace "highest term" by "term of highest degree in the
letter of arrangement."
16. In Ex. 15 perforin the division when both dividend and
divisor are arranged according to the descending powers of a.
17. Divide 4 a^i/^ + 8 a^ + 2/^ + 8 ic^2/ ^J ?/ + ^ x.
18. Divide 2 a^ + A;^ - 5 o^k - 4 ak^ + 6 a'k' by ¥ + a^- ak.
19. (10 x^y^ -^ ii^ -10 xhf -\-5 xy' - 5 x'y -f)-^(a^+y^-2 xy) = ?
20. If the partial quotient, at any stage of the process of divi-
sion, is multiplied by the divisor, and the corresponding remain-
der added, how must the result compare with the dividend ?
21. What check for division is suggested by Ex. 20 ? Is this
check more or less complete than that given in § 25 ? Explain.
22. Divide 2a^ + a;^ + 49a^-13aj-12by a^-2«2 4.7aj4_3.
[Since there is no term in x^ in the dividend, care must be used to keep
the remainders properly arranged.]
Divide (and check the results as the teacher directs) :
23. v^ — 'v* — l-}-2v-\-v^ — v-hjv — l-\- v\
24. 0(^^-41 a-120 by a2 + 4a-|-5.
39] DIVISION OF ALGEBRAIC EXPliEbSIONS 51
25. m* + 16 + 4 m^ by 2 m + m^ + 4.
26. T\x*-ia^'y + lixY + ixfhy^x-\-iy.
27. 1.2 aa;^ -\-a^x^-2a'- 3.4 a-V + 6 aa; by 6 aa; -2 al
28. {2x-{-3a^-l+2x^){l-\-a^-x)hjl + x + x^.
29. a^ - 6^ by (a^ + 63^) (a + 6) + a'b\
30. a^ 4- 6^ + c^ — 3 a6c by a^ -{- b^ -{- c^ — ab — ac — be.
31. a;*-3a^ + a^ + 2a;-l byaj2-aj-2.
fin Ex. 31 the complete quotient is ic'^ - 2 x + 1 + ~ ^ '^ .~|
32. v^ — v-{-7hjv-\-A. 34. 2 s^ — 3 s + 8 by s^ — 4.
33. a^ — 1 by a + 1. 35. a^ + a; — 25 by a; — 3.
36. a*-7a2-9a-6a3-6by3+a2-2a.
37. 3 a;^ 4- 11 a^ + 11 a;^ + 9 a; + 10 by 4 x + 5 4- a^.
38. Divide p^ -{- q^ by p -^ q until 4 quotient terms are obtained ;
divide 1 by 1 — r to 8 quotient terms ; 1 by 1 — mx to 4 quotient
terms ; and a by a — a; to 5 quotient terms.
Divide :
39. cd-d2 + 2c2by c + c?. 42. h^ - 7i^ by h^ + Ic^.
40. oc^ — i/ by x — y. 43. a-" — a;-" by a" — a;".
41. a*-166^by a-26. 44. u^'' -^ 11 u"" -{- 30 by u"" -\- 6.
45. a5'"+'* — a?**?/""^ — a:*"?/** +2/^""^ t>y ^"^ ~ 2/**"^-
46. af"+"-^ - 3 xY" "^ — 5 a;"*" V + 1^ /''-^ by a;" - 5 y.
47. Divide a6c + aoif + a^ + a6a; + 6a;^ + ca;^ + aca; + bcx by
a^ -\- ax -\- ab -\- bx.
Solution. Since x occurs in more terms than any other letter, it will
be best to arrange the work thus (cf. Exs. 24-26, p. 48) :
or^ + (a + ?> +c) x2 + (ab -\- ac ■{- bc)x + abc \c^ + (a + b)x + ab
x^ + (a + h)x/^ + ahx
cx^ + {ac + bc)x + ahc
cx/^ + {ac + hc)x + cthc
I
62 HIGH SCHOOL ALGEBRA [Ch. IV
Divide :
48. 1/ -\- cy -\- cd -\- dy by 2/ + c.
49. -ab-\-ay + y^-byhy y-b.
50. f-\-2 dy' + 2 / + ^^2/ + 4 d?/ + 2 d- by / + 2 ^2/ + d^-
51. a-f{a-\-&)+^y'-y^-2,ay' + aY-2ayhySf-y + a.
52. Divide a^ — 21 by x — a) note that the remainder is what
the dividend would become if a were substituted for x.
53. Divide a^ + Sa^ + l by x — a] note that the remainder
differs from the dividend only in that a replaces x.
54. Divide m^4-7 by m — c and compare the remainder with
the dividend. Similarly, divide v^ — 1 by v — 2 ; 5 m^ — 8 m + 3
by m-3; 2/^-4/+ 32/-1 by y-h-, 2?-* -7^ + 10 by r-1;
by r — 2 ; by r — 3.
55. Divide 2 xy^ + Sx* — 4: x^y^ — 7a^y-\-y^ by a^-\-y^ — xy,' ar-
ranging first according to powers of x, then according to powers
of y, and compare the results.
56. As has just been seen in Ex. 55, the form of the quotient
depends upon the choice of the letter of arrangement when the
division is not exact ; is this the case when the division is exact ?
40. Finite numbers. As we pass from left to right the
numbers of the series 2, 2^, 2^ 2*, etc., increase without end ;
and the numbers of the series 1, J, |^, etc., decrease without
end. Hence we see that, in mathematical operations, there
may arise numbers which are greater, and others which are
smaller, than any fixed number that we can name or even
conceive of; such numbers are called infinitely large and in-
finitely small numbers, respectively. All other numbers are
called finite numbers. An infinitely large number is repre-
sented by the symbol 00.
41. Zero. Operations involving zero, (i) The result of
subtracting any given finite number from itself is called
zero (cf. § 13). Thus if a represents any finite number,
then I a — a == 0.
39-41] DIVISION OF ALGEBBAIC EXPRESSIONS 53
(ii) From this definition of zero and the definitions of
addition, subtraction, etc., already given, it follows that, if k
is any finite number whatever,
then k-\-0 = k-0 = k,
and A; . = . ^ = 0.
J£^.^., ^ . = because k • = k • {a — a) — ka — ka = 0.
(iii) If k is any finite number whatever, then
A; -7- = no finite number whatever.
For, if A; -7-0=/, a finite number, then/- would equal k
(§ 19), but this is impossible (ii).
(iv) 0-^0 =/ [i^iiy finite number
for /•0 = 0.
(v) From (iii) and (iv) above it follows that we must
not divide hy zero^ since doing so leads, at best, to an inde-
terminate result.
EXERCISE XXVII
1. When the values 1, \, \, \, y^g, • • • are assigned to x, how
do the successive values of the fraction 5/a; compare ? Can you
name a number so large that none of these values will exceed it?
Can you name a number so near that none of the series of num-
bers 1, -^, \, ^, yi^, • • • will be still nearer to ?
2. What is meant by an infinitely small number ? by an in-
finitely large number ?
3. Define zero. How does it follow from your definition that
(1) 3-0 = 3? (2) 0-5 = 0?
4. Can the equation ax = be' true if neither a nor x is zero?
Does it require that both a and x should be zero ?
5. What is the value of f ? Why ? What is the value of 0/a,
wh^re a is any number except zero ?
6. May ^ = 5? 1000? -72? ^? Explain. What is
meant by saying that ^ gives an indeterminate quotient ?
7. The quotient ^ cannot be a finite number. W^hy ? Will
it be an infinitely large or an infinitely small number (cf. Ex. 1)?
54 HIGH SCHOOL ALGEBllA [Ch. IV
42 * Some elementary laws. What is the meaning of the expres-
sion 5 + 2 + 8 (cf . § § 4 and 10) ? of 2 +5 + 8 ? Wherein do these
expressions differ?
(i) Although a change in the order in which operations are
performed may, in general, change the result (cf. § 10), yet
some such changes of order do not affect the result. Thus :
5 + 2 + 8=2 + 5 + 8 = 8 + 5 + 2,
5.2.8 = 2.5.8 = 8.5.2,
5 + 2 + 8 = 5 + (2 + 8) = 5 + 10,
5.2.8 = 5. (2. 8) = 5. 16,
and 5.(2 + 8) = 5.2 + 5.8.
(ii) Moreover, based upon our experience with particular sets
of numbers, we have silently assumed, in the preceding pages,
that the above changes may be made with any numbers whatever
without affecting the result. Thus, if a, 6, and c represent any
numbers whatever (positive, negative, integral, etc.), we have as-
sumed (without proof) that :
a + & + c = 6+a + c = c + a + 6, etc., (1)
a 'h ' c = h ' a • c = c • a 'h, etc., (2)
a^h + c = a-\-(h + c), etc., (3)
a 'b ' c = a ' (b ' c), etc., (4)
and a- {b + c) = ab-^ ac. (5)
Of these equations, (1) states what is known as the commutative
law of addition; (2), the commutative law of multiplication; (3),
the associative law of addition; (4), the associative laiv of multipli-
cation; and (5), the distributive law of multiplication as to addi-
tion: all of them taken together are often spoken of as the
combinatory laws of algebra.
These laws are easily verified in any particular cases : through-
out this book we shall continue to assume their correctness. We
wish, however, to point out to the pupil that mere verifications,
however numerous, do not establish a general law.
* This article may, if the teacher prefers, be omitted till the subject is
reviewed. For a full discussion of these laws see El. Alg. Chap. IV.
CHAPTER V
EQUATIONS AND PROBLEMS
43. Equation. Members of an equation. A statement
that each of two expressions has the same value (i.e., repre-
sents the same number) as the other, is called an equation.
These two expressions are called the members of the equa-
tion, the expression preceding the sign of equality being the
first member, and the other the second member.
Thus, Sx—16 = Sx-i-4: is an equation ; 8 a; — 16 is its
first member, and S x -\- 4: its second member.
Remark. In algebraic work, the equation is a most important
instrument ; to it is due the chief advantage of algebra over arith-
metic. We have already seen some evidence of this in § 3, but
much more is to follow. In a recent book Sir Oliver Lodge says :
" An equation is the most serious and important thing in mathe-
matics."
44. Conditional equations. Identical equations. Is the
statement Sx — lQ=Sx-\-4:, true when x = l? when x=2?
when a: = 3 ? when a^ = 4 ? when x = 5? Answer the same
questions with regard to 2x= (x -{- 1)^ — Qa^ + 1).
The equation Sm-\- ^n = 22 is true if m = 4 and n = 2,
but is not true for any other positive integral values of m
and n ; while the equation Sx^ + k — a^ = k-\-2x^is true for
all values that may be assigned to x and k.
An equation which is true for all values that may be as-
signed to its letters is called an identical equation, and also
an identity; while one which is true only on condition that
certain particular values be assigned to its letters, is called
a conditional equation. In the following pages we shall use
the word equation to mean conditional equation unless the
contrary is expressly stated.
55
56 HIGH SCHOOL ALGEBRA [Ch. V
As we shall see later, by performing the indicated opera-
tions the two members of an identity may be reduced to
exactly the same form; hence the name "identical equation."
45. Roots of an equation. Checking. The roots of an
equation are those values which satisfy the equation; ^.e.,
they are those values which, when substituted for the letters
the equation contains, make the two members identical.
Any process by which the roots of an equation are found
is called solving the equation.
The final test of the correctness of supposed roots is to
substitute them for the letters in the equation ; if they satisfy
the equation they are roots, otherwise not. This process is
called checking the roots. Thus, 4 is a root of the equation
8a;-16 = 3a:H-4,
because 8-4 — 16 = 3-4+4. [each member being 16
46. Some axioms and their uses. The following principles,
usually called axioms^ are useful in solving equations :
1. If equals (i.e., equal numbers^ are added to equals^ the
sums are equal.
2. If equals are subtracted from equals, the remainders are
equal.
3. If equals are multiplied hi/ equals^ the products are equal.
4. If equals are divided hy equals, the quotients are equal.
Here, however, as elsewhere, it is not permissible to divide
by zero [cf. § 41 (v)].
The correctness of these axioms rests upon the fact that
equal numbers are in reality the same number, differing at
most in form. Thus, 24 + 11, 7 • 5, and 6^ — 1 are merely
different forms of writing 35.
Suggestion to the Teacher. It is strongly recommended that the teacher
illustrate the physical meaning of an equation, and also the meaning of the
above axioms, by means of a pair of balances (easily made, if not provided
by the school).
44-47] EQUATIONS AND PROBLEMS 57
47. Solution of equations. To show how the above axioms
may be used in solving equations, let it be required to
solve the equation H x—lQ = '^ x -\- 4, ^.e., to find the value of
X which satisfies it.
SOLUTION
Since
Sx-16 = 3x-\-4r,
therefore
Sx
-16 + 16 = 3x + 4+16,
[Axiom 1
i.e.,
8 a^ = 3 a; + 20,
and therefore
8x-3^ = 3a; + 20-3a-,
[Ax. 2
i.e.,
5x = 20,
whence
x = 4..
[Ax. 4
CHECK
On substituting 4 for x in the original equation, that equation
becomes
8. 4- 16 =3. 4 + 4, i.e., 16 = 16 ;
hence the equation is satisfied, and 4 is a root (cf. § 45).
EXERCISE XXVIII
1. Is 2 a root of a^ — 5 07 + 6 = ? Is 3 also a root ? Explain.
How may we check a supposed root of an equation ?
Solve the following equations, give the reasons for each step in
the work, and check the roots :
2. 10a; = 40. 10. 3m + 2 = m4-30.
3. Sy = -32. 11. 7a;-10 = 5x + 18.
4. k + l = 7. 12.-4^ = 3 + ^ — 15.
5. m-9 = 4. 13. 20-12w + 5 = 0.
6. 2i; + 7 = 63. ' 14. 13s-9-2s = 24.
7. 2v-7 = 63. 15. 7x-55 = lS-2x-l.
8. 46 = 5s-4. 16. 6v-(v-3)-12 = 0.
9. -13 = 3a; + 8. 17. 2/'-(/+2/ + 8) = -6.
18. In Exs. 7-13, point out the members of each equation.
Which is the first member of the equation in Ex. 14 ? What is
the other member called ?
58 BIGII SCHOOL ALGEBRA [Ch. V
19. What is meant by solving an equation ? Describe briefly
the process used in solving an equation.
20. Are the equations in Exs. 2-17, above, conditional equations
or identities ? Why ? In which class of equations would you place
2x + S = 2(4:X-{-3)-(6x + 3)? Why ?
21. If 2 a is subtracted from each member of the equation
5x-\-2a = 3x-\-4:b, what is the resulting equation ? What does
this show with reference to removing a term from the first to the
second member of an equation ? Is the same thing true when a
term is removed from the second member to the first? Show
this by adding —3 a; to each member of the given equation.
48. Transposition. Directions for solving equations. Re-
moving a term from one member of an equation to the other
is called transposing that term.
If
x-}-a = b,
then
x-\-a — a = b-
-a.
[Ax. 2
i.e.,
x=h-
-a.
[•.•a-a =
Again, if
x = b-
-a,
then
x-\-a = b-
-a-\-a,
[Ax. 1
i.e.,
x-^a = b.
[... -a-i-a =
Hence, since a msij represent any term whatever, a term
may he transposed from one member of an equation to the other
by merely reversing its sign (cf. also Ex. 21, above).
For solving equations such as those considered in § 47 the
following directions may now be given:.
1. Transpose all the terms containing the unknown number
to the first member of the equation., and all other terms to the
second member.
2. Unite the terms of each member., and then divide both
members by the coefficient of the unknown number.
3. Check the root thus found by substituting it in the given
equation.
47-48] EQUATIONS AND PROBLEMS 69
Ex. 1. Solve the equation 4 s — 15 = 2 s + 11.
SOLUTION
Transposing, we have 4 s — 2 .s = 11 + 15;
uniting like terms, 2 s = 26;
dividing by 2, s = 13.
Check: 4-13-15 =
: 2 • 13 + 11. f each mem
Lber is 37
EXERCISE XXIX
Solve (and check) the following equations :
2. 32/-5 = 22.
3. -10 = 6 a + 8.
13.
^-7 = 12.
4
4. 3(aj-5) = 48.
14.
2^ + ^ = ??.
5. iz-\-2-z^n.
3 6
6. A.-x = -ll-\-2x.
15.
^ ^^^ = m + 10.
3y-7 = A-2y-5.
7. {d-\-lf-d'' = -ll,
a 20-5fc = 3A; + 3.
9. -8a; = 4(aj-2) + 10.
16.
17.
10. 4(-3 + /i') = (2/i-3)2.
XX. 1^-1 = 10.
18.
19.
i(2/-6) = K2/-2).
x-{-l x + 6 o
- 7 + 2 - ^*
[Multiply both members of the
20.
3ia = 5a-9ia-16.
equation by 12 (see Ax. 3).]
21
12-3a; + 20 = 44 + 3a;.
• ^'- f'-ro-''-
22.
x-9 x-5 ^
3 ~ 12 '
23. 14A;-(20-7A:-2)=:6A:+68.
24. (c + 5)(2c-l)-(2c-3)(c + 7)=0.
25. What is meant by transposing a term from one member of
an equation to the other ? What change must be made in a term
thus transposed?
26. State in order the axioms thus far used in solving equa-
tions. Illustrate the use of each. Why does the division axiom
not apply when the divisor is zero? [Cf. § 41 (v).]
HIGH SCH. ALG. — 5
00 HIGH SCHOOL ALGEBRA [Ch. V
27. Point out the fallacy in the following reasoning :
If X = a,
then x^ = ax,
and xF — a^ = ax — a', [subtracting a^ from
each member
i.e., (x + a)(x — a) = a (x — a) ;
therefore 2a(x — a)=a(x — a), [since x = a
and, therefore, 2 = 1. [dividing by a (x — a)
49. Translation of common language into algebraic lan-
guage, and vice versa. Tlie equation a^ — 8 = 3 is an algebraic
sentence ; it may be translated into common (verbal) lan-
guage thus : " X exceeds 8 by 3 " or ^^x is 3 greater than 8."
Similarly, the verbal statement " the excess of a over the
product of 8 and t is 9," when expressed algebraically, be-
comes a — st=9.
In order to use equations easily in the solution of prob-
lems we must learn to translate freely from either of these
two languages into the other.
EXERCISE XXX
Write as algebraic sentences :
1. Nine is 2 greater than x.
2. 2/ is 8 less than 3 x.
. 3. a^ exceeds 2 a by 1. •
4. The excess oi Sx over 6 a; is 2 a?.
5. The difference of two given numbers is five less than three
times their sum.
[Hint. Let a and b be the given numbers.]
6. The product of two given numbers exceeds half the larger
number by 17.
7. Twenty-one is divided into two parts, the smaller of which
is p. What is the larger part? Express by an equation that
the larger part exceeds the smaller by 3.
48-49] EQUATIONS AND PROBLEMS 61
8. Translate into verbal language the equations in Exs. 5-11,
p. 57. In how many different ways may we translate the equa-
tion in Ex. 8 ?
9. A father is now 4 times as old as his son. Eepresent
the age of each 5 years ago ; 5 years hence. Also express by an
equation the fact that 5 years ago the father's age was 7 times
that of his son.
[Hint. Let the son's present age be s years.]
10. Translate into algebraic language the following statement :
a rectangular flower bed whose length is y feet, and whose
width is 6 feet less than its length, contains 40 square feet.
11. If butter costs m cents a pound, eggs n cents a dozen, and
milk r cents a quart, express in algebraic language that
(1) the combined cost of 8 qt. of milk and 6 doz. eggs is $ 1.90.
(2) the cost of 9 qt. of milk is 30 cents less than the cost of
2i lb. of butter.
12. Express as common fractions : 50 % of n dollars ; 26 % of
k bushels ; m % of $ 525. Show that the amount of x dollars at
5 % simple interest for 3 years is cc -f- -^^ x.
13. If the units' digit of a number is 2, the tens' digit 4, the
number itself is 4 • 10 + 2, i.e., 42. What is the number whose
units' digit is 8 and whose tens' digit is 3 ? the number whose
tens' digit is x and whose units' digit is a; 4- 7 ?
14. The smallest of three consecutive integers is a ; what are
the other two ? If n is any integer, 2 n is an even integer ; write
the even integer next higher than «2 n ; next lower than 2 n.
Write the odd integer next lower than 2 n ; next higher than 2 n.
15. A walks 2i miles an hour ; B, 3 miles an hour. How far
does each walk in 3 hours ? in f hours ? How much farther
than A does B walk in 1 hour ? Express by an equation that in
t -\-2 hours B walks 3 miles farther than A.
16. At the rate given in Ex. 15, in how many hours will A
walk 10 miles ? 15 miles ? s miles ? Answer the same ques-
tions for B.
62 HIGH SCHOOL ALGEBRA [Ch. V
17. If I can do a certain piece of work in 6 days, what part of
it can I do in 1 day ? in 5 days ? in a? days ? If I can finish
a job in d days, what part can I finish in 1 day ? in 3 days ?
50. Problems leading to equations. A problem is a ques-
tion proposed for solution; it always asks to find one or
more numbers which at the beginning are unknown, and it
states certain relations (conditions) between these numbers,
by means of which their values may be determined.
In solving a problem the important steps are :
1. To represent one of the unknown numbers involved in the
problem by some letter^ as x.
2. To translate the common language of the problem into
algebraic language.
3. To solve the equation thus founds — called the equation of
the problem.
4. To check the result.
These steps are illustrated in the solutions of the follow-
ing problems.
Prob. 1. The sum of the ages of a father and son is 54 years,
and the father is 24 years older than the son. How old is each ?
Solution. Stated in verbal language, the given conditions are :
(1) The number of years in the father's age plus the number of
years in the son's age is 54.
(2) The number of years in the son's age plus 24 equals the
number of years in the father's age.
To translate these conditions into algebraic language,
let X stand for the number of years in the son's age ;
then, by the second condition,
0? + 24 stands for the number of years in the father's age,
and, by the first condition,
a? + cc + 24 = 54,
which is the equation of the problem.
Solving this equation, we find x — 15, whence a; + 24 = 39.
On substitution in the problem, these numbers are found to satisfy
49-50] EQUATIONS AND PliOBLEMS 63
its conditions (i. e., to check) ; therefore the father's and son's
ages are, respectively, 39 years and 15 years.
Prob. 2. A boy was given 39 cents with which to buy 3-cent
and 5-cent postage stamps, and was told to purchase 5 more of
the former than of the latter. How many of each kind should
he purchase ?
Solution. Stated in verbal language, the given conditions are:
(1) The total expenditure is 39 cents.
(2) There are to be 5 more 3-cent stamps than 5-cent stamps.
To translate these conditions into algebraic language,
let X stand for the number of 5-cent stamps purchased,
then 6x stands for the number of cents in their cost;
and, by the second condition,
a? -f 5 stands for the number of 3-cent stamps purchased,
and 3 07 4-15 stands for the number of cents in their cost;
hence, by the first condition,
5aj + 3a;-f 15 = 39,
which is the equation of the problem.
Solving this equation, we have cc = 3, whence a; + 5 = 8. Sub-
stitution in the problem shows that .these values check. Hence
the number of 5-cent stamps is 3, and the number of 3-cent
stamps is 8.
Prob. 3. If a certain number is diminished by 6, and twice
this diiference is added to 5 times the number, the result will
equal 88 minus 3 times the number. What is the number ?
Solution. To form the equation of the problem,
let n represent the number sought,
then 6 n = 5 times the number,
and 2 {n — 6) = twice the difference of this number and 6,
and 88 — 3 71 = 88 minus 3 times the number.
Hence the given condition becomes
5n-f 2(n-6)=88-3w.
The solution of the equation gives n = 10, which checks; there-
fore 10 is the required number.
64 nWU SCHOOL ALGEBRA [Ch. V
Prob. 4. A number consists of two digits whose sum is 5 ; if
the digits are interchanged, the number is diminished by 9.
What is the number ?
Solution. Let x represent the digit in the units' place ;
then, by the first condition,
5 — a? = the digit in the tens' place,
and 10 (p — x) -\- X =^ the number, [cf . Ex. 13, p. 61.
and 10 X + (5 — ic) = the number with its digits interchanged.
Hence, by the second condition,
1 a; 4- T) - a; = 1 (5 - a?) + a; - 9,
whence x — 2, and 5 — a? = 3.
These two digits are found to satisfy both conditions of the
problem ; therefore the number sought is 32.
EXERCISE XXXI
5. John has 14 cents less than Henry ; together they have
60 cents. How much money has each ?
6. Divide 28 into two parts whose difference is 4.
7. The sum of two numbers is 63, and the larger exceeds the
smaller by 17. What are the numbers ?
8. If 16 is added to a certain number, the result is the same
as it would be if 7 times the number were subtracted from 56.
What is the number ?
9. Of four given numbers each exceeds the one below it by
3, and the sum of these numbers is 58. Find the numbers.
10. Divide $2200 among A, B, and C in such away that B
shall have twice as much as A, and C $ 200 more than B.
11. I take a trip of 90 miles, partly by train, partly by trolley.
If I go 42 miles farther by train than by trolley, how far do I
go by each ?
12. Three boys together have 140 marbles. If the second has
twice as many as the first, but only half as many as the third,
how many marbles has each boy ?
60] EQUATIONS AND PROBLEMS 65
13. After taking 3 times a certain number from 11 times that
number, and then adding 12 to the remainder, the result is less
than 117 by 7 times the number. What is the number ?
14. I spend $ 2.50 for 3-cent and 4-cent stamps, getting 25 more of
the former than of the latter. How many of each kind do I buy ?
15. A man who is 32 years old has a son who is 8 years old.
How many years hence will the father be 3 times as old as his
son (cf . Ex. 9, p. 61) ?
16. The sum of two consecutive integers is 73. What are the
integers (cf. Ex. 14, p. 61) ?
17. Find three consecutive integers whose sum is 51. Show
that the sum of any three consecutive integers is 3 times the
second of these integers.
18. The difference between the squares of two consecutive
integers is 19. Find the integers.
19. Find two consecutive even integers whose sum is 98. •
20. Find the even integer whose square subtracted from that
of the next higher even integer leaves 52.
21. The accompanying diagram represents
the floor of a room. If the perimeter (the dis-
tance around it) is 5 times the width, how wide
is the floor ? how long ? How many square
yards in its area ?
22. The length and breadth of a rectangular floor differ by
5 ft. ; the perimeter is 60 ft. Find the dimensions and area of
this floor ; also make an accurate diagram of the floor.
23. If each side of a square lot were increased by 2 yd., the
area of the lot would be increased by 96 sq. yd. Find the side
of the given lot. Draw an appropriate diagram.
24. A certain rectangle is 5 ft. longer than it is wide ; if each
dimension were increased by 2 ft., the area would be increased by
38 sq. ft. Find the length, the breadth, and the area of this
rectangle.
(aH-6)feet
66 HIGH SCHOOL ALGEBRA [Ch. V
25. Five boys agreed to purchase a pleasure boat, but one of
them withdrew, and it was then found that each of the remaining
boys had to pay $ 2 more than would have been necessary under
the original plan. How much did the boat cost ?
26. A laborer was engaged to do a certain piece of work on
condition that he was to receive $2 for every day he worked, and
to forfeit 50 cents for every day he was idle ; at the end of 18
days he received $28.50. How many days did he work?
27. A number consists of two digits whose sum is 8; and if
36 is subtracted from this number, the order of its digits is
reversed. What is the number?
28. In a certain two-digit number the tens' digit is twice the
units' digit, and the number formed by interchanging the digits
equals the given number diminished by 18. What is the number ?
29. A two-digit number equals 7 times the sum of its digits ;
the tens' digit exceeds the units' digit by 3. Find the number.
30. What principal at 5 % interest yields an annual income of
$250 ? What principal at 4 % simple interest amounts in 5 years
to $2400 (cf. Ex. 12, p. 61)?
31. I make two equal investments, one at 6%, one at 4%. If
the difference in the annual income from the two is $80, find the
total sum invested.
32. Two trains which travel, respectively, 30 and 50 miles an
hour, start toward each other at the same time from two cities
240 miles apart. How long before they meet ?
Suggestion. Let x = the number of hours before they meet ; then
30x + 50x = 240 (cf. Ex. 15, p. 61).
33. Two bicyclists ride toward each other from towns 104 miles
apart, the first at the rate of 12 miles an hour, the second at the
rate of 14 miles an hour. If they start at the same time, how
long before they meet (cf. Ex. 15, p. 61) ?
34. Two bicyclists, A and B, whose rates are, respectively, 12
and 15 miles an hour, start from the same town and rid^ in the
same direction. If A starts 1| hours before B, how long before
B overtakes him ?
60] EQUATIONS AND PROBLEMS 67
35. A walks m miles at the rate of 3 miles an hour, returning
at the rate of 2i miles an hour. If the entire walk is made in
5^ hours, what is the value of m ?
Hint. The first half of the walk can be made in — hours, the second in
^ hours (cf. Ex. 16, p. 61).
5
36. A tourist climbs a certain mountain at an average rate of
2 miles an hour, and descends at an average rate of 3 miles an
hour. If the round trip takes 6 hours, how long is the path ?
37. The diiference of the radii of two circles is 4 inches ; the
sum of their circumferences is 88 inches. Find the radius of
each. (The circumference of a circle equals 27r times its radius ;
and 7r = 3|, approximately.)
38. Divide $351 among three persons in such a way that for
every dime the first receives, the second shall receive 25 cents,
and the third a dollar.
39. Divide 48 into two parts such that twice the larger part
equals 10 times the smaller part (cf. Ex. 34, p. 7).
40. Three times Harry's age equals 5 times the age of his
sister ; the sum of their ages is 24 years. How old is each ?
41. A, working alone, can do a certain piece of work in 3 days;
B, in 6 days. In how many days can they complete it, working
together (cf. Ex. 17, p. 62)?
Hint. Let x = the required number of days ; then - = -+-.
aj 3 o
42. Solve Prob. 41, if A can do the work in 8 days, B in 6 days ;
also, if A can do the work in 4|^ days, B in 4 days.
REVIEW EXERCISE-CHAPTERS J-V
1. Define : negative number, absolute value of a number, coef-
ficient, exponent, term, polynomial, degree of a term, finite num-
ber, equation, root of an equation, identity. Illustrate each of
your definitions;
2. Define and illustrate : inverse operations, multiplication,
division. Point out at least one advantage which the definition
of multiplication given in § 7 (ii) has over that in § 7 (i).
68 HIGH SCHOOL ALGEBRA [Ch. V
3. If distances above sea level are called positive, what would
— 25 feet mean? +35 feet? What is the difference between
these elevations ?
4. A walks east at the rate of 3 miles an hour, B at the rate
of —2 miles an hour. How long before the two are 15 miles
apart? Illustrate by a drawing.
5. Translate into algebraic language :
(1) The number formed by interchanging the digits of a cer-
tain two-place number exceeds the number itself by 18.
(2) A certain number diminished by 5 % of itself equals 76.
(3) The sum of two consecutive even integers equals half the
difference of their squares.
6. Point out at least one advantage in using letters to repre-
sent numbers.
7. How are two or more similar monomials added? State a
rule for subtracting one polynomial from another.
8. How may a parenthesis which incloses several terms, and
which is preceded by the minus sign, be removed without affect-
ing the value of the expression ? Why ?
9. State the law of signs for multiplication ; for division.
What powers of —3 between the 1st and 12th are negative in
sign ? Why ? Find the value of ( - 1)3 . ( - 10)^ -- ( - 5)4.
10. State the exponent law for multiplication ; for division.
By reference to these laws find the value of ^^"^'^' ' ^^"'^^^" .
If c = 9, d = -5, e = -2,/=-2,^ = l^ find the value of;
11. 5c + 3d-e4-/-c--/+2c.3^--e + 8/.
12. 4 c'f -i-6eg^-\-d^-\-5[e- 3/+ c'-3 ed].
13. -^i^llc-(12g'-6efg)^-\-(c'-^d^^{c + dy.
Perform the following indicated operations :
14. (x—5y){4:y — x). 17. (a^ — 1)1
15. (am — en) (3 a -f 5 c). 18. (m^ ~mn-{-7')(m^-{-mn — r).
16. (k-2qf. 19. (62- + c^)(&2-_c-).
50] REVIEW EXERCISE 69
20. (a^« + 2/^-i)(a:-l). 22. (x"^+* -5x''+'+6)--r(x'^+^-3).
21. (r2"' — r"*H-l) (3 + ?•'»-'). 23. {x^' -y^'^) ^ (y^ — x").
24. (1x2+ ia;-i) + (f a.'- 3 x^-r"')- (-2x^ + 1^-1).'
25. 4p - [p2 4. 22)r - (2 q +p') j + (1 -j^r + g).
26. 5m — Sn—\ — 7n-\-m — 5n — 3ml.
27. 2a;-a;-2/ + 22-[-S-(y + 42;)|].
28. a(6 + c — cZ) — 6(a — 2 c + d) — 3 c(— a — d).
If itf=7a6-3Z>2_4a2, ^=36^-4 a^- «?>, P=a-b, and
Q = ?,* _ 4 a^ft - 4 a6^ -h 6 a262 + a^ find the value of :
29. M-\-N. 33. J»f2. 37. Qh-P.
30. Jf-iV: 34. itfP. 38. Q^M.
31. |(4jlf+iV^). 35. MN. 39. 2Q-NP.
32. P^-5N. 36. ^-J-iJf. 40. Q--P'^ + 3P2.
41. Check your work in Exs. 32-37. What two checks may
be used for Ex. 30 ? for Ex. 36 ?
42. In Ex. 44 below, insert the second and third terms in a
parenthesis preceded by + ; place the fourth and fifth terms
under a vinculum preceded by — (see footnote, p. 37).
43. In Ex. 45 below, inclose the first three terms in a paren-
thesis preceded by — ; place the last two under a vinculum pre-
ceded by + .
In each of Exs. 44-49, collect the coefficients of r, s, and t
(cf. Exs. 24-26, p. 47).
44. 2 a^s — 3 ar — 6s — cs — 4 r.
45. -r-\-5fh + 2s+fh-lft.
46. -nH-{-5<^ds-2hH-'ieft-\0<?cls,
47. t -f aV + 6=^s H- r -i- s — 2 c? + 2 hs -{-cH — 2 ar.
48. (a H- 2 c)s -f (c + d)r + 2 cs — 5dr — 7 ds — er.
49. (m^ - l)s - 2(1 - ny + nV -3s- (m* - 2 mhi^)r + 91^^.
50. Multiply x^ -\- {a -{- c)x hj X — c -, hj x^ -j- ex -\- k.
70 HIGH SCHOOL ALGEBRA [Ch. V
51. Divide 2 a^ -f- 2 wV — n^x — mn^ — a^(2 m -f- 2 n) — wa^ —
2 mnic + mna;^ by —x^ + {m-{- n)x + mn (cf . Ex. 47, p. 51). Check
your result by multiplication.
Solve, and check the roots:
52. x{2 x-B)-51 = 2 x{x-\-l). 54. |a;-(|a; + i) = -a; + f
53. ^±2 = ?^±3^_2^. 55. |(.-2)(.-l)=^^-^^
5 3 2 4
56. Four times the number of seniors in a certain school
exceeds the number of freshmen by 15. If the total enrollment
in the two classes is 130, find the size of each class.
57. A certain number is subtracted from 50 and 42 in turn. If
\ of the first remainder equals ^ of the second, find the number.
58. If the population of a town has increased 30 % in the last
10 years, and is now 5200, find the population 10 years ago.
59. A man spends \ of his income for living expenses and
insurance, ^^ for books, -^^ for travel, -^^ for charities. If he
saves $425, what is his income ?
60. A certain rectangle is 8 ft. longer and 5 ft. narrower than
a given square, and its area exceeds that of the square by 5 sq. ft.
Find the side of the square. Draw an appropriate figure.
61. A and B together can do a certain piece of work in 6 days.
If A can do it alone in 10 days, in how many days can B do it ?
62. Silk marked to sell at a gain of 33|^ % has its marked price
reduced 20 %, and then sells for 80 cents a yard. Find its cost.
CHAPTER VI
TYPE FORMS IN MULTIPLICATION — FACTORING
I. SOME TYPE FORMS IN MULTIPLICATION
51. Type forms. Although all exercises in multiplication
and division of integral polynomials can be readily solved
by § 38 and § 39, yet there are a few special cases of these
operations which occur so frequently in practice that it is
well worth one's while to memorize them ; they are often
spoken of as type forms. Some of these type forms are con-
sidered in the next few paragraphs.
52. Square of a binomial. Let a and h represent any two
numbers whatever; then by actual multiplication (§ 33),
(« + ^») (a + 6) = ^2 4- 2 a6 + ^2,
and (a - 6) (a - 6) = a2 - 2 a6 + 52 .
i.e., (a+6)2 = a2 + 2a6 + 62, I
and (a_6)2 = a2-2a6 + 62, II
whatever the numbers represented by a and h.
Translated into common language, I becomes :
The square of the sum of any two numbers equals the square
of the first number^ plus twice the product of the two numbers^
plus the square of the second number.
The student may translate II into common language.
By means of I and II, we can now write down (without
actually performing the multiplication) the expanded form
of the square of any binomial whatever. Thus :
Ex. 1. (m + 3)2 = m^ + 6 m + 9.
Ex. 2. (x — yy = x^~2xy-\- ?/.
Ex. 3. (2 s - 3 ty = (2 sy - 2(2 s)(3 t) + (3 ty = 4 s2 - 12 st + 9 tK
71
72 HIGH SCHOOL ALGEBRA [Ch. VI
EXERCISE XXXII
Expand the following (check your work as teacher directs) :
4. {x^yf. 12. (1+2^)'. 20. (la -2)2.
5. {m + n)\ 13. {l-vf, 21. {\yJ^Qf.
6. Qi-^kf. 14. (2 a; -6)2. 22. (5a^-f)2.
7. (w + it;)l 15. (a. + 7ic)2. 23. (5m + |w)2.
8. (a-p)l 16. (3m^-2)2. 24. (2 a^x + 3 62/^)2.
9. (c-/i)l 17. (2g^-5hy. 25. (5 rs - 3 r^s^)^
10. (x + Sy. 18. (11-7A;)2. 26. (ic'» + r)2.
11. (a - 5)2. 19. (4 b^ -h 1)2. 27. (3 a" - 2 s^f.
28. Expand: (a -\- b - 5)% i.e., ](a -^ b) -5\^', (c + 2H-d)2;
(_2c-(Z + e«)2; (7-ha2-c)2; (3a.-«-p-5)2.
29. Since a — b = a^( — b), show that II, § 52, is included
under I.
30. What must be added to x^-^6x to make it the square of a;+3 ?
31. What must be added to a'* + a262 -f- b* to make it the square
oia'-i-b'?
32. What must be added to 25 — 10 a^ to make it the square of
33. What must be added to ic^ + 2 x*y^ + 4 / to make it the
square of a;'' + 2/?
By the method of § 52 write down the squares of the following
numbers :
34. 16, i.e., 10 + 6. 36. 28. 38. 71.
35. 19, i.e., 20 - 1. 37. 43. 39. 83.
40. Expand (a + 1)2 ; also (—a — 1)2. Compare and explain.
53. Product of sum and difference. If a and b represent
any two numbers whatever, then, by actual multiplication,
whatever the numbers represented by a and h.
The student may translate this formula into common lan-
guage (cf. § 52).
52-54] TYPE FORMS IN MULTIPLWAriON 73
EXERCISE XXXIII
Write the following products by inspection and check results :
1. ix^y){x-y). 12. (4+a«)(4_,,3),
2. (7n-\-n){m — 7i). 13. (2/"» — 11)(^'" + 11).
3. {^x + y){^x-y). 14. {ax'' + W){ax''-W).
4. {x-2y)(x + 2y). 15. \{x-y) + z\\{x-y)-z\.
5. (4a + 15Z/)(4a-15?>). 16. K« + &) + cn(« + &) -cj.
6. (G /) — 5 q) (6 ji) + 5 q). * 17. (m + n +p)(m + n — p).
7. (2a;2/— 7)(2a;?/ + 7). 18. (c - d + 5)(c — d — 5).
8. (4m2-3n)(4m2 + 3w). 19. j2-(a;4-2/)n2 +(^4-2/)S.
9. (9 + 5pV)(9 — 5pV). 20. (7 + m + rz)(7 — m — w).
10. (a^ + i/)(a^-i/). 21. {a-h^x){a-\-h-x).
11. (10mn-6)(10mn + 6). 22. (2 A: - Z + 3)(2 A: + Z-3).
23. (9lx''-4.f)-^(^x-2y) = ? Why?
24. (16a2-2562)-f-(4a + 56) = ? Why?
25. (a;^-2/')^(a;^-2/0 = ? Why?
26. (a;«-2/*)-(ar^-/)=? 27. (a;i« - ^/S) -^ (a^^ + 2/') = ^
28. Find, by the above method, the product of 22 by 18, i.e.,
of 20 + 2 by 20-2; of 17 by 23; of 42 by 38; of 56 by 44.
54. Product of binomials having a common term. By ac-
tual multiplication,
(2^+3)(:r+5) = a;2 + 8:r4-15 = rr2^(3 + 5)2:4-15;
and (a:+3)(a:-5) = a:2-2a;-15 = a;2 + (3-5>-15.
And, in general,
(jr + a)(jr + 6)= jr2 + (a + 6)jr + ab,
whatever the numbers represented by a, h, and x.
Translating this formula into words, it becomes:
The product of two binomials having a term in common equals
the square of the common term, plus the algebraic sum of the
unlike terms multiplied by the cornmon term, plus the product
of the unlike terms.
74 HIGH SCHOOL ALGEBRA [Ch. VI
EXERCISE XXXIV
Write down the following products (check as teacher directs) :
1. (a + 5)(a + 7). 16. {xy - ^)(xy + U).
2. (a-5)(a-7). 17. (-8-fmV)(2 + mV).
3. (a + 5)(a-7). 18. (s4- 7'2)(3s + r^).
4. (a_5)(a + 7). 19. \{l + m) -2\\{l^m) -^,
5. (2/-c)(2/ + 2c). 20. 5(Z + m) + 8n(^ + m)-15i.
6. (a^-f.4)(ar' + 5). 21. (m -n- 5)(m-n- 9).
7. (a^ + 4)(^2_5>)^ 22. (s-« + 4)(s-^-4).
8. (a^_4)(aj2_5)^ 23^ (,.p_ 10)(r^ + 15).
9. (a^_4)(aj2-|.5). 24. (a;^- + 3)(a^2- -7).
10. (3 4-m)(5 + m). 25. (3 a^-a;)(2.T + 3 a^).
11. (6 + a)(c + a). 26. (&^ + 2/)( -c + 2/).
12. (2a; + l)(-5 + 2aj). 27. \^{axY + 2\\^{a.xf -\-l\.
13. (a-&)(a-c). 28. (5 - 4 i«y ) (5 + a^y ) .
14. (4 + 3a)(-6+3a). 29. .(m2-c)(- 7 m^- c).
15. (4s2-5)(4s2-|-i). 30. (3j9-g-7)(3p-g + 7).
31. When 6 = a, what does the formula of § 54 become ? What
does it become when b =— a? Are the formulas of §§52 and
53 only special cases of (x -{- a) (x -\- b) = a^ + (a -\- b)x + a&?
55. Product of two binomials whose corresponding terms are
similar. By actual multiplication we obtain
^x -1y
15 x^ + 20 xy
— ^ xy —%y^
15x^-{-Uxy-Sy^
Here the term 14 xy is the algebraic sum of the " cross
products " 5x'4:y and — 2 y • S x.
With a little practice the final product of two such bi-
nomials may be written down by inspection, i.e.^ without
first writing the partial products.
54-66] TYPE FORMS IN MULTIPLICATION 76
EXERCISE XXXV
Write the following products by inspection :
1. (3a; + 2)(4a;-3). 4. (o- 11)(3 a- 1).
2. (5m-l)(2m-3). 5. (3x-^2 y){Ax-\-3y),
3. (2r + 5)(r-5). 6. (x-S y)(5 x-h 6y).
7. In each of the above products, how is the first term ob-
tained ? the third ? the second ?
8. What is meant by the expression " cross products " as used
in § 55 ? Illustrate from Ex. 3, above.
Write down the following products by inspection, and check
results as the teacher directs :
9. (7-2m)(7-m). 18. (ia-2)(|a+4).
10. (3 - 4 a) (4 -f 3 a). 19. {x-{- a){x-^b).
11. (9x — 2y)(x-\-y). 20. (3x+ c){x-^d).
12. (2a-4 62)(5a-6 62). 21. (3x-c)(x-d).
13. (7(y'-{-d'){3c^-\-Sd^. 22. (3 x-{- c){5 x + d).
14. (a"'-2e)(a'"-e). 23. (3 x-c){5 x-d).
15. (6ic^-f4)(3a;^-2). 24. (ky -\-l)(ny-l).
16. (11-7 cd^){6-{- 3 cd^). 25. (Jcy -^a){ny + b).
17. (ic + 2)(ic + l). 26. Qcy-a){l-cy).
56. The square of any polynomial. By actual multiplica-
tion it is found that
(a 4. 5 + c)2 = ^2 4- 52 + c2 + 2 a6 + 2 a^ + 2 he,
(^a-^b-{-c-\-dy = a'^-{-P + c'^-{-d^ + 2ah-\-2ac + 2ad
+ 2bc-{-2bd-h2cd,
(^a-]-b-^c+d+ey=a^-^h^-{-(P + d^-{-e^ + 2ab-^2ac-{-2ad
+ 2ae-{-2bc-{-2bd+2be-\-2cd-\-2ce-{-2de,
and so on for any polynomials whatever; that is, The square
of any 'polynomial whatever equals the sum of the squares of
all the terms of the poly)fiomial, plus twice the product of each
term by all the terms that follow it (for proof see § 209).
HIGH SCH. ALG. — 6
76 HIGH SCHOOL ALGEBRA [Ch. VI
EXERCISE XXXVI
Expand by inspection (check as teacher directs) :
1.
{c + d + ey.
12.
(4a3-&_5)2.
2.
(m + n — sf.
13.
(5-^x-fy,
3.
(a-b-cf.
14.
(-5-x + yy.
4.
(m + r + l)2.
15.
(a — b-\-c — dy.
5.
(^m-r-Sy.
16.
(ax-\-by-\-czy.
6.
(2x-\-y-^zy.
17.
(ia-ic + iey.
7.
(2x-j-3y-zy,
18.
(mn — np—pqy.
8.
{2x-3y-\-zy.
19.
(abx — acy — bczy.
9.
l-(2x + 3y^l)y.
20.
(2x-3y + 4.z-ay.
10.
(3c2 + d-4)l
21.
(x^-\-x + iy.
11.
(4.a'-\-b' + Sc'y.
22.
(/+m+w+p+^4-r+s)2.
23. Could any of the above products have been found by
means of formulas alrfea^dy used (cf. § 52, also Ex. 28, p. 72) ?
24. Give a rule for writing down the square of any polynomial
whatever. What does this rule become when the polynomial is
a binomial (cf . § 52) ?
57. Cube of a binomial. The cube of a binomial is another
product which, because of its frequent occurrence, should be
memorized. By actual multiplication we obtain
and (a-by = a^-3aH-{-Zab^-b^
whatever the numbers represented by a and b.
By means of these formulas (which the pupil should trans-
late into words) we may write by inspection the cube of
any binomial whatever.
Note. § 52 and § 57 are particular cases of what is known as the
binomial theorem; this theorem is considered in § 112.
Ex.1. (x-\-2y = a^-\-3 x''2-\-3x-2^-\-2^=x'-{-6x^-{-12x-\-S.
Ex. 2. (2 a-5 by = (2 a)«- 3 (2 ay • (5 b) +3 (2 a) • (5 by- (5 by
= 8 a«- 60 a'b + 150 ab' - 125 b'.
EXERCISE XXXVII
Expand the following
: expressions :
3. (x-^yf.
7. (c + l)«.
11.
4. (m-ty.
8. (a -3)3.
12.
5. (2x-yy.
9. (d' + c^y.
13.
6. (z-Syy.
10. (2yz-5y.
14.
56-68] FACTORING 77
(l + 2m)3.
(3a2-2 67^
(_5_v)3.
15. What is the difference in meaning between "the cube
of the sum of two numbers" and "the sum of the cubes of two
numbers " ? Illustrate, using 3 and 4 as the two numbers.
16. Give a rule for finding the cube of the difference of two
numbers.
17. Expand (c-\-dy, also (—c—dy. If we change the signs of
an expression, do we change the signs in its cube ? Why ?
II. FACTORING
58. Definitions. In a broad sense, any two or more num-
bers whose product is a given number are factors of that
number. Thus, since J -1^. 15=6, therefore ^, |^, and 15
are factors of 6; so also are ^, 18, and ^; the important
factors of 6 are, however, 2 and 3.
In order to exclude fractional and other unimportant fac-
tors, we shall (as it is customary to do) define factors thus:
The factors of a number or algebraic expression are its
rational* integral exact divisors.
JEJ.g.^ the factors of 3 a; (a^ — b^} are S, x, a-\- 6, and a — b,
as well as the product of any two or more of these.
Observe, too that if 3, a + 6, etc., are factors of any given
expression, then — 3, — (a-f-J), etc., also are factors of this
expression.
A factor (or expression) is said to be prime if it contains
no factors except itself and 1 ; otherwise, it is composite.
* An expression \\raUonal with regard to a particular letter if it contains
no indicated root of that letter (see § 113).
78 HIGH SCHOOL ALGEBRA [Ch. VI
By factoring a number (or expression) is usually meant
the procetis of separating it into its prime factors.
Factoring an expression, as will appear later, often greatly
simplifies algebraic work ; it is therefore important that the
pupil should early master those cases of factoring which
present themselves most frequently.
59. Factors of a monomial. The literal factors of a mono-
mial are evident by inspection, and the factors of the numer-
ical coefficient are found as in arithmetic.
E.g.y the factors of 30 a^x^y^ are 2, 3, 5, a, oi?, and t/^. (The ic-and
^/-factors are as evident in the forms ni? and y^ as from x-x- x and
yy-)
60. Monomial and polynomial factors of a polynomial. If a
polynomial contains a monomial factor, the latter is readily
discovered by mere inspection.
E.g., in 12aVH-4a6a^?/ — 8aicy, it is seen that each term contains
the factor 4 ax^, hence (see § 38) the other factor is Sax-\-by—2y^;
i. e., 12 a V + 4 abx^y — 8 ax^y^ = 4 aar^ (3 ax -{-by— 2 y^).
To factor a polynomial completely/ requires (1) the removal
of all monomial factors, and (2) the factoring of the poly-
nomial thus freed from its monomial factors. The simpler
cases of (2) are considered in the next few articles; (1)
may always be accomplished as above.
EXERCISE XXXVIII
Factor :
1. Qa'x^. 3. 42 s¥. 5. 408 mVt/l
2. WmjA^ 4. 210 2/V. 6. 572 a'(fwv\
7. The expression 5 a — 10 6 + 30 a^ has what monomial fac-
tor? what polynomial factor? How do you find the former?
the latter ?
Separate the following expressions into their monomial and
polynomial factors, and check your results :
58-61] FACTORING 79
8. 17a;2-51^. 14. 3 a' - 6 a'b + a'b\
9. 4 ar' — 6 x-y. 15. 7nIn'^-\- m^w^ + m^/i^.
10. 4a-^62-26a-6l 16. 3 r^ - 12 r^s^ + 6 rs^
11. 10 mhi^ — 15 m%\ 17. ac — bc — cd — abed.
• 12. -16x--2abx. 18. 32 .^ry - 28 a^y + 12 .ti/.
13. 15a;^-10ar^ + 25a^. 19. Uxyz^-2 afyh''-{-Sxyh\
20. 60 mhih^ — 45 m^nh'^ -\- 90 m*n^r^.
21. 12a;267/-18a.7y«6 + 24a;y?>*.
22. 14 a^mii^ — 21 aSnhi^ — 49 a*mn^.
23. 35 c^cZar^ + ^ c\? V - 55 c^d V.
24. 51i«?/V-68ar^/22_^85a;y^l
25. 52 a%V- 65 ^6=^6^ + 91 a2?>V.
26. Write (m + 7?.)^ — 3(m + 7i)^ + (m + n) as the product of two
factors, one of which is m + 71.
27. Write 2(3 0- - 1)2 - 5(3 a; -1) + 4(3 0^-1)3 as the product
of two factors; also 6 (2 -a)^ - 8 (2 - a)^ - 12 (2 - a)«; also
x\a - c)-(l-3x') (a-c)-(a- c).
28. If — 5 mhi^ is one factor of 10 m'^71^ — 15 mhi^, what is the
other (cf. Ex. 11) ? Factor again the expressions in Exs. 12-16,
in each case taking the monomial factor as negative.
61. Factoring by means of type forms. Expressions of the
type a^ + 2 a6 + 6^. Factoring being the inverse of multipli-
cation, it follows that to every case of multiplication there
corresponds a case of factoring. Ease in factoring, as in
every inverse process, depends upon a ready knowledge of
the corresponding direct process.
Thus, if we promptly recognize the form
a2 + 2 a5 + 52, [see § 52
then we can as promptly write down its factors, viz. :
a-\-h and a-{-h.
So, too, the factors of
^2 _ 2 a5 + 52
are a — h and a — h. [see § 52
80 HIGH SCHOOL ALGEBRA [Ch. VI
The expressions 6 mn + m^ -f 9 w^ and 4 a;^ + 25 — 20 rr be-
long to this type form, for, in each case, two terms of the
trinomial are the squares of certain numbers, and the third
term is twice the product of these numbers. These expres-
sions may, therefore, be written as (m + 3 9i)(m + 3 w) and
(2 2: — 6) (2 a? — 5), or as (m + 3 ^)2 and (2x — 5)^, respectively.
EXERCISE XXXIX
Factor the following expressions :
1. x'-'Zhx-^W. 8. a'V'-2ah-\-l.
2. u^ + 2uw + 'u^. 9. l-12?/ + 36i/2.
3. a^-6a; + 9. 10. aJ^-4a^ + 4.
4. 2/'-42/H-4. 11. 30a^ + 225 + a;i». *
5. l-\-2a + o?. 12. 9 a;2 - 12 a;2/2 + 4 2/V.
6. m^- 10 m + 25. 13. 6 a5cd + 9 c^d^ _^ a^ftl
7. 49 - 14 s 4- s'. 14. 4-36a^&2^81a^6l
15. What first suggests to you that a;^ + 9 ?/^ + 6 xy may be the
square of a binomial ? How do you test the correctness of this
supposition ? When is a trinomial the square of a binomial ?
16. Write out a carefully worded rule for factoring expressions
of the types ar + 2ab-\-W and a^-2ab + h'^'? How do we
find the terms of the binomial ? How do we determine the
sign by which they are to be connected ?
17. Is a* + 2 c^Jf — W the square of a binomial ? Explain.
Factor the following expressions, and check your work.
18. 5 a7? — 80 aa? + 320 a. [Remove monomial factor first.]
19. 7 n^ + l^ahri'^l o?h''n. 25. a^P-6cM" + 9 6^
20. 18a3t/-60a% + 50a6Y 26. ic^-ic + f
21. 27 cV- 36 0^^^7134-12 c^c^V. 27. l + f^s^ + fs.
22. — a^ + 2 0^2/ — 2/^. 28. m2p+2 + 2 mP+V+3 + 7i2'+« .
23. — m* H- 2 m^n^ — m^n^ 29. (a -f x)^ + 2(a + cc) + 1.
24. a:2n4.4ajy4.42/6. 30. i6_8(ic + 2/) + (» + 2//.
31. 9(m — n)2 — 6a7(m — n) + a;^
1.
/-^^.
2.
y'-^z\
3.
4 2/2 _ 49 62.
4.
2^a?W-lQ>.
5.
9 /-I.
6.
225 a;^- 9/.
(Jl-62] FACTORING 81
62. Expressions of the type a^ — 6^. From § 53 it follows
that the factors of aP' — b^ are a -\- b and a —b.
Again, the expression 25 n^ — 9 t^ is of the above type,
and its factors are 5^4-3^ and 5 w — 3 ^.
EXERCISE XL
Factor the following expressions :
7. a^x-h^x. 13. 49-36a^/.
a 36aV-81dl 14. m^^-n^-.
9. i»2n_4 3^5 49a^/-16
10. 121a^-36 6^ 16. 64a^/-81.
11. 64 0^2/'" -144 21 17. 2S^o^z^-f^z.
12. {x + yf-\. 18. 4.d?-{x-yy.
19, In factoring the difference of two squares {e.g., a^—b^),
how are the terms of each factor found ? How are these terms
connected in the first factor ? in the second ?
20. Write a rule for factoring the difference of two squares.
By rearranging and grouping terms factor the following ex-
pressions, and check your work :
21. b^-2bc-d^-{-c\ 28. - a^+ 6*-2 a^-l.
[i.e.,62_26c+c2-d!2,i.e.,(6-c)2-d2]. 29. - 18 A: + 81 + A;^ - 25 ^^^^
22. c^ + 2cd-{-d'-e'. 30. -9 ^^ + 49-12 wv -4^^^
23. x^-b'-2xy-^y\ 31. S cH^ - 4: -{- c* -\- 16 d\
24. x^ + 4:xy-4:z' + 4.yK^^^ 32. s^ -4.7^ -^f-2 st.
25. m2-6m + 9-p2. / 33. -524- ^2_4^_4 ^^^
26. l-s'-2st-t\ 34. 36c2-aV-36 + 12aa;.
27. 25-m24-2m7i-n2. 35. - 22 a;?/ + 121 - 2^ + aj^^/^.
36. Supply the required factor in each of the following:
a'-b' = (-a-{-b)'(?)', 16x'-9y' = (-4rX-Sy).( ? ).
May the factors in Ex. 4 be written ( — 5a6— 4)(— 5a6H-4)?
Explain (cf. §§ 18, 58).
82 HIGH SCHOOL ALGEBRA [Ch. VI
63. Expressions of the type x^+ mx-\- n. From § 54 it fol-
lows that the factors oi aP' -{- (^a -\- b) x -\- ab are
x-\- a and x -\- b.
Again, since the expression A;^ + 7 A; + 12 may be written
in the form A;^ + (3 + 4)A: + 3-4, therefore its factors are
A: + 3 and k + 4:.
So, too, j92 4-2jt?-15=jt?2+(5-3)jt? + 5.(-3)
= (^ + 5)(p-3).
From these illustrations we see that we can separate the
trinomial x^ + mx -\- n into two binomial factors whenever
we can separate n into two factors whose sum is m. Hence,
whenever such an expression as x'^ + mx + n can be factored,
its factors may be found by a few trials — the number of
trials never exceeding the number of pairs of factors of n.
EXERCISE XLI
I. If the expression cc^ + 5 aj — 36 is the product of two bino-
mial factors, what is the product of the unlike terms in these
two binomials ? Have these terms like or unlike signs ? Why?
What is the sum of these unlike terms ? Is the larger of them
positive or negative ? Why ?
Factor the following expressions and check the work :
ax — 90 al
nx^^i2x
[i.e., x(x2- 17 a; + 72)].
2x'-Qx + 4..
Uv^-S2v'\
8. r- 122? + 35. 18- ax'-i-Ta^x + ea^
9. m2 + 15m + 50. 19- 66 + 39/ + 3/.
10. Jc'-Sk-AO. 20. -a2-27 + 12a.
II. 'y2-7v-18. 21. a2&2_7a& + 10
12. f + 13 ^ - 30. • ^ U-e-, (aby - l{ah) + 10].
2.
x--^x + 2.
13.
0?
3.
x'-\-x-Q.
14.
v'
4.
:^-x-2.
15.
:^
5.
s2+_i2s4-36.
\i.
6.
f^^y^h.
16.
2i
7.
o2 + 7a-30.
17.
v'
63-64] FACTORING 83
22. 4ic2 + 4a;-3 30. ^-'"-24r + 63.
[i.e., (2a:)2 + 2(2 x) - 3]. 31. ^^2 _ ^^ ^^^ _^ 28 n\
23. 4a;2-8a;-21. 32. s'^-st-4t2f.
24. Oo.'^ + eaj-S. 33. -12a;2/2 + ar/ + 32 22.
25. 9x2-21a;-8. 34. (m + ri)^^- 7(m + n) + 6.
26. 9x' + 21x + U. 35. (s-A;)2-26(s-A:) + 69.
27. 16ar^-56ic + 33. 36. Q-y-y\
28. 15 + 32a;H-16a;2^ 37. r^ - {b - f)r - bf.
29. 25a:2_8_;l^()^ 33^ ic2 + (3a- 2 5)a;-6a&.
64. Expressions of the type kx^ + mx + n. Every trinomial
of this type which is the product of two binomials, may be
readily factored by an extension of the method of § 63.
For example, to factor 6 x^ — 11 x — Sb^ we proceed thus:
62:2_ii^_35^ 1 (36 2;2- 66 a; -210)
= i[(6^)2-ll(6.'c)-210]
= 1 (6 ^ - 2 1) (6 2; + 10) [§ 63
= (2x-l}(Sx+5).
The given expression is first multiplied by 6 so as to make
the first term an exact square, and the factor ^ is then
inserted so as to keep the value unchanged.
Note to the Teacher. The above method may, if the teacher prefers,
be replaced by the following ; in that case § 64 should follow § 67.
Let 6 x^ — 11 a; — 35 = (ax + b) (ex + d), wherein a, &, c and d are to
be determined ;
then 6 aj2 -11 X - 35 = acx^ + (ad + bc)x + bd,
whence — 11 = ad + be and 6 ( — 35) = ac . &d (i. e. , ad • be).
If, therefore, we separate 6 (—35), le., — 210, into two factors whose
sum is — 11, we shall then have found ad and be ; these factors are — 21 and
10, hence we have
6 a;2 - 11 X - 35 = 6 x2 + ( - 21 + 10) x -35
= 6x2-21x+10x-35
= 3x(2x-7)+5(2x-7)
= (2x-7)(3x + 5).
When this method is used with young pupils, special care will be needed to
keep the work from becoming merely mechanical.
84 HIGH SCHOOL ALGEBRA [Ch. VI
EXERCISE XLII
1. In factoring 3a^ + 13a;-fl4 by the method of § 64, what
multiplier should be used ? Why ? What divisor must then
be used ? Why ?
Factor the following expressions and check your work :
2. 3a^ + 13aj + 14. 19. 10-19a:4-6a^.
3. 6a2-lla + 4. 20. 56 x-\- 15 + 20 a^,
4. 322 + ^-10. 21. Sc^-10cd-3(^.
5. 4a^ + 16a; + 15. 22. -28 + 39s-8s2.
6. lOy^-lSy-S. ^ 23. - 30 i2_;i^9 ^.^5
7. 9a^ + 7x-2. 24. 12j92_ 28^ + 11.
8. 10ar^ + a;-2. 25. 16 ar' + 4 a^^^^ _ 3Q ^^4^
9. 12i»2_j_43._5^ 26. 4: at)'- 73 abc-\- IS ac\
[Multiply and divide by 3.] 27. — 14: y — 16+15 /.
10. lSs^-9s-5. 28. 15a^'* + 16a;"2/ + 4/.
[Multiply and divide by 2.] 29. 14 k^ — 27 A;^'' — 20.
11. 12m2 + 7m-10. 30. 3(a + &)' + 10(a + &) - 8.
12. 20m2-7m-6. 31. 5(c-dy -7(c-d) -6.
13. 2a^ + a-55. 32. 15 a.-^^ - a;^ - 28.
14. 8^2_^7^-18. 33. sx'+(7 s-t)x-7t.
15. 10/ + 7 2/ -12. 34. cz^+(fc-d)z-fd.
16. 8^2 + 14 71 — 15. 35. locP — Ikx — mx + km.
17. 6i)2-29p + 35. 36. 6ay^ + 2aby — Scy -be.
18. 10 62 + 37 6-12. 37. 90 a;^/^^ - 98 a^a??/^ + 8 a^a^y.
38. Is a product altered when two of its factors are changed
in sign? Explain (cf. § 18, also Ex. 36, p. 81). Change the
signs in each factor found for Ex. 2 above, and thus write the
factors of 3 x' + 13 a? + 14 in a new form. Similarly, in each of
Exs. 3-8 write the factors in a new form.
65. Squares of polynomials. Cubes of binomials. These
types may be recognized by comparing them with the for-
mulas of § 56 and § 57.
04-65] FACTORING 85
Thus, since the expression a^ + z^ — 4:2/z + 2xz-\-4^^ — 4ixy
consists of three square terms and three double products,
it may be the square of a trinomial. On rearranging its
terms thus : x^ -\- ^y'^-\-z'^ — 4:xy -{-^xz — -^yz, and compar-
ing with § 56, we see that the given expression is the square
oi x — 2y -\- z.
Again, the expression 12 am^ — 6 a^m — 8 m^ + «^, consisting,
as it does, of four terms, two of which are cubes, may be the
cube of a binomial ; further examination shows that it is the
cube oi a — 2 m.
EXERCISE XLIII
1. Is a^ — 2a6 + c^ + 26c — 2ac -\- h^ the square of a trino-
mial ? What suggests to you that it may be ? How do you find
the terms of this trinomial ? Which of them are alike in sign ?
Which unlike in sign ? Why ?
Factor, and check your results as the teacher directs :
2. m^ + w^ -h s^ + 2 mn — 2 ms — 2 ns.
3. 4: x^ -\- y^ -\- 2 yz -{- 4: xy + 2- + 4 xz.
4. ^v^-{-2kx + x'-Q>kv-&vx-^'k?,
5. 6ac + 8 6c + 9a- + c2 + 24a6 + 1662.
6. 4c2-f 9a2-12ac-fl6ftc-24a6-fl662.
7. l-\-2r — 2m + m^ — 2rm + r^,
8. 2lm — 2ln +/ — 2lp-^m- + l--2mn — 2mp + n^-\-2np.
9. If an expression {e.g., 3 pq- — (f -\- 2)^ — 3 p^q) is the cube of
a binomial, how do you find the terms of this binomial ? By
what sign do you connect them ? Illustrate.
Factor, and check by § 25 :
10. a^-Sa''y-{-3ay'--y\ 13. - y^ - 12 x'y -^ 6 xy^ -\- S x\
11. m^-f-{-3mf-37nH. 14. - 27 yh -\- 27 y^-z^ + 9 yz\
12. 3a*-f 1-f 3a2 + a«. 15. S - c^ -12 c' -{-6 c\
16. a;«-2a^ + 10a;2 + cc*-10aj3 + 25.
17. 216 - 108 s¥ -f 18 s¥ - sH\
18. 25-^6m^n-10n-\-9m*-30m^-\-nK
86
HIGH SCHOOL ALGEBRA
[Ch. VI
66. Factoring the type forms jr^— /" and x" +/". By actual
division we obtain the following results :
(^x^ — a^^ -^ (^x — a') = X -{■ a,
(ar^ — a") -7- (x — a^ = x"^ -^ ax -{- a^^
(^x^ — a^ ) ^ (x — (i) = x^ 4- ax^ + oP'x + a^,
(a^ — a^) ^{x — a) = x^-\-ax^ + a?x^ + a^x + aS etc.
I.
11.
III.
(^x^ — a^) -^ (x -{- a) = X — a,
(a;^ — a^) -^ {x + «) not exactly divisible,
{x"^ — rt^) -h (x -]- a) = x^ — ax^ + a^x — a^
(x^ — a^ ) -V- (^x -\- a) not exactly divisible, etc.
(^x^ + «2) _^ (2; — a) not exactly divisible,
(^x^ 4- rt!^) -^ (a; — a) not exactly divisible,
(a^* + a^) -r- (x— a) not exactly divisible, etc.
(x^ + a^) -7- (2: + «) not exactly divisible,
T V J (^^ + «^ ) ^ (a: + «) = a;2 — a:r 4- ^^
(2:* + a*) -f- (2; + a) not exactly divisible,
(^x* 4- a^) -=- (2: 4- «) = ^* — «a;3 4- oP'x^ — (^x 4- a*, etc.
These quotients illustrate the following principles (for
proofs see Exs. 17-19, p. 94):
(i) From I, x^ — a^ is always exactly/ divisible hy x — a;
the quotient terms are all positive.
(ii) From II, x^ — a^ is exactly divisible by x -\- a only ivhen
n is even; the quotient terms are alternately positive and
negative.
(iii) From III, x^-\-a^ is never exactly divisible by x— a.
(iv) From lY, x"-\-a^ is exactly divisible by x-\-a only
ivhen n is odd ; the quotient terms are alternately positive and
negative.
(v) The order of the letters and exponents is the same in all
the quotients ; the exponent of the first letter decreasing., and
that of the second increasing^ in passing toward the right.
66] FACTORING 87
EXERCISE XLIV
Write the following quotients by inspection and then verify
them by actual division :
1. :^— ^. 7. •" '^•'^ . 13. ■
x-\-y
2.
x^-
-r
X
-y
T^-
-f
X-
-y
a'-
-b'
a
-b
m^
-n«
m
+ rt
u'-
-v'
u-
— V
u'-
-v'
^jj-jf^ T, x^y
x + y
x^ 4- y^
x + y
[i.e., W^-OTh .
L x2 - y-i J
x'
-f
(x^r - (y^r
X2 - y^
^10
+ P
s^
+ ^^
^ao
-?/"
s'
-/
x''
_y2
x'
-t
x''
-f'
9. '£±^. 14.
m-\-s
4. •.^^. 10. ^^!±A'.
a-\-b
5. :^^. 11. (■^7' + 0/y. 16.
i^ + f
u-\-'\} (J? — & x? — y^
18. In Exs. 5-11, above, express the dividend as the product of
the quotient and the divisor.
19. Of which of the following binomials is r — s a factor :
,.8 _|_ ^8 . ^10 _ ^10 . ^.7 _ ^7 . ^.11 _|_ gU 9 Answer the same question for
the factor r + s.
Write each of the following as the product of two factors :
20. m^ — n^. 26. x^-^y*^, 32. l^p'^ — q'^,
21. d^' + e^ 27. r^ — s^. 33. 32.^'^4-l.
22. x'^-y'^. • 28. 2/^ + 8 34. 8-27r^.
23. F-Z^. [Le.,?/3 + (2)3]. 35^ Ho x' -'^l.
24. 2/3 +z3. 29. 0^3 + 27. 36_ 27v^-64m;«.
25. aio + &''. ^^' 8^'-l- 37. / + 32a.'io.
[Cf.Ex. 14.] 31.^^-32. 38. 64r-r^
39. Factor a^ — & in two ways : (1) by taking out tlie factor a — c,
(2) by using § 53 (cf. Ex. 12, above) and then refactoring the two
factors thus found. Which is the better plan to use when the
prime factors of a^— c^ are sought ? Show that this plan is advis-
able in general, e.g.^ with a^ — y^ and p^ — q-^.
88 HIGH SCHOOL ALGEBRA [Ch. VI
Resolve the following expressions into their prime factors :
40. x*-y\ 45. aiV^-2/i«. 50. a^-\-y^
41. a^-b^ 46. 64a^-l. 51. x^^-xy^.
42. a^-b^ 47. a«-81. 52. Sas^'-Saf^
43. m«-l. 48. SI a*b*- 16 xy. 53. 64ic« + 2/^
44. r^^—n^, 49. a;^ — 2/^ 54. /4-1-
67. Factoring by rearranging and grouping terms. A re-
arrangement and grouping of the terms of an expression will
often reveal a factor which could not be easily seen before.
E.g., ax— 3 by-{-bx — Say = ax-^bx — 3by — Say
= x(a-hb)-Sy(a-^b)
= (a-\-b)(x-Sy),
i.e., ax — 3by + bx — S ay = (a-\-b) (x — 3y) .
Again, a;(a; + 4) — y(j/ -f- 4) = a;^ + 4a; — / — 4?/
= xF - y'^ -\- 4:(x — y)
= (x-y)(x-{-y)-{-4:{x^y)
= (x-y){x + y-\-4:).
I.e., x(x + 4:)-y(y + 4.) = (a;-?/) (a; -h2/ + 4).
EXERCISE XLV
- Factor the following expressions and check your work :
1. cx — cy-{-Sx — Sy, 12. m^ — n^ — (m — ny.
2. ay-^1cx-\-ax-i-ky. 13. 'Sxy(x^-{-y)-^16(x^ + f).
3. p3_p2_^7^_7^ 14_ x^ _ xy^ — ax^ -{- ay\
4. p3_y_7^_^7^ 15, ab-\-bx'' — x''y'^ — ay'^.
5. ac + bd — ad— be. 16. a^ — 9 a;^ + 4 c^ — 4 ac
6. 9cy — 6cx-12mx-\-lSmy. [i.e., (a2-4ac+ 4c2)- 9 a;^].
7. aV-acd-a6c + 6d. 17. - Uk^ - A9 b^ -\- A k' -\- 121.
8. 7mr-3rs + 21ms-9s2. 18. ac^ + M^ - ad^ - fec^.
9. 5a^-«2 + 2-10a;. 19. 1 + ds - (c^ + c(^y.
10. 5a^-hl-x^-5x. 20. (a + 1)2 -(4 a + 3)2.
11. aa^ + l + a + ar\ 21. (p^ - q^ - (p^ - pqy.
00-68] FACTORING 89
22. a^x + ahx + ac 4- b^y + a6?/ + &c.
23. (ar' + 6a; + 9)2-(aj2 + 5a; + 6)^
24. a^-a2 + 2/=^-62 4.2a;2/-2a6.
25. h'-m^-\-10m + k^-25-2hk.
26. (a; + 2/)^ + 12(a; + 2/)-85 (cf- Exs. 34-35, p. 83).
27. x2 + 4a;^ + 4i/2-f.3a;H-62/ + 2 (cf. Ex. 26 above).
28. 4: x"^ + 10 X — 6 — 5 a — A ax -\- a\
29. Show that by changing the signs of two of them at a time
the factors in Ex. 10 may be written in three different forms
(cf. Ex. 38, p. 84). Is the same true in Ex. 18 ?
68. Factoring by means of other devices. It often happens
that the factors of an expression will become apparent by
adding a certain number to, and subtracting the same num-
ber from, the given expression ; this, of course, leaves the
value of the expression unchanged.
Ex. 1. Find the factors ofx^ + x^ + l.
Solution. If the second term in this expression were 2x^
instead of x^, then (§ 61) the expression could be written (x^ + 1)" ;
this suggests that x^ be both added and subtracted, which gives
x*-{-x^-{-l=x' + 2a^-{-l-x'
==(^^^l+x)(x^ + l-x), [§62
i.e., x* + x'i-l=y' + x + l)(a^-x + l).
EXERCISE XLVI
2. Find the factors of a* + a^b^ + b\
Suggestion. By the method of Ex. 1,
a4 _,. ^2^2 + 54 = (^4 _^ 2 a-^62 + 54 _ ^252
= (a2 + &2)2_(a5)2.
3. Find the factors of a^ - 4 a; - 32.
Suggestion. Here the first two terms, plus 4, form an exact square ; this
suggests the following arrangement :
x2_4x-32 = ic2-4x + 4-32-4
= (x-2)2-36.
90 EIGH SCHOOL ALGEBUA [Ch. VI
4. What must be added to ic^H-3a^+4 to make it an exact
square? What must then be subtracted to leave the value un-
changed ? Factor the given expression.
5. Can the sum of two squares be factored (cf . § 66)? Is a;^ -f 4
the sum of two squares ? Can it be factored ?
6. What must be added to a;^ -f- 4 to make it (a^ + 2)^? Is the
added term a square ? Factor x"^ + 4.
Factor :
7: p^^q^\ 16. 4a8-21a46^ + 96^
a x^-f-64 2/^ * 17. ^x^-lOxY + ^y^'
9. m^ + mhi^^-n\ 18. ^ a"" ^2Q> a'h'' + 25 h\
10. x''^a'x^ + a\ 19. o? + 2 ab -d''-2hd,
11. a^ + a;y-f/. 20. 4a^ + 81.
12. a^ + 6a;4-5. 21. a^/-h4aj/.
13. 9 s^ -f 30 si 4- 16 ^^ 22. m^-\-4:m7i\
14. a^6^ + «'&Vd2 + c4d^. 23. a' + Sa'-12S.
15. 9 a;^ + 8 a^2/^ + 4 2/^ 24. 5 na;^ — 70 wa;^ + 200 n.
25. Find the four factors oi x* i-y' + z^-2 a^/ - 2 a^2;2 _ 2 ^^V.
69. General plan for factoring a polynomial.
1. By inspection, find and remove all monomial factors.
2. By comparison with type forms, by rearrangement and
grouping of terms, or by some other device, separate the
resulting polynomial factor into two factors.
3. Then, if possible, separate each of these factors into
two others, and so continue until all factors are prime.
Note. By the above plan the simpler expressions can usually be factored.
For determining the binomial factors of longer polynomials, see § 71.
EXERCISE XLVII
Factor :
1. 4aa^-4a/. 4. s^ + 16 s^ -]- 15 s.
2. 49A;-fc3^ 5. x*-Sx^-^15x\
3. fp—p\ 6. mV + my.
68-69] FACTORING 91
7. tv - nw - nv -\- tw. 14. k* — 17 k^ -\-16.
. 8. v^ -7?;--2i; + 14. 15. Sx-^ — lOxy + 37f,
9. 1 - 24 .s -h 144 si 16. 49mV + 42wms-h9n2.
10. a''b'-4.abhj-^4:by. 17. c- + a;2-2 ex- 1.
11. p^ — Sr\ 18. &* + 6y + 2/^
12. 1-2/'. 19. 2u^-Uu^ + 70-107L
13. c2-5c-14. 20. 4.c'-25a^-\-b^-4:bc.
21. Give two methods for checking an exercise in factoring.
Illustrate, using Ex. 15 above.
Factoi: and check as the teacher directs :
22. 71^ — 1. 38. a^x^ — oc^,
23. q*p — t^*. 39. s^^ — t^^.
24. 216 + 2/3. 40. Tr'-.QOT.
25. a« + 4. 41. a^^ + l.
26. 3(a;-2/)^-27. 42. (a -6)^-03.
27. rsv^ — ar^st — 4: cr^ . 43. /c^^ + 4p^
28. x^-\-ax-ay-yx. 44. 12 4-s(^2_4) _3 ^2^
29. m(d2-3)+d2_3^ 45; x^''^^-{-2x''-'by + by,
30. m^" — 4: m^b -\- 4. b\ 46. m^ — 1 — 3 m(m — 1).
31. fc(Z2-4)-/2 + 4. 47. 3a2p-28rg-21ri94-4aV
32. mV + 4-5mV. 48. (5a + 2/)2-7(5 a +?/) + 10.
33. 7(a; + a)-ll(a;2-tt2). 49. 8 -12 mn + 6 mV-m%l
34. y^-y^L 50. m2 + 6m/i-16.T2/ + 9nl
35. -a;y-;2(3 2/-a;-3 2;). 51. (a? -1/)^- 2 2/ + 2 aj+l.
36. x^ — y^ — 3xy + 3a^y\ 52. m^n^ + 2 mVrV + mV W.
37. 6x'' + 12a^~lS. 53. (a2 + 5a4-4)2-(a2-5a-6)l
54. a;2-2x2/ + l+/ + 2(a;-2/).
55. (c-3)«-l-3(c-3)2 + 3(c-3).
56. m^~2mn-[-n^-s--{-2st-f.
57. {c'-2cd-\-dy-(Sc'-cd-2d'y.
58. «2 + 9/ + 2522_6iC2/-10^^ + 302/«.
59. 2(a262_aV-6V) + a^ + 6* + c*.
HIGH SCH. ALG. — 7
92 HIGH SCHOOL ALGEBRA [Cii. VI
60. a¥ ^ yiy. _ ^2g _^ ^2^ _|_ 52^2 _ ^2g^
61. a2_2a6 + 6'-2ac + 26c + c2-2ad + 26d + 2cd + d2.
62. iB2-9a;-|-14 = (aj-7).( ? ) = (7-a;).( ? ).
63. c3-r3=(c-r).( ? ) = (r-c).( ? ).
64. 7'3_36r = r(r+6).( ? ) = -r(r + 6).( ? ).
65. Write the four factors of a;* — 10 a^ + 9 in seven different
ways (cf. Ex. 29, p. 89).
70* Remainder theorem. In Ex. 53, p. 52, it was seen that
if oc^ -^ 3 X -\- 1 is divided by a; — a the remainder is a^ + 3 a 4- 1 ;
i.e., the remainder is what the dividend would become if a were sub-
stituted for X. (Cf. also Ex. 52, p. 52.)
And this relation between dividend and remainder is not acci-
dental ; it is true for all such expressions. For, let
Ax"" + Bx""-^ + Cx""-^ H [-Hx -\-K •
be any polynomial in x, let it be divided by x — a, and let Q and
Rj respectively, represent the quotient and remainder; then
Ax^ + Baf-^ + Cx^-'-^ ...j^Hx + K= Q(x-a) + R.
Moreover, since the second member of this equation, when
multiplied out, must be exactly like the first, therefore this equar
tion is true for all values that may be assigned to x; but if the
value a be given to x, the equation becomes
Aa'^ + 5a"-i + Ca^-^ + - -- + Ha -\- K=: R,f
hence, in every such division, the remainder may be obtained by
simply substituting a for x in the dividend.
71.* Application of the remainder theorem to factoring. By
means of the remainder theorem (S 70), and without actually
performing the division, write down the remainder resulting
from dividing a^— 3a^ + 3a;-f2 by a; — a. Also write the re-
mainder when aj^ — 3a^-f-3a; — 2 is divided by a; — 2. What is
* Articles 70 and 71, with Exercise XLVIIJ, may, if the teacher prefers,
be omitted till the subject is reviewed.
t Since, in that case, Q(x — a) becomes Q'(a — a), i.e., zero; and M is
the same as before substituting, since it does not contain x.
60-71] FACTORING 93
the value of this last remainder? Does this show that a;— 2 is
a factor of y? -'d x^ + Sx-2'!
Binomial factors of many polynomials may be found in this
way, for, from § 70, it follows that if
^a" + BoT-"^ + (7a"-2 -\ +Ha + K^ 0,
then, and then only, is Ax"" + Bx""-^ + Ca;"-^ -\ h ^a; + /f ex-
actly divisible by ic — a ; for in that case, and in that case only,
is the remainder zero.
Thus, we know that a? — 3 is a factor of y? — 2y? — ^:X -\-Z
because 3^ — 2-3-— 4.34-3 = 0; and a;-f-l, ?.e., a?— (— 1), is a
factor of a;2 + 7 a; -f 6 because (- 1)2 ^_ 7(_ i) + 6 = 0.
Again, if a; — a is a factor of a^ — a;^ — 2 a; + 8, then a is a factor
of 8 ; hence, in seeking such factors of a?^ — a;^ — 2 a; -f 8 we need
try only 1, — 1, 2, — 2, 4, — 4, 8, and — 8 in place of a.
When, by any process whatever, any factor of an expression
has been discovered, this factor may be divided out; the remain-
ing factors may then be more easily found.
EXERCISE XLVIII
1. If a;^-f 6a^— 12 a; + 5 is divided by x — a, what is the
remainder ? Without performing the division, find the re-
mainder when the divisor is x — 2; also when it is a; — 1 and
when it is x-\-l. Which of these divisors is a factor of the
given expression ?
2. If the expression x^ — 3a^ — a;-f-3 has a factor of the form
X— a, what are the four possible values of a ? Find all the
binomial factors ofa;^ — 3ar^ — a;-f3.
By the above method, factor the following expressions :
3. a^_7a;-f6. 7. T<? -\-4.k^ -Ilk -30.
4. a;3-9a^4-23a;-15. 8. w^ - 15 w;^ + 10 lo -f 24.
5. a.-3 + 14ar-f-35a;-f-22. 9. a^-\-l a" -\-2 a- ^0.
6. a.-3-lla;^-f-31a^-21. 10. c^-S c^- 29 c + 105.
11. .^•* - a.-*^ - 7 a^ + a; 4- 6.
12. /-102/' + 402/^-80/-f 80y-32.
94 HIGH SCHOOL ALGEBRA [Ch. VI
13. If X — A: is a factor of any given expression, what does the
value of that expression become when x = k ^ Why ? If any
given expression becomes zero when x = k, is x — k a factor of
the expression ? Why ?
14. By means of the remainder theorem show that a—b, b — c,
and G — a are factors of a(b^ — c'-^) + b(c^ — a^ + c{a^ — b^).
15. Write the remainder when (2x — Saf + (Sx — ay is divided
by ic — a ; also the remainder when (x— y -\-zy — 'if-\-x^ is divided
by a; — 2/ ; and hj x-{-y, that is, hj x—{ — y).
16. What is the remainder when or"*— ctMs divided hjx — a?
Why? Write the remainder when x'' -\- a^ is divided by a; — a;
when ic^ + a^ is divided by x -\- a.
17. By means of the remainder theorem, show that x^ — a" is
exactly divisible by a; — a ; also that x'' + a" is not exactly divis-
ible by £c — a (cf. § Q^Q).
18. By means of the remainder theorem, show that x"" — a"* is
exactly divisible by x + a only when n is an even positive integer.
19. By means of the remainder theorem, show that x"^ + a"* is
exactly divisible hy x-\-a only when n is an odd positive integer.
72. Solving equations by factoring. Factoring greatly
simplifies the solution of certain kinds of equations. The
following examples illustrate the procedure.
Ex. 1. Given ic^ — 5a; + 6 = 0; to find its roots, i.e., to find
those values of x for which this equation is satisfied (cf. § 45).
Solution. By § 63 the first member of this equation is the
product of ic — 3 and a; — 2 ; hence the equation may be written
thus:
(a;-2)(a;-3)=0.
Now this equation is satisfied if either
x-2 = or a?-3 = 0, [§ 41
i.e., if either a? = 2 or a; = 3.
On substitution these values are found to check; they are,
therefore, the roots of the given equation.
71-72] FACTORING 95
Ex. 2. Given oc^ = 8 x -\- 4: ; to find its roots.
Solution. On transposing, tliis equation becomes
x^-3x~4:={),
i.e., (x_4)(a; + l)=0; [§63
hence either ic — 4 = or a^ + 1 = 0,
i.e., x = 4: or£c = — 1;
and these numbers check, therefore the roots are 4 and — 1.
Ex. 3. Solve the equation 6 a?^ — 11 cc = 35.
Solution. On transposing and factoring (§ 64), this equation
becomes
(3x+5)(;2x-7) = 0',
hence 305 + 5 = or 2a; — 7 = 0; .
therefore the roots are — | and J.
Remark. Since the roots of the equation (x — a)(x—b)=:0
are a and b, therefore an equation which shall have any given
numbers as roots may be immediately written down; thus the
equation whose roots are 3 and 8 is
(x-3){x-S) = 0, I.e., a^- 11 a; + 24 = 0.
Similarly, the equation whose roots are 2, — 1, and 5 is
(x-2)(x-\-l)(x-5) = 0, i.e.,a^-(jx'-{-Sx-\-10 = 0.
EXERCISE XLIX
4. What is meant by a root of an equation (cf . § 45) ? May
an equation have more than one root ?
5. What values of x satisfy the equation (x—2)(x—3')=0?
Can any values of x other than 2 or 3 sat^* sfy this equation ?
Explain. How many roots, then, has this equation ?
Solve the following equations by factoring, and check the roots :
6. 2/--62/ + 5 = 0. 11. ^2-4 = 0.
7. x^-4:x-21 = 0. 12. m2-36 = 0.
8. ^-13s-\-A0 = 0. 13. /-7?/ = 0.
9. a;2-2a; = 15. 14. c2 + 22c = -121.
10. A;2 + 47(; = 45. 15. '^^-3^-50 = 38.
96 HIGH SCHOOL ALGEBRA [Cii. VI
16. 32/- + ?/-, 10 = 0. 22. a;--3aa;-54a2 = 0.
17. 6ar^ — a;=l. 23. s^ — (c + d)s -f cd = 0.
18. 4y2_27 = 12'y. 24. 8a^ + 10.T = 3.
19. 82/2 + 15 = -26y. 25. 36 = -x^+13a:2^
20. 5if2— 7a; = 0. 26. ^-\-x~ — x = l.
21. 1222 = _42. 27. 2a^ + 5.^ = 2a; + 5.
28. What are the roots of {x -l){x -2){x -^2) = 0? Explain.
Determine by inspection the roots of (a;+l)(3 a;— 2) = 0.
29. Determine by inspection the roots of :
(1) (5a;-3)(a;-l)=0.
(2) (2/-f)(22/ + 9) = 0.
(3) m(3 m + 1)(4 m - 3) = 0.
(4) («-a)(2aj-lla)(4a; + 5a) = 0.
30. Write an equation whose roots are 5 and 2. Also one
whose roots are 3, 1, and 7.
31. Write the equations whose roots are : 1 and — 5 ; f and 6 ;
a and 65 3, — 1, and o ; a, —a, and 2a; 1, 2, 3, and 4.
The following problems lead to equations whose roots may be
found by factoring. Solve and check each problem.
32. Find a number such that if 3 and 5 are subtracted from it
in turn, the product of the two remainders is 24. How many
solutions has this problem ? Explain.
33. The sum of two numbers is 12, and the square of the
larger is 1 less than 10 times the smaller. Find the numbers (cf.
Ex. 18, p. 6). •
34. Tlie difference between two numbers is 2, and the sum of
their squares is 130. What are these numbers ?
35. One side of a rectangle is 3 feet longer than the other. If
the longer side be diminished by 1 foot and the shorter side in-
creased by 1 foot, the area of the rectangle will then be 30 square
feet. How long is this rectangle ?
Note. The equation of this problem has two roots, one positive and one
negative ; but only the positive root will satisfy the problem itself, for it is
implied that the dimensions of the rectangle are positive.
72] . FACTORING 97
36. How may $ 128 be divided equally among a certain num-
ber of persons so that the number of dollars received by each
person shall exceed the number of persons by 8 ?
37. The senior class of a certain school present the school with
a picture whose cost is $12. If each senior contributes 3 times
as many cents as there are members in the class, how large is the
class ? How much does each member pay ?
38. A rectangular orchard contains 2800 trees, and the number
of trees in a row is 10 less than twice the number of rows. How
many trees are there in a row ?
39. If the dimensions of a certain rectangular box which con-
tains 120 cubic inches were increased by 2, 3, and 4 inches,
respectively, the new box would be cubical in form. Find the
dimensions of this box (cf. § 71).
CHAPTER VII
HIGHEST COMMON FACTOR — LOWEST COMMON MULTIPLE
I. HIGHEST COMMON FACTOR
73. Definitions. A factor of each of two or more algebraic
expressions (or numbers) is called a common factor of these
expressions. The highest common factor (H. C. F.) of two
or more expressions is the product of all the prime factors
(§ 58) that are common to these expressions; it is, therefore,
the factor of highest degree common to the given expressions.
Thus, the H. C. F. of 9 a%V and 6 abh"^ is 3 ah\ because
when this factor is removed from the given expressions they
have no common factor left.
So, too, 2x(^a- 1)2 is the H. C. F. of 6 aV (a - 1)* and
Sx(a-iy(is-ty,
Two or more algebraic expressions which have no common
factor except unity are said to be prime to each other.
74. Highest common factor of two or more monomials. The
H. C. F. of two or more monomials may, obviously, always
be found by inspection.
E.g., to find the H. C. F. of 12 a^b^xy, 6 a6V, and 9 ab^x\
Inspection shows that 3, a, b^, and x are the only factors com-
mon to the given monomials; hence the H. C. F. of these mo-
nomials is 3 ab^x.
A rule for writing down the H .C. F. of several monomial
expressions may be formulated thus : To the H. C. F. of the
numerical coefficients annex those letters that are common to
the given monomials, and give to each of these letters the lowest
exponent which it has in any of the mono^nials.
98
73-75] HIGHEST COMMON FACTOR 99
75. H. C. F. of polynomials whose factors are known. By
first writing any given polynomials in their factored forms
their H. C. F. may be found by inspection.
For example, to find the H. C. F. of 4 ax^ — 20 ax -\- 24: a and
6 abx^ + 24 abx — 126 ab, we write :
4:ax'-20ax-\-24:a = 4.a{x-2)(x-3), [§ 63
and 6 aba^ + 24:abx-126 ab = ^ ab(x-\-7) (x-S)-,
hence their H. C. F. is 2 a (x—3).
EXERCISE L
Find the H. C. F. of each of the following sets of expressions :
1. 3 a'b' Sind 6 ab\
2. 15 afy% 24 xY, and 18 xSj.
3. 16 x'fz', 52 yh% and'39 x'f-
4. 195 a*6V and 260 a'bc\
5. 96 ?/V, 100 ?/V, and 56 t/V.
6. 104 x'^y^^'z^' and 364 x^'^f^'z^'.
7. (c + d)\c - d) and (c + d)(c - d)\
a 6(c + d)2(c-d)2and 15(c-(^)2(c+c?).
9. 24 a^x (y - zf and 56 a'^b^ (y - z)*.
10. a^ — b^, a(a — b), and a^ — 2ab-\- 6^
11. a;2_^7a; + 10and a.-2 + 12x + 20.
12. 'm? —m— 12 and di? — 4 m — 21.
13. 15 (2/2; — z) and 35 (/s; — yz).
14. Is — (tt — b), I.e., 6 — a, a common factor of the expressions
in Ex. 10 (cf. Ex. 36, p. 81)? May we then call the H. C. F.
of these expressions either a — 6 or 6 — a?
15. Show that the H. C. F. of rn? — mn and n^ — mn is either
m — 71 or 71 — m.
In each of Exs. 16-25 find two forms of the H. C. F. :
16. 1^ — s^ and ^ — rl
17. 5 a — as and 3 s' — 75.
18. p^ - 125 and p^ - 10 p^ + 25 p.
100 HIGH SCHOOL ALGEBRA [Ch. VII
19. a^ 4- a^ and 3 a^ + 3 a^x — 5 aoiy^ — 5 a.*^.
20. 28^2-17?! -3 and 4n2 + 5/i-6.
21. 5-19 A;-4A;2 and A;2^2A;-15.
22. a^a; — x — y-\- a^y and a^a? + 4 a^a; — 5x.
23. 12 a6^a; + 4 a6^a;^ — 40 aft^, 18 a^mx^— 54 a%a; + 36 a^m, and
6 aV?/ — 6 a^xy — 12 a^y.
24. ?^v — w^, u^— 5v -\-5u — uVj and 3 w^ — 10 iii; + 7 v^.
25. 15 aV 4- 15 a'6V + 15 ft^aj^ and 3 {a' - ab^ + ^>')-
Find the H. C. F. of each of the following sets of expressions:
26. 2a^-a;-3 and2a^ + llar^-a;-30.
Suggestion. Find the factors of 2 x^ — x — 3 and determine by trial which
of these are factors of 2 cc^ + 11 a;^ _ ^ — 30 also. This plan may be used
whenever any one of a given set of expressions is easily factored.
27. (a;+3)(aj2-4) anda;^-h4a^ + 2a.'2-aj + 6.
28. a^ + 1, Sa^-4.a^-\-4.a-l, and 2 a^ + a^- ^ + 3.
29. b^-S, b'^b' + 2b-4., ^ndb'-\-2b'-b'-10b-20.
30. Of what is the H. C. F. of two or more expressions com-
posed ? State a rule for finding the H. C. F. of two or more ex-
pressions which may easily be separated into their prime factors.
31. Is the H. C. F. as above defined the same as the greatest
common divisor (G. C. D.) in the arithmetical sense ? What is
the H. C. F. of a^(a; - 1)^ and xia^-l)? Is this also the G. C. D.
of these expressions for all values of x? Try a; = 3, also a; = 4.
76.^ H. C. F. of polynomials neither of which is easily factored.
The H.C.F. of two or more polynomials can always be found by
what is known as the Euclidean (division) process. This process
is essentially the same as that used in arithmetic to find the
G. C. D. of two numbers.
The steps in the arithmetical process are : (1) Divide the larger
number by the smaller ; (2) if there is a remainder, divide the
smaller number [I'.e., the divisor in step (1)] by this remainder;
* Articles 76, 77, and 78, with Exercises LI and LIT, may, if the teacher
prefers, be omitted till' the subject is reviewed.
:(J]
HIGHEST COMMON FACTOR
101
(3) divide the remainder in (1) by the' remainder in (2); (4) so
continue, dividing each remainder by the one following, until
there is no remainder; (5) the last divisor is the G.C.D. sought.
This work may be more compactly
arranged thus :
2639
2866
Thus, to find the G. C. D.
of 1183 and 2639.
1183)2639(2
2366
273)1183(4
1092
91)273(3
273
The last divisor, 91, is the G. C. D. of the given numbers.
Similarly, the H. C. F. of a:* + 3 a;^ + 2 ic-^ - x - 5 and x^ + x^ - 2 may be
found thus :
1183
1092
91
QUOTIENTS
273
273
QUOTIENTS
x* + 3 x3 + 2 x"^
x*+ x3
X
2x
2 x3 + 2 x2 + X
2 x3 + 2 x2
X- 1
x + 2
a;3 4x2-2
X3-X2
2x2-2
2 x2 - 2 X
x2 + 2 X + 2
2x -2
2x -2
Hence x — 1, the last divisor, is the H. C. F. of the given polynomials.
EXERCISE LI
By the above method, find the H. C. F. of each of the following
pairs of expressions :
1. a^ + 5i»4-6and4a^ + 21a;2 + 30a; + 8.
2. 6a2-13a-5andl8a3-51a2 + 13a + 5.
3. 5 m''' — 2 m — 3 and 15 m^ — 6 m^ — 4 m -f- 3.
4. c3_2c2-2c-3 andc^-c3-3c2^4c-2.
5. 12x'-Sa^-55a^-2x-^5and6a^-x'-29x-15.
6. lSx^-\-75a^-{-17x'-2Sx-lSand6a^-}-2Sx^-3x-10.
7. S0y' + 16y* + 16f-Sy^-3y-2a,nd20f + 4:y'-y-3.
8. 4.k'-\-201(^-10k'-^SJc-\-S5sind21i^ + llk'-25,
9. 5n*-10n3 + lln2-67i-hl and
10
5n*-7n^ + ldn'-Un-{'2.
102
HIGH SCHOOL ALGEBHA
[Ch. VII
11* Proof of principle involved in § 76 (see footnote, p. 100).
The success of the method employed in § 70 is due to the follow-
ing considerations :
Let A and B represent any two polynomials in ic, the degree of
A being at least as high as that of B, and let q and R represent
the quotient and remainder respectively, when A is divided by B\
then A = qB^R. [Ex. 20, p. 50
This equation shows that: (1) every divisor common to B and
II is a divisor of A also (why ?), and (2) every divisor common to
A and ^ is a divisor of H also (why?); hence the H. C.F. of B
and R is the same as that of A and B.
If now B is divided by R, giving p and M as quotient and
remainder respectively, then, by reasoning as above, we see that
the H. C. F. of M and 7^ is the same as that of B and R, and
therefore the same as that of A and B.
Suppose now that this series of divisions is continued ; then, by
the above reasoning, the H. C. F of ^ and B is the same as that of
any ttvo successive remainders.
If now the last one of this series of divisions is exact, i.e., if the
final remainder is zero, then the H. C. F. of the two preceding
remainders is the last divisor itself ; hence the last divisor is the
H.C.F. of A and B, which was to be found.
Remark, The H. C. F. of two expressions is evidently not
altered by multiplying (or dividing) either of them by any num-
ber which is not a factor of the other ; this fact enables us to
avoid fractional coefficients in the division process.
Thus, to find the H. C. F. of 3 x^ -|- 8 x'^ -h 3 x - 2 and a;^ - 2 a;^ + x + 4 :
3x3 + 8x2 + 3x- 2
3x3-6x2 4- 3x + 12
14)14x2-14
x2-l
x2-f-X
-x-1
-x-1
x-2
x=^ - 2 x2 4- X -f- 4
- 2 x2 -}- 2 X + 4
- 2 x2 -H 2
2)2x4-
X4-1
Before beginning
the second division
the factor 14 is sup-
pressed (see Remark
above), and later 2
is suppressed also ;
fractional coeffi-
cients are thus
avoided.
Hence x 4- 1, the last divisor, is the H. C. F. of the given expressions.
77-78]
HIGHEST COMMON FACTOR
103
As a further illustration, let us find the 11. C. F. of
x4 + 4 x^ + 2 x2 - X 4- 6 and 2 x^ + 9 x^ + 7 x
6.
xH4x3 + 2x2-x+6
2
X, +1
2x-l
2x3+ 9^2_^ 7a^_6 i
2x3+10x2+12x
2xH8xa + 4x2-2x+12
2x4+9x3 + 7 x2-6x
-x-i- 5x-6
-x2- 5x-6
- x3-3x2+4x+12
-2
2x3+6x2- 8x-24
2x3+9x2+ 7x- 6
_3)_3x2-15x-18
x2+ 5x+ 6
Before beginning
the division the fac-
tor 2 is introduced
so as to avoid frac-
tional coefficients in
the quotient ; later
— 2 is introduced
for the same pur-
pose ; and finally
— 3 is rejected.
Hence x2 + 5 x + 0, the last divisor, is the H. C. F. of the given expressions.
78.* Supplementary to § 77 (see footnote, p. 100). (i) If the
polynomials whose H. C. F. is sought contain monomial factors,
these should be set aside before the division process is begun.
Monomial factors that are common to the given polynomials must,
of course, be reserved ; all others may be rejected.
(ii) The H. C. F. of three or more polynomials is found by first
finding the H. C. F. of any two of them, then the H. C. F. of that
result and the third polynomial, and so on until all the poly-
nomials have been used.
(iii) To find whether polynomials which involve more than one
letter have a common factor containing any particular one of these
letters, they need only be arranged according to powers of that
letter, and divided as already described. By a repetition of this
process all the common factors of such polynomials, and hence,
their H. C. F., may be found.
[For fuller discussion of H. C. F. see El. Alg. pp. 116-121.]
EXERCISE Lll
Find, by the Euclidean method, the H. C. F. of :
1. ar^-3a;2^3^_j^ andaj4-2ar^+-2a;2-2ic + l.
2. c^ + 4c3-12c2-f c + Gaud c^- 0^-2 C2+-C + 1.
3. 5z^ + lSz^-3Sz-\-10sind2z^-{-5z^-22z-\-W.
4. 2ii^-{-Sx'-\-2x-2sindx' + 2a^-{-x'-2x-2.
5. r^-2r^ + 2r2-4andr^+-2r^-7'3-2.
104 HIGH SCHOOL ALGEBRA [Ch. Vll
6. 1 — 4 m^ 4- 3 m'' and 1 — 5 m^ + 4 m'* + m — m^.
7. 63 + A:^ - 9 A; - 7 A;2 and 40 A; + A:^ - 5 Ar^ + 111 - 23 fe2^
8. ar^- 4 a;3_ 2 a^ _ 8 H- a;4 and 2 a^+ 9 ic^-a^- 4 ar^ + 14 a; -16.
9. 8a^-22a^ + 17a;-3and6x=^-17a^ + 14aj-3.
10. 2a;2-3a;-35anda;4 + 14a!--9cc3 + 35a;-25.
11. What is meant by the H. C. F. of two expressions A and B ?
If a is not a factor of A, how does the H. C. F. of A and a • B
compare with the H. C. F. of ^ and B ? Explain.
12. If a is a factor of A, but not of B, how does the H. C. F.
of A and a • B compare with the H. C. F. of ^ and B? In intro-
ducing and suppressing factors during the process of division
(§ 77), what precaution must be exercised, and why ?
Find the H. C. F. of the following expressions :
13. m* — 3 m^ + 1 and m^ — 2 m - 2 — m*^ — m^ + 2 m^.
14. a' + 2a'-5a'-10anda'-{-a^-a'-2a-2.
15. a^-4x'-2-\-3x-3i^-{-Ba^Siudx-\-2s(^-{-2-5x'.
16. s^-2s^-2s3-lls=^-s-15 and
2s^-7s^ + 4s3-15s2 + s-10.
17. a;^ + 3 ar^ - 2 «2 _ 6 a; and 4 iK^ - a^ + a^ + 4 a;^ - 12 + 4 a^.
18. 21 ax — 17 ax^ — 5 aar^ 4- ax^ and 5 aa^ — 34 aa^ — 7 ax.
19. 7 mV— 49 m-^x + 42 m^ and
14 a^mx^ + 14 a^mx^ — 56 a^mx — 56 a^m.
20. 48 s^to* - 162 ^tx^ + 54 s^^ and
18 ^fu -9 s fux - 48 ^fux" -f 24 sH'uo^.
21. 4a;*-12a^2/ + 5a^2/^-fl2a;i/3-92/*and
12 a;^ - 36 a:^^^ + 11 a;y + 48 aJ2/3 _ 36 2/4.
22. x^-x^y-llxy^-4:f and a^^+a^y - 12 a^/ - 30 a;^ - 8 /.
23. The H. C. F. of any number of expressions must be a
factor of the H. C. F. of any two of these expressions. Why ?
Must it be the H. C. F. itself of any two of these expressions ?
Explain.
78-79] LOWEST COMMON MULTIPLE 105
FindtheH.C.F. of:
24. a^ + 4a3 + 4a2, a^b-4:ab, and a*6 4- 5 a^fe + 6 a%.
25. Sx*-9a^+6a^, a^-9x^-\-26 x-24:, and a:^- 8 0^24. 19 a; -12.
26. a + a^a; — 2 a^, a + 3 a^a; + 4 ax^ + 2 a;^, and
2a3 4-3a2a;4-2aa^-2a^.
II. LOWEST COMMON MULTIPLE
79. Multiples of algebraic expressions. A multiple of an
algebraic expression is another algebraic expression that is
exactly divisible by the given one ; hence it contains all the
prime factors of the given expression. A common multiple
of two or more algebraic expressions is a multiple of each of
these expressions.
j5^.^., 12 a^Q^{]f'— 1) is a common multiple of 3 aVQy + 1)
and 2a%(^ — 1).
The lowest common multiple (L. C. M.) of two or more
algebraic expressions is the algebraic expression of lowest
degree which is exactly divisible by each of the given expres-
sions ; hence it contains all the prime factors of each of the
given expressions, but no superfluous factors.
E.g.^ a common multiple of 2 a%'^o[^ and 3 a^a^y^ must con-
tain the factors 2, 3, a^, 5^, x^^ and ?/* ; it may contain other
factors also, but it need not do so. Therefore 6 a%^j[^y^ is the
lowest common multiple (L. C. M.) of 2 a^h^a:^ and 3 a^j(^y^.
So, too, the L.C.M'. of 12 m\x^-lc^) and 8 JV.2(s+0(^-^)^
is 24 m%\s -\-t')(x — k')^(^x -|- ^), — show that this last expres-
sion contains all the necessary^ but no superfluous^ factors.
The procedure for finding the L. C. M. of two or more
expressions whose prime factors are known (or easily found)
may be formulated thus :
To the L. C. M. of the numerical coefficients annex all the
different prime factors that occur in the given expressions^ and
give to each of these factors the highest exponent which that
factor has in any of the given expressions.
106 HIGH SCHOOL ALGEBRA [Ch. VII
EXERCISE LIII
Find the H. C. F. and the L. C. M. of :
1. 8 a'h'', 24 a^6V, and 18 ahc\ 5. x" - ?/' and aj^ + 2 ic?/ + if.
2. 15a^6^ -20a26V, and 30 ac^ 6. 21 a;^ and 7 ^^^(x -h 1).
3. lQ>o?h\ 24:aMc, and 36 a^ft^dl 7. ic^-l and a^ + a;.
4. 18a^6r^, 12 pV^, and— 54a6y. 8. 4:X^y — y Siud 2 x^ -\-x.
9. Is 12a^6''(a^ — 1/^) a common multiple oi 2 a^b {x — y) Siud
3 ab\x -y)? Is it their L. C. M. ?
10. What factors must an expression contain in order that it
may be a common multiple of two or more other expressions ?
that it may be their L. CM.?
11. Are both 6 ax^ and — 6 ax"^ multiples of 3 a; ? Explain.
If a multiple of an expression has its sign reversed, does it
remain a multiple of the given expression ?
12. Does a change in the sign of an expression affect the de-
gree of the expression ? If the L. C. M. of several expressions
has its sign reversed, it may still be regarded as their L. C. M.
Why ? (Cf. Exs. 14-15, p. 99.)
FindtheL. C. M. of:
13. a + &, a — b, o? + W, and a* + b\
14. 3 + a, 9 - a^ 3 — a, and 5 a + 15.
15. a^ — if, :ii? -\-xy + 2/^, and a^ — xy.
16. 4a + 4&, Q>o?-24.b\ and a--3 a& + 2 ftl
17. a? -f 2/^, ^y — y^, and a;^ — 2/^.
18. y'^-^y + Q and i/^ - 7 2/ + 10.
19. a^— (a+6)a;4-aft and a;-— (a— 6)a; — a&.
20. 3s2-7s + 2and6-s-s2.
Hint. 6 - s - s^ = - 1 (s2 + ^ _ 6).
21. c2-4c4-4, 4-c2, andc^-16.
22. 3p«-13p4-14 andl3p-5i)2-6.
23. r^" — s^" and (.s« — r^'f.
24. (m + ny—p^ and (m + w +i))^.
79-81 j LOWEST COMMON MULTIPLE 107
25. 63^262-46-8, 86-12 + 62-63, and 6^ + 4 6^-3 6-18.
26. Find the L. C. M. of each of the sets of expressions in
Exs. 19-25, p. 100.
80.* The L. C. M. of two algebraic expressions found by means of
their H. C. F. The use of the H. C. F. in finding the L. C. M. may
be shown as follows :
Let it be required to find the L. C. M. of 3x* — a:r^ — x'^-\-x — 2
and2ar^-3a;2-2a; + 3.
By § 76 it is found that the H. C. F. of these expressions is
x^ —1', they may, therefore, be written thus :
Sx*-x^-x^-\-x-2={x''-l)(3x'~x + 2),
and 2aj3-3a;2-2aj-f-3 = (x2-l)(2x-3),
wlierein Sx- — x-\-2 and 2 x — 3 have no common factor. Hence
the L. C. M. of the given expressions is
(x'-l)(3x^-x + 2)(2 X - 3).
This shows that the L. C. M. of the given expressions may he
found by dividing their product by their H. C. F.
Obviously, the L. C. M. of any other pair of expressions may
be found in the same way ; hence.
To find the L. C. M. of two algebraic expressions, divide either
of the given expressions by their H. C. F. and multiply the other
expression by the resulting quotient.
81.* The L. C. M. of three or more expressions. The L. C. M. of
three or more algebraic expressions whose factors are not easily
found, may be obtained by first finding the L. C. M. of two of the
given expressions, then the L. C. M. of that result and another of
the given expressions, and so on.
EXERCISE LIV
Find the L. C. M. of :
1. a;3_6a^_^lla;_6anda.'3-9a^ + 26a;-24.
2. a^- 5x--4i» + 20anda^ + 2ar^-25a;-50.
*Articles 80 and 81, with Exercise LIV, may, if the teacher prefers, be
omitted till the subject is reviewed.
HIGH SCH. ALG. — 8
108 HIGH SCHOOL ALGEBRA [Ch. VIl
3. 2 2/3 - 11 / -h 18 ?/ - 14 and 2 2/^ + 3 / - 10 2/ -h 14.
4. 6a^x-o a?x - 18 aa; - 8 a; and 6 o?h - 13 c^h - 6 a6 + 8 5.
5. 4 a;^ - 17 ^f 4- 4 / and 2y^-Qi?y-?, xY -^xf-2y^.
6. 2a;^-9a^ + 18i»2-18a; + 9and3a;*-llar'H-17a^-12a;+6.
7. If ^, B, and stand for any three given expressions, and
if i»f is the L. C. M. of A and B, while iV^is the L. C. M. of M
and (7, show that iVis the L. C. M. of A, B, and (7; that is, show
that N contains all the factors necessary in such a multiple, and
no superfluous factors.
Find the L. CM. of:
a s*-2s^-i-s^-l, s'-s^^2s-l, ands*-3.s2-fl.
9. c3 + 3c2-6c-8, c^-2c--c + 2, andc^ + c-G.
10. a52-4a2, ar5 + 2aa^^-4A-}-8a^ anda^-2aa^ + 4a2a;_8a3.
11. a3-j-7a2 + 14a + 8, a^ + Sa'-e a-S, and a3 + a'-10a+8.
12. Ar^-9 A:24.23 k^W, k'+k'-17 k+W, and ]i^+7 k'+7 k-15.
CHAPTER VIII
ALGEBRAIC FRACTIONS
82. Definitions. An algebraic fraction is an indicated divi-
sion in which the divisor is an algebraic expression : the
dividend may be either an algebraic or a numerical expres-
sion. (Cf. § 8.)
Here, as in arithmetic, the fraction A-i-B is usually written
A
in the form — or A/B ; A and B are called the terms of the
B
fraction, A being the numerator and B the denominator.
If A is exactly divisible by B, then A/B is, in reality, an
integral expression, but is written in the form of a fraction.
E.g.^—- , — -•,and — are algebraic fractions; while
ab—oj^ m—2n -.a^— a;^ . , .
, , and are integral expressions written
a 1 a— X
in fractional form.
If both terms of a fraction involve the same letter, and if
the numerator is not of lower degree than the denominator
(in this letter), then the fraction is said to be improper ;
otherwise it is proper. An expression that is partly integral
and partly fractional is called a mixed expression.
E.g.^ ^ — and — — — are improper fractions, and
X — 1 a
4 2: — 3 H — ^^—— is a mixed expression.
x — \
83. Operations with fractions. The reduction of fractions,
and the various operations with fractions (addition, subtrac-
tion, etc.), are essentially the same in algebra as in arith-
luetic.
109
110 BIGH SCHOOL ALGEBRA [Ch. Vlll
A
Thus, if — and ~ are any two fractions whatever, then
^ ^ B D~BI)' ^ ^ B^I)~b'~C'
These formulas state the rules for finding the product and
quotient, respectively, of two fractions ; the pupil may trans-
late each formula into verbal language.
(i) The proof of (1) follows directly from the definition
of a fraction (cf. §§ 82, 8).
Thus, let — = a;and — =?/,
then A = X'B •Awdi 0=y'D, [§§82,8
hence A- C= xByB = xy'BD, [Ax. 3
AO
and therefore — — = xy [Ax. 4
BB
^4 O fsince A/B = x
~ B' B' L and C/B = y
A C^AO
B' D BB"
which was to be proved.
(ii) To prove (2) above, let - - - = t,
B B
then | = ^'|' [§§82,8
hence A.^=t.^.R [Ax. 3
B Q BO •-
n
i.e.^
I.e.,
^'{b'o)^*' ^^^^ ^^^^^
and therefore A^^=4.^
B B B O'
which was to be proved.
Kemark. The reciprocal of any given number is 1 divided by
that number; e.g., the reciprocal of 3 is ^.
Hence it follows from (ii) that the reciprocal of a fraction is
that fraction inverted.
Note. Observe that the validity of § 42 is assumed in the proofs of (i)
and (ii) above.
83-85] ALGEBRAIC FRACTIONS 111
84. Reducing an improper fraction to a mixed expression.
This change in form is made in algebra precisely as it is
made in arithmetic.
E.g.f just as -ijQ=3|^, i.e., 3 + J, so, too, since a fraction is an
.,.",,... x^-^-'Jaf + B ' ,-,, 4r-2x
indicated division, — -^ -— = a? + 1 + — -.
EXERCISE LV
Reduce each of the following improper fractions to an equal
integral or mixed expression, and explain your work :
a^ — 2ab-\-ac
8.
2.
a
3ar^+9a; + 2
3a;
^ 2a;'^ + 4aa; + 2a^ __
x-{-a
2s-l.
11.
12.
ar^-ar^-2ar^-
-2a;-l
a^-x-
1
6-hQc-oc^-
-2c3 + c*
c'-3
8a;''-10a;2_3a; + 5
4x^^-3
3a,6_^2a;-5
a;» + 2a;4-l
15^^-13 i;2-
8i;-l
A: + 2 Sv'-^ + S'y + l
g^ + g^ + l 7fe^-l
a + 1 * Jc' + k + l
a^4-7a;2_5 I8a;^-a^-2a;^- 7
a;2_i • ■»•*• ^3_:3^_^1
^3 2 a 4- 1
15. Is ^^^-^ a proper or an improper fraction ? Why ?
16. Write the reciprccal of 11 ; of —a; of — |; of each frac-
tion in Exs. 1-5 (cf. Remark, § 83).
85. Reducing fractions to lowest terms. In algebra, as in
arithmetic, a fraction is said to be in its lowest terms when
its numerator and denominator have no common factor.
Hence, to reduce a fraction to its lowest terms, divide
both numerator and denominator by their H. Q. F. Instead of
dividing at once by the H. C. F., we may, of course, divide
112 HIGH SCHOOL ALGEBUA [Ch. Vlll
by any common factor, then by another, etc., until all com-
mon factors are divided oat. Multiplying or dividing both
terms of a fraction by any given number leaves the value
of the fraction unchanged; for, whatever the algebraic ex-
pressions represented by A, B, and m,
^ = i.^ [§83 (i)
Bm B m l^ \ J
= — , [since m/m = 1
which was to be proved. -"
3 ax^ 3 ax
E.g., ~ = , which is in its lowest terms ; so, too,
4 bxy 4 by
, ^~^ — = (a? + l)(a;- l) ^ x±l^ ^^.gj^ .g .^ .^g lowest terms.
a^-2a; + l {x-l){x-l) x-l'
EXERCISE LVI
Beduce each of the following fractions to its lowest terms :
a^ — ah m^ -\-2 mn -f- w^ „ m? -\-if
a^ — h^ m® + n^ x^ + x^y"^ + y^
34«^6V 2a^ + 3a; + l ^ 3a^-2a-l
Sla^ft^c* ' a^ + 5x + 4 * ' l + a-a^-d"'
c'-d^ {r-qY-s^ a*-a^-20
{c-df' ' (r-q-sf' ' a'-9a'-{-20'
10. May equal factors be canceled from the numerator and
denominator of a fraction ? May equal parts (or factors of parts)
be thus canceled ? Is 1^L±^ equal to -^? Is P^^^^,
obc-^x obc ox — on^
equal to ^ + ^y ? Explain fully.
Eeduce the following fractions to their lowest terms, and
check your work by § 25:
n ^-t\ T« xy-zy-x + z
^^' '^r~t' ^®- f^^
c3-17c^ + 72c 50-40m + 8m^
c2(cd4-16-2c-8d)* ' 125-8m«
85-86] ALGEBRAIC FRACTIONS ' 113
16. ^^:^ 21. ^"-^
(a -«)(«- 6) l + Ai^"*
(s-2)(8-3) . (a^-5)(a.- + 2) ,
(2-s)(3-s)(s-4) • aj3-7a^ + 2a; + 40
18 p^-llp + 24 3a^ + 8a -3
56+i)-i)^ 3a3 + 17a2 + 21a-9
^!±l!. 24 10^ + 20 a:'-a?- 2
* y'-x^' ' 3a^ + 6a^ + 21a;+42'
86. Reducing fractions to equal fractions having given de-
nominators. Since multiplying both terms of a fraction
by the same number does not change its value (§ 85), there-
fore any given fraction may be reduced to an equal fraction
whose denominator is any desired multiple of the given
denominator.
E.g., to reduce - — - to an equal fraction whose denominator
shall be 12 cx^y, multiply both terms of the given fraction by
12 cx^y -7- 4:X^, i.e., by 3 cy,
EXERCISE LVII
Find the required part in each of the following equations :
1 ^ = L. 1 m-2 _ (m-2y
3.
4 12
Sab^ ?
4 12^'
2 cd 16 cH^
8.
6t
9
3c-
2d'
1 ?
Scd*
4:X
9
1
7x^-5
a-b
a'-b'
10.
5. ^ = , ; ^ ' 11.
a-\-b ? "■ 2(2a;-5) -6x{5-2x)
9m
9
— r
9
r2 + 3 r^ + 10r2+21
2u-v
?
^f +2 V Su
^j^5uv-2v''
u^ — uv-\-v^
_2{u' + if)^
lu^-1
9
3m-8
9
2a;-5 -
-2.TH-5
3m-8
9
114 ' HIGH SCHOOL ALGEBRA [Ch. VIII
13. If the denominator of a fraction is multiplied by any given
expression, what must be done to the numerator in order to pre-
serve the value of the fraction ?
14. Change { to an equal fraction whose numerator is
J9^ — 9
— 3j9 — 2; to one whose numerator is 3jp^''H-5p^4-2j9; to one
whose denominator is 9p— p^; to one whose denominator is
3y4_p2_27p_9.
87. Reduction of fractions to common denominators. To
reduce any given fractions to equal fractions having a com-
mon denominator it is necessary only (1) to choose some
common multiple (§ 79) of the denominators of the given
fractions as the new denominator, (2) to divide this com-
mon multiple by the denominators of the given fractions in
turn, and (3) to multiply both terms of the given fractions
by the respective quotients (cf. § 86).
3 h hn
E.g., to reduce -^^ — and — - to equal fractions having a com-
2 ax 3 x^
mon denominator, we choose 6 ax^ as the new denominator, and
find, by § 86, that
3h _ 9 hx T hn _ 2 dbn
2 ax 6 ax? 3 x- 6 ax^
The lowest possible common denominator is, of course, the
L. C. M. of the given denominators.
EXERCISE LVIII
Reduce the following to equal fractions having the lowest pos-
sible common denominator:
1. -, — T' '^"^^ -r—^'
111 inr 5 m"*
^ 3a + l T 3i»-f4
2. — and ■
4 6
^ 9-3 a ,34-5a^
3. and — ^ „ -
16 h 20 h^
4.
« + ^and^-^
a — h a-\-h
5.
^, ¥, - 1^
6.
rfyl|."0|^
86-88]
ALGEBRAIC FRACTIONS
115
7. -^:z^and ^ + ^
ar + aic 2 aa^ — 2 a^a;
and
9. — !-^^ and ' -^
10.
a;2 + a-2/ + t/'^
and
11.
12.
13.
14.
15.
16.
(m — l)(m — 2) m — 3 * x + y'x — y^ x^ — xi/
Hint. First multiply both terms
of by — 1, so as to arrange
and
2 — m' m + 2' m- — 4
2 - m
the denominators in the same order.
m
and -^
m — n rr — m'
m 4-^*'
6x
2 J 3
. and
ar^-1
3 a
and
3 a — 6 X
^ ^-2 and ^ + ^ ,
5 — a' a^ — 8 a + 15' a^ — 6 o + 5
and
(aj-l)(a;-3) (x-8)(3-a^)
17. Show that
Hint.
equal
(x-S)iS-x)
-7
(x-S)(x-S)
Cf. Hint, Ex. 11.
(c-l)(2-c) (c-l)(c-2)
2-a; x-2
18. Show that _ ^^ ox — / Kx/ ox-
(5 — x-)(aj — 3) (a;-5)(a;— 3)
Reduce to equal fractions with the lowest common denominator ;
? , ? , and — (cf.Ex.l6).
(a; _ 7) (a? - 2)' (2 - x)(x - 4)' (4 - aj)(a; - 7) ^ ^
1 2.-3
19.
20.
and
(.y _ 2)(v - 5) ' (v - 5)(3 - ?;)' (v - 3)(7 - v) '
21.
a+ 5
a-2
and
a + 1
a2_ 4 a + 3' 8 a- a2_ 15' 6a -5- a^
88. Addition and subtraction of fractions. From § 38 it
follows that
a h a + h -.a h a — h
- -4- - = and = ;
c c
c c
116 HIGH SCHOOL ALGEBBA [Ch. VIIl
that is, ill algebra, as in arithmetic, the sum (or difference^ of
two given fractions which have a common denominator is a
fraction whose numerator is the sum (or difference^ of the given
numerators, and whose denominator is the common denominator
of the given fractions.
^ m^ 2h ^ m^^2h . 2 a b^c ^ 2a-l?c
'^'"^ax Sax Sax ' ^^ 6(x-l) 5(a^-l) 5(a:-l)*
If the given fractions have unlike denominators, they must,
of course, be reduced to equal fractions having a common
denominator (§ 87) before they can be added or subtracted.
3 7
Ex. 1. Find the sum of and
x — 2 x-\-l
Solution. The L. C. M. of the denominators is (x — 2) (a; -j- 1)
and, by § 87,
and
3 ^ 3(a;+l) ^ 3a; + 3
x-2 (x-~2)(x + l) (aj-2)(a! + iy
7 ^ 7(a;-2) ^ 7a;-14
x + 1 (x + l)(x-2) (a;-fl)(a;-2)'
7 Sx + 3-{-7x-U 10a!-ll
+
x-2 aj + 1 (x-{-l){x^2) (a; + l)(a;-2)
7 3
Ex. 2. Subtract from
x-^1 x — 2
Solution. Proceeding as in Ex. 1, we obtain
_3 7 ^ 3a; + 3- (7a;-14) ^ -4a; + 17
x-2 x-\-l (^x-\-l)(x-2) (^x + l)(x-2)
Note. The minus sign before the second fraction means, of course, that all
of this fraction is to be subtracted, hence the need of the parenthesis in the
numerator of the next fraction.
EXERCISE LIX
Simplify the following expressions and check your results :
3 ^4.^ 5 a?-l x-^3 x-\-7
■ 3 "^6 • ' 2 5 "^ 10 *
. a + S a-{-5 c-^-d b_
5 7 * ' d 2d'
88] ALGEBRAIC FRACTIONS 117
- 1 +^. 16. 1 3
x+y x — y 2s^—s — l 6 s- — s — 2
- X a; # _ _ ct^ — aa.' H- 0^ a + a;
1 — a^ 1 + a^ a^ + ax + a;^ a — x
10. ^Lz^H-^^I^ + ^r:i^. [Suggestion. x = ^.]
ah he ac L 1 J
11. r-j_^r+^__r^--^> ^^ _a c^ ^ s.
2rs s — as^ — w
(a5-2/)^ a;^ + 4a;?/-5/ a^-7a; + 12 a^-5a;+6
13. ft + ^-c a-6 + c ^ 21. 1 I ^-?/ a:^-a;y ^
a? — ih — cf {a—hy—c^ ' x-\-y x^—xy-\-y^ 3^-\-'if
14. J~-^ 22. -i- + -^ 1.
a;2_i aj2_^_2 s(s — t) t{s + t) st
15 a? 4- 7 a; + 2 ^^ 2a;-3c 2a;-c ^^
a^_3a;-10 a^4-2a;-35' * a;-2c x-c
Remark. Since a fraction is a quotient, its sign (the sign
before the fraction) is governed by the law of signs in
division; hence, whatever the expressions represented by a
and hy
__a_ — g _ a
h~ b ~-h'
Exercises in subtraction may therefore be changed into exercises
in addition ; the results of such exercises are often called algebraic
sums (cf. § 16).
E.g., Ex. 23 may be written ^^~^^ + ~^^"^^ + 3 a;.
X— 2 c x~c
24. Write Exs. 6, 8, 11, 13, 14, 15, 19-22, above, as exercises
in addition.
* Cf . Ex. 2, Note.
118 HIGH SCHOOL ALGEBRA [Ch. VIH
Write the following as exercises in addition, and find the alge-
braic sum in each case :
a a + 1 a + 2 s—1 2(s + 1)
26. h « — :i 30. 1
ic — 1 1 — X a — 1 a(a — 1)
27. -i L.-.!^. 31. 1 1
a + 6 a-6 5'^ - a^ 2a^-a;-l 3-aj-2a^
d cd cc?2 ^^ 2h-a 3x(a-b) , ?^-2a
G + d {c-\-df (c + df x-b b'^-x" x + b
Simplify :
{a-b){a-c) {b-c){b-a) (c-a){c-b)
[Hint. The given expression, witli the letters in alphabetical order, is
^i + -I + J .1
(a-6)(a-c) (,b-c)(a-b) (o-c)(6-c)-l
34. "" + * + <'
(a — b)(a — c) (b — a)(b — c) {c — a){c — b)
35 a;-l 2(a^-2) x-S
(x-2)(x-3) {3-x){x-l) (x-l)(2-x)
36. ., . ^ .. „ - „ . V „ „ + '
x^ — 5 xy -\- 6 y^ x^ — A xy -{- 3 y- x' — 3xy-^2y'
37. ~ H f- ^
x^ — 5x-{-6 3a7 — 2 — ic- 4a; — 3 — a;'-'
38. ^^ [ ^^ I «^
(a-c)(a-6) (6_c)(6-a) (c-a)(c-6)
89. Reducing mixed expressions to improper fractions.
Since an integral expression may be written in the frac-
tional form with the denominator 1, therefore reducing mixed
expressions to improper fractions is merely a special case of
addition.
E.g.,x + 1+ 1 ~»' + l . 1
a; — 1 1 a; — 1
x—1 x — 1 x — 1
88-90] ALGEBRAIC FRACTIONS 119
EXERCISE LX
By the method of § 89 simplify the following expressions :
af — l a + 2b
, -i 2x „ 9 oX'^-j-oi^ — x-\-l
2. x-\-l • 7. X — or — XT ■ —
x — 1 l-\-x-{-x~
c^— 2c + l 1 — 2a; + a;-
a4-& 2aH-36
_ -, 2 1— V^ -.r* -1 I, ax-\-bx-\-ab
5. 1 — y — if — ^ • 10. 1 — aa; — 6ir :; —
1 — 2/ 1 — ax
11. May the numerator in the answer to Ex. 1 above be found
by multiplying x~l by a^ — 1 and adding x'^ to the product ?
Explain this method fully (of. § 84, also Ex. 20, p. 50).
12. By the method of Ex. 11, solve Exs. 2-5, and 8-10, above.
90. Product of two or more fractions. In § 83 it was
shown that, whatever the expressions represented by A^ B,
(7, and i>,
A O^AO
B ' J) BD'
This principle is easily extended to finding the product of
any number of fractions ;
A ^ ^ ^ = A^ ^ Oi^ACE a^ACEa
^*^*' B' d' f' H BB' f' H BBF ' H BBFH
Hence, the product of two or more fractions is a fraction ivhose
numerator is the product of the numerators of the given fractions^
and whose denominator is the product of their denominators,
Ex. 1. Find the product of ^^^ ~ ^ and , ^^ ^^ •
'6xy a (or — 1)
SOLUTION
a(x — l) 6x _ Gax{x — \) _
3xy a(ar — 1) Saxy{x^ — 1) y{x + l)
[§85
120 HIGH aCHOOL ALGEBRA [Ch. VllI
Remark. Observe that the factors 3, a, x, and a^ — 1 might
have been " canceled " even before the multiplication was actu-
ally performed. Pupils should cancel wherever possible, and thus
simplify their work.
EXERCISE LXI
Find the following products, and simplify your results :
2. ^ . ^. 9.
xyz ac^ m- — mn ef—f^
g Qxy l^yh\ ^^ x-1 . a; + l .
* 82 * 9a^ * ' a^ + 2a; + l x-1
^ - 5 iH^ Sr^t ^^ x-1 x2 + 4a; + 3
12iu^ lOs^v^ 3aj2-|.8i» + 5 ar^-1
— Ix^yz f 6 fey^ Y /-^ — ?-8 + s^ j? — <f
IS xyz'- '\ 11 x'zj ' {p-qf *r^ + ?V + «^*
4myi^ —am (a — xY x^-\-xy + y^
1 m^n^ x^ — y^ d^ — 2 ax-\-x^
7. Isx^y^ . t . 4-. 14. i—^r-r . a±b+l,
y 9xY (a + 6)2_l a-b-1
^m+2 * ^m •52* * 2s^ * si + i^ 's2_g^*
16 ^ + 5 a4-3 gg-Ta + lO .
a-2 * a-5 ' 2a- + 5a-3*
17 a.'^ + y' a* + a'b' + b* x^-f
' a^-^b^' a^-b^ 'x*-2afy''-\-y*'
1^ _ f 3
18- -^ — r • r^r-: = ? May i^ be canceled in this example ? Ex-
plain (cf. Ex. 10, p. 112).
19. Simplify [x-\-2y ) • — ^ by finding the product of each
V y) a-\-x
term of the multiplicand by the multiplier, and then adding the
partial products (cf. § 32).
20. Simplify [ a; + 2 ?/ — - ) • — ^ by first reducing the multipli-
V • 2/7 a + a;
cand to an improjjer fraction.
9(V01]
ALGEBRAIC FRACTIONS
121
Simplify, and check your results :
22. (5s2-36s+7)
25.
26.
27.
28.
29.
+
(f - ?>
1-6-
3s-21*
23.
24.
a + 25
2b-a
1 ^'"
V
x' + fj x'-f
Sa(a-b)
V af-9x-^20j\'jf-6x + 9 J\ x J
b(x^
a
9x-^
-hx + y
3 a' + 3 a'bx + 3 a*by
ax — 3by—Sbx-\-ay
1 ^a + b^(a-\-byi\ (a-\-by\a
a_b__c 2\A 2c V 5a
6c ac a6 « A a + 6 + cy l^ — a — b
(a + 6)2fa + 6 + c
30. What doesr-Y mean [cf. § 9 (ii)!? Show that (^-Y
Ve/ \/-i \o/
and that ("^ = i^?!^' = ^^ .
fey f rs' Y / m - 2 n V / -Sc V / a + 6
31. Raise the following fractions to the powers indicated :
b-c\\ r 2.<i/V Y
/ ' \—'Smyip)
32. Write the following fractions as powers of other fractions :
6^ + 26 + 1 . _t_. s^-^sH-^-^sf-f
a? ' 16aj8' 64 + 48i/ + 122/' + 2/'*
91. Division of fractions. In algebra, as in arithmetic, to
divide by any fraction gives the same result as to multiply hy
the reciprocal of that fraction [of. 83 (ii)].
a^x
E.g., -^1^ -
a'^x by
by- by b-y^
ex
af_
bey
Note. If the divisor is an integral (or mixed) expression, it should be
written in fractional form (that is, as an improper fraction) before proceed-
ing as above.
122 HIGH SCHOOL ALGEBRA [Ch. VllI
EXERCISE LXII
Simplify the following :
Ud'b' 2a'b' /^H-8^ + 7 ^ ^
a" -121 . a + 11 ,^ / _^^^^V «' i\ . '^
2- ^ — r -^ — ^ • 11. « + /i .> '> - i - 2
• ^V ' -5q^' ' (a+6)2-9 ' a + 6 + 3
4. l^!i£±fU-^. , 13. ^l±^^(t^-^2tv + 2v^y
7'.s — .9^ r — s —or
^ oc^ — a" (x — ay , ^ -y^ — 25 5 — 'i?
5^ J.- N z_ , 2,4. r- •
a^ -j-a^ x^ — a^ v^-\-v-\-l v^—1
^ Ux^-7x 2aj-l „ x^-1 x''-12x+?>n
15.
12x^-\-24:X^ x'^ + 2x lO-^'Sx-a^ x^ + 3x +
„ p(^ Q„4 5 r)V ^ /h 2cd \ d — c
7. OO^rrs^-? '- — 16. 1 ' •
a. (.^-3.-10).-^;. 1. ^14-.-^^
a^-b* , (g - by r'-l , 1 - r^
• a^ + a2^2_^6* • a'-b' ' t^-dr ' 3-r'
0^^-5^-36 V ^'-y
20. 2r^-21r-ll ^/^_r-3
^^-^-ISr-T • V r-7
^.
22.
i»2_49 a;2_^_42
5 m^n — 5 ri^
urn + 2 mrr + n^ V wi
23 2a^ + 13.T + 15 . 2a?^ + lla; + 5
4x^-9 ■ 4x2-1
2^ x^-nx- + l^ . / a^2_3^_4
9rt^-34a2 + 2o V 3a2 + 8a + 5
91-92] ALGEBUAIC FRACTIONS 123
{p — qY P — Q p^-\-q^ p^ — q'^
25- f:^=^,^^~^^ + z^ ■ P-'P\^1 (cf. § 10).
I'oTj—WjfY ^ (^ — ^y^ P^^ — 4 py—^p^y+px
' \p-q J
p^ — pqj \p—qj ^ -\- ^ xy -\- ^ if
92. Complex fractions. In algebra, as in arithmetic, a frac-
tion whose numerator or denominator, or both, are them-
selves fractional expressions, is called a complex fraction.
1 1
a X
^'Q--> ^ and r- are complex fractions.
X
Since a complex fraction is merely an indicated quotient,
it may be simplified by means of §§89 and 91.
^g ^ _ ^ _aP — 1 X _ ^ — 1
'.,. .1" x^-\-2x-\-l~ X ' x^-\-2x-{-l~ x + 1
X X
In many cases, however, a complex fraction is most easily
simplified by first multiplying both its terms by the L. C. M.
of their own denominators.
Thus, in the above example, multiplying both terms by x
gives o^ ^ T") which (by § 85) equals ~ , as before.
X^ -}- ij X -\~ J~ X -J- JL
EXERCISE LXIII
Simplify the following expressions :
^ (m — nf -4-i
s—p _m- g. s s
1 =— . 3. . 9. •
■ s +p m^ — n^ 1 _ I
p^ mn V s
c-^d (x-S)(x-5) 1_1
^ c— d x — 7 6 ^ -^
(^-d^ x-n i^_j.
c^-fd^ (a!-3)(.T-7) e" f
HIGH SCH. ALG. — 9
124
7.
8.
c
HIGH SCHOOL AH
1 + 7
9 ^
a 6 c
9EBRA [Ch. VIII
3 a-c
11. «-f
. ah'
x — 6
x-2 '
X
a
(a - 6)^
2a-3&+c
^ ,.4
oca
a
1+- + J
a —
3
18. "^^
6 a-\-h ^^
1 ^
a^-b' a^ + W
a-{-h a—h
15
& ' a + 6
a — b a + b
1
1 I ^
a «-l
^ + 1 + 0.
a;
a+1
" 1 / 1 ^
* To simplify such an expression we begin at the end and work backward.
The first step here is to add ^ to 1 , then divide 4 by this sum, then add this
quotient to 1, and finally divide 3 by this sum, obtaining the result j\.
CHAPTER IX
SIMPLE EQUATIONS
93 Introductory remarks and definitions. As we have
already seen in Chapter V, an algebraic problem states a re-
lation between numbers whose values are known (called
known numbers), and others whose values are at first un-
known (called unknown numbers). It is by means of this
relation, translated into an equation, that we can find the
values of the unknown numbers.
Besides the numerals 1, 2, 3, ••• the letters a, 5, <?, •••are
often used to represent known numbers ; unknown numbers
are usually (though not necessarily) represented by the later
letters of the alphabet, such as ic, ^, and z.
A literal equation is one in which one or more of the
known numbers are represented by letters; while in a nu-
merical equation all known numbers are represented by the
numerals 1, 2, 3, etc. An integral equation is one whose
members are integral in the unknown numbers (cf. § 34);
known numbers may appear as divisors and the equation
still be integral.
By the degree of an integral algebraic equation is meant
the highest number of unknown factors which it contains in
any one term.
E.g., of the equations (1) 4a;-5y=:10, (2) —-8 = -, (3) 4ta'^ = 2+x,
a h
(4) 3 x2 - 9 a3 = 4 y2^ and (5) |- -3 a;y = 5 y, all are integral, (1), (2), and (3)
are of the first degree, (4) and (5) are of the second degree, (2), (3), and
(4) are literal, and (1) and (5) are numerical.
Equations of the first degree are usually called simple
equations, and often also linear equations (cf. § 140, Note) ;
125
126 HIGH SCHOOL ALGEBRA [Ch. IX
while equations of the second and third degrees are called
quadratic and cubic equations, respectively.
94. Equations having fractional coefficients. Equations
having fractional coefficients may be solved as follows:
Ex. 1. Given -— — ^ = « 5 *^ ^^^ ^•
Solution. On multiplying both members of this equation by
6 (Ax. 3), it becomes 5 oj — 48 = 3 a;,
whence 2 a; = 48, [Ax. 1
and therefore a; = 24. [Ax. 4
On substitution in the given equation, 24 is found to check; it
is therefore a root of that equation.
Multiplying both members of an equation by a common
multiple of its denominators is usually spoken of as clearing
the equation of fractions.
EXERCISE LXIV
Solve the following equations, checking the root in each case :
_ 2x — 4 3a; — 7 o
O. — = Zi.
2.
X 4:X f,
2==Y"^'
3.
2a; 5a; „
3 4 ~
4.
3a; = | + 25.
5.
7a; 5 w 2a;
10 6 15"
6.
4. ._23.
9 6 36
7.
7?i 4- 8 m -f 4 ^
6 11 ~ •
[Cf. §88, Ex. 2, Note.]
5
7
9.
V+s=
:91
-lOv.
10.
T-'i-
= -
-i + 2ia,
,, X 2x , 2x-3 ^
12. l„-~+-^ — .
,, a; — 3 , 7x 4x — S
3 18 5
14. Are the above equations integral or fractional ? Why ?
93-95] SIMPLE EQUATIONS 127
15. How is an equation cleared of fractions ? Upon what
axiom does the process depend ?
16. In each of Exs. 5-7 name three factors, any one of which
might be used to clear the equation of fractions. How may we
in each case find the least factor which can be used ?
17. Name the degree of each equation in Exs. 22-28 below.
Which of these equations are simple ? quadratic ? cubic ?
Solve the following equations, and check as the teacher directs:
^3 7,a;-4^^^ 4.x + 7^ 25. a:^-x=6. (Cf . § 72.)
5 " 7 *
. /o ox X 26. x'-15x' + mx = 0.
19. -2 a; + 4 -(3 a; 4- 2)=--.
20 Mzz^ = ^±2_3^-2 27. v'-^2v' = v + 2.
5 G 7 '
^, Ax 21X-25 4.(3x-2) 28. ^±i ^ _A_ = ^"zi^S^
21. -3- = — 12 2~ ^ -^-^ 1^
22. 3s-15_2g-^^3 -^^ 29.
8 4 2
23.
1.25+ .5 a ^ .25 g- 2.375
.25 1.125
7
' -
k-i
5
;22-
-2^-
15
:2| =
^-3
3
2a:
3
+ 4
161
: —
— X
a-32 2a-3 _15-3a _3 _ lb^-x x
2 4 ~ 8 ' ^^- 2 32'
31. J[c2- i(c2_3) + 6c-7] = 9fi.
- (i-0(i-^^)-(i-'^)(3-^^)=-^-
95. Equivalent equations, (i) Two equations are said to
be equivalent if every root of either is a root of the other
also. Thus, the several equations in the solution of Ex. 1,
§ 94, are equivalent ; each has the root 24, and that only.
The method of solution used in Ex. 1, § 94, consists (1) in
deducing from the given equation, by means of the axioms,
a succession of new and simpler equations, and (2) in finding
the root of the last and simplest of tliem all.
128 Jliail SCHOOL A LG Eli HA [Ch. IX
That the root of this last equation (24, in this case)
happens to be a root of the given equation also is due to the
fact that applying the axioms to equations usually produces
equivalent equations.
(ii) Although the axioms are correct, their application to
equations does not always result in equivalent equations.
E.g., given the equation 3 a; — 4 = 1.
Multiplying both members (Ax. 3) by x — 2, we obtain
3a---10aj+ 8 = a;-2.
Simplifying, 3a^- 11 a; + 10 = 0, »
i.e., (a;-2)(3aj-5) = 0;
whence (§ 72), aj = 2 or a; = f ;
but 2 is not a root of the given equation.
The axioms must, tlierefore, be used with caution, and
results should always be checked.
(iii) The following changes in equations will, however,
always produce equivalent equations (cf. El. Alg. p. 143).
(1) Transposing and uniting terms (Axioms 1 and 2).
(2) Multiplying and dividing by any expression whicli is
not zero, and which does not contain the unknown number
(Axioms 3 and 4).
96. Literal equations. Literal equations in one unknown
number, and of the first degree, may evidently be solved by
the method already employed for numerical equations.
E.g., given the equation — — = 3; to find x.
hah
On multiplying through by ah, to clear of fractions, the given
equation becomes
ax-hx-2W = a^-Zah. [Ax. 3
Hence ax — hx = a- — 'S ah -\-2 h^, [Ax. 1
i.e., {a — h)x = or — 3 a6 + 2 ft^j
and, therefore, x = ^'-3a6 + 2 h^ r^^ ^
a — b
= a-26.
96-96] SIMPLE EQUATIONS 129
Check. On substituting a — 2 b for x in the given equation,
we obtain
a — 2b a — 2b-\-2b ^a g
b a b '
in which the first member readily simplifies, and becomes - — 3;
hence a — 2 6 is a root of the given equation.
EXERCISE LXV
Solve, and check as the teacher directs :
1.3cx — cl = 2d. 11 i_l_i
2. (a-b)x = 3b-Sa. * e /
3. 2 ax = 0? — ex. 12_ _1 i_ _- J
4. «6- + c.s = a2-c2. ^-/ ^+/
c 7 ^
a
^ -|^ 14. (6 — c)aj — (a4-&)a; = c? — 20.
^' d^^~ ■^^d' 15. ^^^lA^4.^ = 16 + i^.
7. z-Saz=(l-Say. b a a
a cx-c' + id' = 2dx. 16- 3d(aj + 3cd)=c(c2-a^).
9.
cz + d z + 9cd 17. ^-2a&_-^^^-3c
3(^
ab
10. n^a;-a; = n-a;-e. la ^nl^ + ^ ^ ^^!±iA\
2 6 a a6
19. Solve Ex. 1 for c ; for d. Similarly, solve Exs. 2, 5, 9, and
11 for each letter in turn, and check your results.
Solve the following equations :
20. b(c — x)-\-a(b — x) — b(b — x) = 0.
21. ah + bh = 3ab-3 s{a'b + ab"").
22. (a —b)(x — c) — (b — c)(x — a) = (c — a)(a; — &),
^^. — 1 1 = 7H 1 r-i-
b a c b c a
24. What is a literal equation ? a numerical equation ? To
which class does 2 u; — 13 + ax = 14 a? belong ?
130 HIGH SCHOOL ALGEBRA [Ch. IX
25. Each member of the equation m- — 5 m — 24 = — 3 (m — 8)
is divided by m — 8 ; are the quotients equal ? Why ? Show-
that the new equation is not equivalent to the given equation.
97. Fractional equations. Equations containing expres-
sions that are fractional in the unknown number (§ 34) are
called fractional equations. The methods already employed
apply to such equations also.
3 15 1
Ex. 1. Given the equation - — - = [■-'•> to find the value of x.
X 2 3x 6
Solution. Clearing of fractions by multiplying each member
by 6 Xj the L. C. M. of the denominators, we obtain
18-3x = 10 + »,
whence x=2;
moreover, this value of x checks, hence 2 is a root of the given
equation.
Ex.2. Given = — ? — ; to find a;.
2(a;-l) 7(x + l) x + l 7(0^2-1)'
Solution. On multiplying each member by 2 • 7 (a;-|-l) • {x—1),
to clear the equation of fractions, we obtain
3-7(a; + l)-2(a;-l) = 8-2.7 (x-l)-20,
le., 21a; + 21-2a; + 2 = 112a;-112-20,
whence ^ = |>
which checks, and is, therefore, a root of the given equation.
7 af — 1
Ex. 3. Given - + — = x: to find x.
6 ar — 1
Solution. On multiplying this equation by 6 (a^ — 1), to clear
of fractions, we obtain
7 (x"" -1)+ 6 (x^ -1) = 6 x(a^ -1),
i.e., 7x^-7-\-6x^-6 = 6a^-6x,
whence 7 a^ + 6a;-13 =0,
i.e., (a;-l)(7a;H-13) = 0,
and the roots of this equation are 1 and — J/. [§ 72
On substituting these values of x in the given equation, it is
found that — -y- checks, but that 1 does not check ; hence — 4^ is
(and 1 is not) a root of the given equation.
96-97] SIMPLE EQUATIONS 131
Note. This shows that clearing an equation of fractions may introduce
extraneous roots, i.e., roots which do not belong to the given equation.
~3 _ 1
In this example the fraction might have been reduced to its lowest
x'^ — 1
terms before clearing the equation of fractions. In that case the multiplier
6(x + 1), instead of 6(x^ — 1), would have sufficed to clear of fractions;
the unnecessary factor x — 1 brought in the extraneous root 1.
No extraneous roots are brought into a fractional equation unless an
unnecessary factor is used in clearing of fractions (cf. El. Alg. § 99). Such
roots, if introduced, are always discovered in checking.
EXERCISE LXVI
Solve and check the roots :
4 0^-3 a;-f >^ ^ a; + 26 ^ _3 ?_ =0
73 6 ' '.T + 1 x-l~ '
ic-l,ir-2 11-13a; o^/-l i 1
23 12 y + 1 y
X 16 8a; 10 ^y by
10. ^(2-a;)-|(3-2a:) = ^±i^.
11. -M_H--^_=-A_ + 3.
x^ —1 x—1 x-\-l
12. Clearing Ex. 11 of fractions, we obtain a;^ — 2a;— 3 = 0;
are the roots of this equation, viz., 3 and — 1, roots of the given
equation also (cf. Ex. 3, Note) ?
13. Define a fractional equation. Which of the equations in
Exs. 4-11 are fractional ? Explain.
Solve the following, and check as the teacher directs:
10a;4-17 5a;-2 12a;-l
14.
18 9 11 a; -8
Hint. Multiply both members by 18, and combine similar terms ; then
multiply both members of the resulting equation by 11 aj — 18.
5 (^-4-4) 7 a;- 3 ^ 3 (3 a; + 1)
11 3a;H-2 22
132 HIGH SCHOOL ALGEBRA [Ch. I.\
3-2/ 8^ 2/ + 3 8(2/ + 3)
17 g-5 g-lQ ^g-4 g-9
'^ + 5 2; + 10 2 + 4 ;2 + 9'
Hint. Simplify each member before clearing of fractions.
lo> — — — ——^— •
x + 2 x + 'd x-\-Q> x-^7
19 ^~^ I -'^ — '^ _ ^ — 5 , g; — 3
03 — 2 0?- 8 x — 6 X— 4:
0^ + 2 a^-2 ^ 10-2a^ 2a; + l 8 ^ 2a;-l
^°* oj + l ic-1 x'-l ' ' 2x-l 4.0^-1 2x4-1'
21 1 + ^_ = ^±^. 25 ^-^II^-?: = 0.
• 2 2(07 + 1) cc + 6 X bx a
2 s 5 s- 3 1 ^^ 2c , 6 c(a-2a;)
22 — • =-• 26. \--=— -•
3 10s2_l 3 a x a{2-x)
23. 1 ^ 6 (1 - a?) ^ - g; ^7. ^ + ^a^ =^-^.
1 — x X x — 1 ' x^ — cx-^ax — ac c — x
3v 15 ^ 10 g
v + l 3v2 + v-2 3v-2
2 5 m ^ m + 29 _ ^
• ^_5 3m + 2 (m-5)(3m+2)
^^ 07 + 7 a, x — a x-^7 a a — x
30. — --: h —
aj + 6a x — 3a x-\-a 2a-\-x
a(b — x) o{c — x) a {c — x)
17+3 ,^18 21 _^ 100^5
„ « . X X X 3
^^- -§- + — 5- -9- + -T5—
33. If C represents the circumference of a circle whose radius
n
is Rf then - — = 7r(cf. Ex. 37, p. 67) ; solve this equation for O;
2 7?
for i? . Taking tt = 3|, find the value of it when 0—56.
34. v = --
t
37. II'=-ii'.s'.
35. at = v.
38 F^^".
36. Z> = f
39. V =u — gt.
PROBLEMS
07] SIMPLE EQrATlONS 183
Solve each of the following equations for each letter it contains:
40. i(F-32) = a
41. s = ig(2t-l).
42. - + -=-•
^ J?' /
1. Three fourths of a certain number exceeds | of it by 25.
What is the number ?
2. The sum of a certain number, its half, and its third is
.36. Find the number.
3. If f of a certain number diminished by J of that number
equals 3 more than i of the number, what is the number ?
4. The sum of two numbers is 18, and the quotient of the less
divided by the greater is \. What are the numbers ?
5. Divide the number 32 into two parts such that -^ of the
larger shall equal ^ of the smaller.
6. Divide the number 80 into two parts such that -| of the
smaller shall exceed ^ of the greater by 2.
7. Divide the number 25 into two parts such that the square
of the greater shall exceed the square of the smaller by 75.
8. Wliat number must be added to each term of the fraction
YY so that the resulting fraction shall be equal to J?
9. If a certain number is added to, and also subtracted from,
each term of the fraction |, the first result exceeds the second
by i; find the number. How many solutions has this problem ?
10. B's present age is 18 years, which is | of A's age ; after
how many years will B's age be | of A's age ?
11. The combined cost of a table and a chair is $ 11, of the
table and a picture, $ 14, and the chair and the picture together
cost 3 times as much as the table. What is the cost of each ?
12. Divide a line 28 inches long into two parts such that the
length of one part shall be | that of the other.
184 HIGH SCHOOL ALGEBRA [Ch. IX
13. A field is twice as long as it is wide, and increasing its
length by 20 rods and its width by 30 rods would increase its
area by 2200 square rods. What are the dimensions of this field
(cf. Exs. 23-24, p. 65) ?
14. An orchard has twice as many trees in a row as it has
rows. By increasing the number of trees in a row by 2, and the
number of rows by 3, the whole number of trees will be increased
by 126. How many trees are there in the orchard ?
15. An officer in forming his soldiers into a solid square, with
a certain number on a side, finds that he has 49 men left over ;
and if he puts one more man on a side, he lacks 50 men of com-
pleting the square. How many men has he?
16. A boy was engaged at 15 cents a day to deliver a daily
paper, with the added condition, however, that he was to forfeit
5 cents for every day he failed to perform this service ; at the
end of 60 days he received $ 7. How many days did he serve ?
17. A man was hired for 30 days on the following terms : for
every day he worked he was to receive $ 2.50 and board ; for
every day he was idle he was to receive nothing, and was to pay
75 cents for board. If his total earnings were $49, how many
days did he work ?
18. The square of a certain number is diminished by 9, and the
remainder is divided by 10, giving a quotient w^hich is 3 greater
than the number itself. Find the number (two solutions).
19. If a certain number is subtracted from each of the four
numbers 20, 24, 16, and 27, the product of the first two remain-
ders equals the product of the second two. What is the number ?
20. Find a fraction whose numerator is greater by 3 than one
half of its denominator, and whose value is f .
21. The numerator of a certain fraction is less by 8 than its
denominator, and if each of its terms is decreased by 5, its value
will be i ; what is the fraction ?
22. What principal at 4% interest for 3 years amounts to
f 784 (cf. Ex. 12, p. 61) ? Solve the same problem if the amount
is $ 10,140.
97] SIMPLE EQUATIONS 135
23. I invest $6000, part at 6 %, part at 5%, thus securing
a total yearly income of $ 325 ; how large is each investment?
24. A gentleman made two investments amounting together
to $ 4330 ; on one he lost 5 % , on the other he gained 12 % . If
his net gain was $ 251, how large was each investment ?
25. In a certain quantity of gunpowder, made up of saltpeter,
sulphur, and charcoal, the saltpeter weighs 6 lb. more than i of
the whole, the sulphur 5 lb. less than ^ of the whole, and the
charcoal 3 lb. less than ^ of the whole. How many pounds of
each constituent does this gunpowder contain ?
26. A boy bought some apples for 24 cents; had he received
4 more for the same sum, the cost of each would have been 1
cent less. How many did he buy ?
27. Knowing the time consumed by an automobile in making
a run of a given number of miles, how can you find the average
speed ? How, from the distance and the rate, can you find the
time ? How, from the rate and the time, can you find the dis-
tance ? Illustrate your answers (cf . Exs. 15-16, p. 61).
28. A tourist ascends a certain mountain at an average rate
of 1^ miles an hour, and descends by the same path at an aver-
age rate of 4^ miles an hour. If it takes him 6J hours to make
the round trip, how long is the path (cf . Exs. 35-36, p. 67) ?
29. A north-bound and a south-bound train leave Chicago at
the same time, the former running 2 miles an hour faster than
the latter. If at the end of 1|- hours the trains are 141 miles
apart, find the rate of each.
30. In running 180 miles, a freight train whose rate is f that
of an express train takes 2 hours and 24 minutes longer than the
express train. Find the rate of each.
31. If the freight train of Ex. 30 requires 6 hours longer than
the express train to make the run between Buffalo and New York,
how far apart are these two cities ?
32. An express train whose rate is 40 miles an hour starts
1 hour and 4 minutes after a freight train and overtakes it in
1 hour and 36 minutes. Find the rate of the freight train.
136 HIGH SCHOOL ALGEBRA [Ch. IX
33. An automobile runs 10 miles an hour faster than a bicycle,
and it takes the automobile 6 hours longer to run 255 miles than
it does the bicycle to run 63 miles. Find the rate of each.
How many solutions has the equation of this problem ? Is
each of these also a solution of the problem itself ?
34. A steamer now goes 5 miles downstream in the same time
that it takes to go 3 miles upstream, but if its rate each way is
diminished by 4 miles an hour, its downstream rate will be twice
its upstream rate. What is its present rate in each direction ?
35. A steamer can go 20 miles an hour in still water. If it
can go 72 miles with the current in the same time that it can
go 48 miles against the current, how swift is the current?
Hint. Let x = the rate of the current (in miles per hour) ; then 20 — jc =
the steamer's rate upstream, and 20 + x its rate downstream. (Why ?)
36. A man rows downstream at the rate of 6 miles an hour,
and returns at the rate of 3 miles an hour. How far downstream
can he go and return if he has 2^ hours at his disposal ? At
what rate does the stream flow ?
37. At what time between 2 and 3 o'clock are the hands of a
clock together ?
Hint. Make drawing, or use model of clock face. Let x = the number
of minute spaces over which the minute hand passes after 2 o'clock before
the two hands come together ; then — = the number of minute spaces over
which the hour hand passes in the same time (why?); and ic = -^ + 10.
(Why?) ^^
38. At what time are the hands of a clock together between
8 and 9 ? between 5 and 6 ? 6 and 7 ? 11 and 12 ?
39. At what time between 3 and 4 o'clock is the minute hand
15 minute spaces ahead of the hour hand ?
40. At what time do the hands of a clock extend in opposite
directions between 4 and 5 ? between 2 and 3 ? 7 and 8 ?
41. The tens' digit of a certain two-digit number is ^ the units'
digit, and if this number, increased by 27, is divided by the sum
of its digits, the quotient will be 6^1^. What is the number
(cf . Prob. 4, p. 64) ?
07] SIMPLE EQUATIONS 137
42. Divide 72 into four parts, such that if the first is divided
by 2, the second multiplied by 2, the third increased by 2, and the
fourth diminished by 2, the results will all be equal.
43. M can do a certain piece of work in 8 days, and N can do
it in 12 days ; in how many days can the two do it when working
together (cf. Ex. 41, p. 67)?
44. Two plasterers, A and B, working together, can plaster a
house of a certain size in 12 days, while A, working alone, can
plaster such a house in 18 days. In how many days can B alone
do the work ?
45. A reservoir is fitted with three pipes, one of which can
empty it in 4 hours, another in 3 hours, and the third in 1 J hours.
If the reservoir is half full, and the three pipes are opened, in
what time will it be emptied ?
46. The first of three outlet pipes can empty a certain cistern
in 2 hr. and 40 min., the second in 1 hr. and 15 min., and the
third in 2 hr. and 30 min. If the cistern is f full, and all three
pipes are opened, in what time will it be emptied ?
47. A can do a piece of work in 6 days, and B can do it in 14
days. A, having begun this work, had later to abandon it ; B took
his place and finished the work in 10 days from the time it was
begun by A. How many days did B work?
48. A certain number is increased by 1, and also diminished
by 1 ; it is then found that twice the reciprocal of the second
result minus 3 times the reciprocal of the first result equals \.
What is this number ? How many solutions has this problem ?
49. A picture whose length lacks 2 inches of being twice its
width is inclosed in a frame 4 inches wide. If the length of the
frame divided by its width, plus the length of the picture divided
by its width, is 3^, what are the dimensions of the picture ? How
many solutions has the equation of this problem? Is each of
these a solution of the problem also ?
50. A gentleman invested ^ of his capital in 4% bonds
(i.e., bonds yielding 4 % interest per annum), f of it in S^ %
bonds, and the remainder in 6 % bonds, purchasing all these bonds
at par. If his total annual income is $ 3412.50, find his capital.
138 HIGH SCHOOL ALGEBRA [Ch. IX
51. At what time between 9 and 10 o'clock is the hour hand
20 minute spaces in advance of the minute hand?
52. A pedestrian finds that his uphill rate of walking is 3 miles
an hour, and his downhill rate 4 miles an hour. If he walked 60
miles in 17 hours, how much of this distance was uphill ?
53. A wheelman and a pedestrian start at the same time for a
place 54 miles distant, the former going 3 times as fast as the
latter ; the wheelman, after reaching the given place, returns and
meets the pedestrian 6| hours from the time they started. At
what rate does each travel ?
54. In a mixture of water and listerine containing 21 ounces
there are 7 ounces of listerine. How much listerine must be
added to make the new mixture J pure listerine ?
Hint. Let x = the number of ounces of listerine to be added. Then
7 +x
(Wliy?)
21 +rc 4
55. In an alloy of silver and copper weighing 90 oz. there are
6 oz. of copper ; find how much silver must be added in order
that 10 oz. of the new alloy shall contain but f oz. of copper.
56. If 80 lb. of sea water contains 4 lb. of salt, how much
fresh water must be added in order that 45 lb. of the new solu-
tion may contain 1 J lb. of salt ?
57. If a mixture of water and alcohol is y% pure alcohol, how
much water must be added to one gallon of the mixture to make
a new mixture ^ pure alcohol ?
58. Solve Prob. 57 if the given mixture is 80 % pure alcohol
and the required mixture 50 % pure alcohol.
59. How much alcohol must be added to one gallon of a mixture
40 % pure to make a new mixture 75 % pure ?
60. What fractional part of a 6 % solution of salt and water
(salt water of which 6 % by weight is salt) must be allowed to
evaporate in order that the remaining portion of the solution may
contain 12 % of salt ? that it may contain 8 % of salt ? 10 % ?
07-98] SIMPLE EQUATIONS 139
61. A physician having a 6% solution of a certain kind of
medicine wishes to dilute it to a 3| % solution. What percent-
age of water must he add to the present mixture ?
62. If the specific gravity of brass is 8^,* while that of iron is 7|-,
and if, when immersed in water, 57 lb. of an alloy of brass and iron
displaces 7 lb. of water, find the weight of each metal in the alloy.
63. If, on being irtimersed in water, 97 oz. of gold displaces
5 oz. of water, and 21 oz. of silver displaces 2 oz. of water, how
many ounces of gold and of silver are there in an alloy of these
metals which weighs 320 oz. and which displaces 22 oz. of
water ? Find the specific gravity of the alloy ; also of gold.
98. General problems. Formulas. Interpretation of results.
A problem in which the known numbers are represented by
letters, instead of by arithmetical numerals, is often called
a general problem ; it includes all those particular problems
which may be obtained by giving particular values to these
letters. Some problems of this kind are given below.
Prob. 1. A yacht was chartered for a pleasure party of 12, the
expense to be shared equally ; 3 members of the proposed party
being unable to go, the share of each of the others had to be
increased by $ 2. How much was paid for the yacht ? How
much was each to pay under the original arrangement ?
SOLUTION
Let X = the number of dollars each member was to have paid,
then x-\-2 = the number of dollars each participant did pay ; hence
12 X and 9 (a? -f 2) each represent the number of dollars charged
for the yacht ;
therefore 12 a; == 9 (a; + 2),
i.e.y 12 a; = 9 a; + 18,
and therefore • x = 6j and 1 2 a; = 72 ;
hence the amount each was to have paid is $ 6, and the rental price
of the yacht is $ 72.
* This means that a given volume of brass weighs 8f times as much as an
equal volume of water.
HIGH SCH. ALG. — 10
140 BIGH SCHOOL ALGEBRA [Cii. IX
Prob. 2. Substitute p, q, and d, for 12, 3, and 2, respectively,
in Prob. 1, and solve the problem thus formed.
SOLUTION
Let X = the number of dollars each member was to have paid,
then x-\-d = the number of dollars each participant did pay; hence
2}x and (/) — q) - {x-\-d) each represent the number of dollars
charged for the yacht ;
therefore px =z {p — q) {x -\- d)= px -\- pd — qx — qd ;
whence x = ^^^ ~ -^^ , the amount each was to pay,
and px=:p • ^^~^\ the rental price of the yacht.
Remark. The solutions of Probs. 1 and 2 are alike except
in this: In the solution of Prob. 1 the numbers given in that
problem (12, 3, and 2) have, by combining, completely lost their
identity before the result is reached ; but in the solution of Prob.
2 the given numbers {p, q, and d) preserve their identity to the end.
For this reason the result in Prob. 2 may be used as a formula,
by means of which the answer to Prob. 1, or to any like problem,
may be immediatel}^ written down.
E.g., substituting 12, 3, and 2 for p, q, and d respectively, in
the solution of Prob. 2, gives the answer to Prob. 1.
The solution of Prob. 2, therefore, includes that of Prob. 1.
The first problem, and all like numerical problems, are merely
particular cases of the second, which is called a general problem.
Prob. 3. Divide m golf balls into two groups, in such a way
that the first group shall contain n balls more than the second.
Solution. Let a? = the number of balls in the first group.
Then m — x = the number of balls in the second group,
and, therefore, by the condition of the. problem,
x = m — x-\-n]
whence x = -, the number in the first group,
2
m — x = m
second group,
and m — x = m ^~- = — — ^, the number in the
98] SIMPLE EQUATIONS 141
As in Prob. 2, so here, the general solution may be employed to solve any
particular problem of the same kind. For example, if m = 30 and n = 4,
then the two groups contain, respectively, — ^^ and — ^^^— balls, i.e., 17 and
18 ; while, if w = 10 and w = 2, then the two groups contain 6 and 4 balls,
respectively.
If, however, m = 10 and n = 14, then the number of balls in the two groups,
as given by the above solution, is — ^ — and - — ^^- — , respectively, i.e., 12
and — 2 ; but since there cannot be an actual group containing — 2 golf
balls, therefore this last problem is impossible, and the impossibility is indi-
cated by the negative result.
Eemark. Some problems admit of negative results, and some
do not, just as some problems admit of fractional results, while
others do not. The nature of the things with which any particu-
lar problem is concerned will always make it evident whether or
not fractional or negative solutions are admissible.
Prob. 4. Two boys, A and B, are running along the same
road, A at the rate of a, and B at the rate of h, yd. per minute ;
if B is m yd. in advance of A, and if they continue running at the
same rates, in how many minutes will A overtake B ?
Solution. Let i)j = the number of minutes that must elapse
before A overtakes B. Then by the conditions of the problem,
ax = hx 4- m,
whence x = , the number of minutes before
A overtakes B. ^ ~
As in the two previous problems, so here, the general solution may be
employed to solve any particular problem of the same kind.
QO
E.g.,\ia = 280, h = 270, and m = 90, then x = — = 9; i.e., A will
'' ' ' ' ' 280 270
overtake B in 9 minutes.
Again, if a = 280, b = 280, and m = 90, then x = = '— ; i.e., an
'280 - 280
infinite number of minutes will elapse before A overtakes B ; in other words,
A will never overtake B. Compare § 41 (iii), also Ex. 7, p. 53.
QO
But if a =280, b = 290, and m = 90, then x = — = -9; i.e., the
280 - 290
two boys are together — 9 minutes from the moment they were observed,
i.e., the two boys loere together 9 minutes ago.
Let the pupil show that this interpretation of the negative result accords
fully with the physical cuuditions of the problem.
142 HIGH SCHOOL ALGEBRA [Ch. IX
Prob. 5. The present ages of a father and son are respectively
50 and 20 years ; after how many years will the father bfe 4 times
as old as the son ?
Solution. Let x = the number of years from now to the time
when the father's age shall be 4 times that of the son. Then, by
the conditions of the problem,
50 + a; = 4(20 + a;),
whence a? = — 10.
This means that 10 years ago the father's age was 4 times the
son's.
N.B. The general problem of which Prob. 5 is a particular case, may be
stated thus : The present ages of a father and son are, respectively, m and n
years ; after how many years will the father be p times as old as the son ?
EXERCISE LXVII
6. The sum of two numbers is a, and the larger exceeds the
smaller by b. What are the two numbers ?
7. By substituting in the formula obtained from the solution
of Prob. 6 above, solve Probs. 6' and 7, p. 64. Could Prob. 16,
p. 65, be solved by means of the same formula ?
8. Is Prob. 9, p. 64, a particular or a general problem ? Why ?
Make a general problem which shall include this one as a par-
ticular case. Solve the new problem and thus find a formula by
which Prob. 9, p. 64, may be solved.
9. Answer the questions in Ex. 8 above, supposing them to
have been asked with regard to Probs. 4 and 12, p. 133.
10. Which of the following admit of fractional results : Probs.
14, 15, 18, p. 134; Probs. 24-26, p. 135?
11. Do any of the problems mentioned in Prob. 10 above admit
of negative results ? Explain.
12. By a slight change in the wording of Prob. 5 above^ make
an equivalent problem whose answer shall be positive. This an-
swer should agree with the interpretation of the negative result
oriven in Prob. 5.
98] SIMPLE EQUATIONS 148
13. By slightly changing the wording in the last particular
case under Prob. 4 above, make an equivalent problem whose
answer shall be positive.
14. What principal at c % for t years will earn i dollars simple
interest ? By substituting in your answer, find the principal
when c = 5, I = 270, i = 3 ; also, when c = 3i, i = 224, t = 8.
15. A father is now m times as old as his son; in p years, the
father's age will be n times that of the son. Find the present
age of each. Also interpret your result when m is less than n.
Is p positive or negative in this case ?
16. Solve the equation of Prob. 2 above for d, and then find
the value of d corresponding to p = 12, g = 2, ic = 4. May d be
fractional in value ? negative ? Explain.
17. M can do in a days a piece of work which N can do in 6
days. In how many days can they do it when working together ?
Use this answer to solve Prob. 43, p. 137.
18. A merchant has two kinds of sugar worth, respectively, a
and h cents a pound. How many pounds of each kind must he
take to make a mixture of n pounds worth c cents a pound ?
19. How many solutions has Prob. 18 if a = 5 = c? ifa = 6
while c differs from a ? Does the answer to Prob. 18 show these
facts [cf. § 41 (iii) and (iv)] ?
20. An alloy of two metals is composed of m parts (by weight)
of one to n parts of the other. How many pounds of each of the
metals are there in a pounds of the alloy ?
21. A bell made from an alloy of 5 parts (by weight) of tin to
16 of copper, weighs 4200 lb. ; how many pounds of tin and of
copper in the bell ? How is Ex. 22 related to Ex. 21 ?
22. At what time between n and n + 1 o'clock will the hands
of a clock be together ? By means of your answer write down
the answers to Prob. 38, p. 136.
23. At what time between n and w + l o'clock will the hands
of a clock be pointing in opposite directions if n is less than 6 ?
if n is greater than 6 ? if n equals 6 ? By means of your
answer write down the answers to Prob. 40, p. 136.
CHAPTER X
SIMULTANEOUS SIMPLE EQUATIONS
I. TWO UNKNOWN NUMBERS
99. Indeterminate equations. A simple equation in one
unknown number has but one solution (i.e., one root, cf.
Chapter IX), but an equation that contains two or more un-
known numbers has many solutions.
E.g., in the equation 3x+2y = 6, which, when solved
for y, becomes ^ o
^' b — ox
we see that if the values 1, 2, 3, — 1, etc., are assigned to x, then
// will take the corresponding values 1, — ^, — 2, 4, etc. That is,
this equation is satisfied by the pairs of numbers :
a; = 11. x = 2 1. » = 3 1. ^ = -11. ^^^
An equation, such as the one just now considered, which
has an infinite number of solutions, is for that reason called
an indeterminate equation.
EXERCISE LXVm
By the method of § 99 find five solutions of each of the fol-
lowing equations :
1. a; + 3 2/ = 7. 3. 5 a; + 3?/ = 11. 5. 2^y = 5 + 32;.
2. a; 4- 2/ = 5. 4. 5 m + 2 ti = 15. 6. v — vj — 1.
7. How many solutions has each of the above equations ?
Why ? What are such equations called ?
8. How many positive integral solutions (i.e., solutions in
which both x and y are positive integers) has the equation
3a; + 22/=ll?
Hint. Solve the equation for y, and thus show that x cannot exceed 3.
144
99-100] SIMULTANEOUS SIMPLE EQUATIONS 145
9. By the method of Ex. 8 find four positive integral sohi-
tions of the equation 2x-\-y = 9. How many such solutions has
this equation ?
10. If possible, find positive integral solutions of the equations
in Exs. 1-6 above.
Show that the following have no positive integral solutions :
11. 2x-4:y = l. 12. 3x + 6y = 5. 13. 9x-{-3y = 17.
14. Find three solutions of the equation 2x — 5y + Sz = 6',
also, three solutions of the equation 2 x + 3y -\-4:Z = 20.
15. A farmer spent $22 in purchasing two kinds of lambs, the
first kind costing him $3 each, and the second kind $5 each.
How many of each kind did he buy ?
Hint. Let X = the number of the first kind, and y = the number of the
second kind ; then Sx -\- 5y = 22, where x and y are positive integers.
16. A man spends $300 for cows and sheep costing, respec-
tively, $4:5 and $6 a head; how many of each does he buy?
17. In how many ways may a 19-pound package be weighed
with 5-pound and 2-pound weights ?
18. How many pineapples at 25 cents each, and watermelons
at 15 cents each, can be purchased for $2.15?
19. Divide a line which is 100 feet long into two parts, one of
which shall be a multiple of 11 feet, the other of 6 feet.
20. Find the least number which when divided by 4 gives a
remainder of 3, but when divided by 5 gives a remainder of 4.
100. Simultaneous equations. Independent equations.*
The equations Sx-\-2i/ = 5
and x—2y = l^
have, individually, an infinite number of solutions (cf . § 99) ;
they also have 07ie solution, viz., x=3 and «/= —2, in common ;
I.e., these values of x and y satisfy each of the given equations.
xA set of equations, like those above, having one or more
* If time permits, read §§ 137-140, also § 142, in connection with §§ 100-101.
This plan will make the definitions, and also the operations, more concrete.
146 HIGH SCHOOL ALGEBRA [Ch. X
solutions in common, is usually called a system of simul-
taneous equations.
Simultaneous equations are often called consistent equa-
tions, while two equations which have no solution in com-
mon are called inconsistent equations. Thus, x-\-^ = 4: and
2 a; -f 2 ^ = 9 are inconsistent equations.
Two or more equations, no one of which can be derived
from the others, are called independent equations. Thus,
3 a?+^ = 11 and 7 x — i/=9 are independent ; but 3 a;-|-^ = ll
and 6 a; H- 2 ?/ = 22 are not independent, the second being
obtained by multiplying each member of the first by 2.
101. Solving simultaneous equations. The solving of a
system of simultaneous equations is the process of finding
the solutions which these equations have in common.
x-\-y = 4., (1)
Ex. 1. Solve the equations ,
^ 'x-y = 2. (2)
Solution. Adding these two equations, member to member
(Ax. 1), gives 2i. = 6,
whence x = 3.
Substituting this value of x in Eq, (1) gives
whence 2/ = 1-
Moreover, these numbers, viz., x = S and y = l, when substi-
tuted in the given equations, check; therefore they constitute a
solution of these equations.
3x-\-2y = 26, (1)
Ex. 2. Solve the equations ,
^ '5x + 9y = SS. (2)
Solution. On multiplying both members of Eq. (1) by 5, and
of Eq. (2) by 3, these equations become, respectively,
15aj-f-10i/ = 130, (3)
15ic + 27?/ = 249; (4)
and (Ax. 2) subtracting Eq. (3) from Ex. (4) gives
17 2/ = 119,
whence y=T,
100-102] SIMULTANEOUS SIMPLE EQUATIONS l-iT
Substituting this value of y in any one of the equations con-
taining both X and y gives
a; = 4;
and since these numbers, viz., x = 4 and y = 7, check, therefore
they constitute a solution of the given system of equations.
102. Elimination. Any process of deducing from two or
more simultaneous equations other equations which contain
fewer unknown numbers is called elimination. Such a process
eliminates (i.e., gets rid of) one or more of the unknown
numbers, and thus makes the finding of a solution easier.
That particular plan of elimination which was followed in
the examples given in § 101 is known as elimination by
addition and subtraction. It is evident, moreover, that this
method is applicable to any pair of such equations. The
procedure may be formulated thus :
(1) Multiply the given equations hy such numbers as will
make the coefficient of the letter to he eliminated the same (in
absolute value) in both equations.
(2) Subtract or add these last two equations (according as
the terms to be eliminated have like or unlike signs).
(3) Solve the resulting equation for the unknown number
which it contains.
(4) Substitute that value in any one of the earlier equations
and thus find the other unknown number o
(5) Check the results.
Note. Number (2) above is permissible only because the letters have the
same value in both equations (of. § 101).
EXERCISE LXIX
Solve each of the following systems of equations and check
the results :
{2x
4.
-2/ = 5, g ja; + 3y=ll,
H-32/ = 17. ' |3a;-4.y = 7.
= 15,
= 33,
|a;4-2?/ = 9, {2v-\-du
1 2 a; + 2/ = 15, ' [4^ + 9?^
148 HIGH SCHOOL ALGEBRA [Ch. X
3a; + 7?/ = 6. [12x-9y = 0.
2x-\-5y=:S, f5s + 6f = 17,
8. ^ _ _ . _ 11.
7 a: + 10 2/ =-17. [6.9 + 5^ = 16.
15 oj + 77 2/ = 92, f 4 m - 15 7i = 32,
9. i 12. '
5aj-32/=2. [10m-9w = -34.
13. What is meant by saying that two equations are simul-
taneous ? consistent? inconsistent? independent? Show the
appropriateness of these terms.
14. If in two simultaneous equations the coefficients of the
letter to be eliminated are prime to each other (cf. Ex. 11), what
is the simplest multiplier for the first equation ? for the second ?
Answer the same questions when the coefficients under considera-
tion are 7iot prime to each other (cf. Ex. 12).
Solve the following systems of equations and check the results :
5p-{-3q=6S, {3m-2n = 7,
15. r. . ' 19.
2^ + 5^ = 69. [4 m — 7w=— 47.
22 .T- 8 ^ = 50, (4r-f 5s=-19,
26x-\-6y = 175. ' [2 r-\-3 s= -10^-^.
15 it- -f 14 ?/ = - 45, f 35 a; - 27 ?/ = - 19,
25 a^ - 21 2/ = - 75. [ 21 2/ -f- 40 a; = 82
18 t^ + 10 V = 59, f 28 a; - 23 y = 33,
18. <! _ _ _ 22. ' -^
12w-15v = 28i. [63 a; -25 2/ = 199.
103. Other methods of elimination. Besides the method of
elimination described in § 102, there are several other methods
that serve the same purpose ; two of these, which are often
useful, will now be explained.
(i) Elimination by substitution.
3x-Ay = 7, (1)
Ex. 1. Solve the system of equations , ^ ^ ^
^ ^ '2x + 3y = 16. (2)
102-108] SIMULTANEOUS SIMPLE EQUATIONS 149
SOLUTION
From Eq. (1) x = ^^^; [§ 99
o
on substituting this expression for cc, Eq. (2) becomes
2(^) + 32,= 16, (3)
whence (§ 94) 2/ = 2;
on substituting this vahie in either Eq. (1) or Eq. (2), we obtain
X = r).
Moreover, these values, viz., x = 5 and y = 2, check ; therefore,
they constitute a solution of the given system of equations.
This method of elimination is known as elimination by
substitution ; it is manifestly applicable to any such system
of equations as the above.
The student may solve, by this method, the system
I 8 M - 4 u = 19,
1 5 w + 2 V = 10,
being careful to check the result, and then write out a "rule" for applying
this method to all such exercises.
(ii) Elimination by comparison.
r. . . n . f3a;-42/ = 7, (1)
Ex. 2. Solve the system of equations <
-^ ^ \2x-{-3y = 16. (2)
SOLUTION
From Eq. (1), x = I^til, and from Eq. (2), x= l^Llzll . Now,
since a? is to have the same value in each of these equations,
therefore Tj^^lfi-l^. ^3^
Solving Eq. (3) gives y = 2,
whence, substituting this value in either of the given equations,
X = 5.
Moreover, these values, viz., x = 5 and y = 2, check ; therefore,
they constitute a solution of the given system of equations.
This method of elimination is called elimination by com-
parison ; it is applicable to all such systems of equations.
150 niGll SCHOOL ALGEBBA [Ch. X
The student may solve, by this method, the system
I 8r + 5s = 3,
ll2r-7s=48,
and then write out a " rule " for applying this method to all such exercises.
EXERCISE LXX
Solve the following systems of equations, using first elimina-
tion by substitution, and then that by comparison ; observe which
method is easier in the different exercises :
• 5a;-2^ = 10.
4. ^ - 7.
llx-lOy
= 14,
ox+1 y=
41.
21 2/ H- 20 a;
= 165,
772/-30«^
= 295.
8 ^ - 10 y =
14,
6^ + 35 'y =
:41.
4x + 2/ = 34,
42/ + a^ = 16.
^ |2a^ + 72/ = 34,
■ l5a; + 92/ = 51.
9. Using the method of elimination by comparison, solve each
of the systems of equations in Exs. 3-6, p. 147.
10. Using the method of elimination by substitution, solve each
of the systems of equations in Exs. 7-10, p. 148.
11. Show that elimination by comparison is merely a special
case of elimination by substitution.
Solve Exs. 12-21 below ; use the simplest method of elimination
in each case, giving reasons for your choice of method ; and check
your results as the teacher directs :
7ic + 42/ = l, r8?/-21v = 5,
±2. { ^ , ' 15. i '
9a; + 42/ = 3. [6ii + 14'y = -26.
^^ 3a; + 52/ = 19, ^^ |34a^-152/ = 4,
5a;-42/ = 7. * [ 51 a; + 25 2/ = 101. •
14.
f aj - 11 ?/ = 1, I 39 .'c - 1 5 2/ = 93,
jlll7/-9a? = 99. ^^' [6535 + 17 2/ = 113.
103-104] SIMULTANEOUS SIMPLE EQUATIONS
151
18.
19.
19s + 85^ = 350,
17 s + 119 ^ = 442.
j8.9-llw = 0,
[258-17 w; = 139.
20.
21
I
3a;-ll^ = 0,
19a;-19 2/ = 8.
aj + 9 2/+42 = 0,
25 2/ + a; 4- 20 = 0.
104. Equations containing fractions. The solution of si-
multaneous equations containing fractions is illustrated by
the following examples :
-2
Ex. 1. Given
3
13__2/
4-¥ = 4i
; to find a? and y.
Solution. On multiplying these equations by 12 and 6, re-
spectively, and collecting terms, we obtain
4 a; + 3 2/ = 29,
and 3 ic + 4 2/ = 27 ;
whence (§ 101) x = 5 and y = S.
Moreover, when substituted in the given equations, these values
check ; they are, therefore, the solution of those equations.
Ex. 2. Given
3 s r r (?
1-5 = 0.
16
3s)
> ; to find r and s.
Solution. On multiplying these equations by r (r — 3 s) and
3, respectively, they become r + 4 (r — 3 s) == 16,
and r_3_3s = 0;
whence (§ 101) r = 4 and s — \.
When substituted in the given equations, these values check ;
they are, therefore, the solution sought.
Ex. 3. Given
1 + 1 = 3
?-? = !
to find u and v.
162
HIGH SCHOOL ALGEBRA
[Cn. X
Solution. Instead of clearing of fractions here, it is better
to treat - and - as the unknown numbers ; we may even substi-
tute a single letter for each of these unknown fractions.
Thus, on substituting x for - and y for -, the given equations
u V
become
x + y = 3,
and
2x-3y=l,
respectively.
Whence
x = 2 and y = lf
[§103
i.e.,
whence
i = 2and ^ = 1;
U V
u = ^ and v = l',
[§97
and these values are found to check.
EXERCISE LXXI
Solve the following systems of equations^ and check the results ;
eliminate before clearing of fractions when practicable :
4.
7.
M=i^'
4 2
3 + 3 '
2.
X y
6 2
6i.
-4-^-7
2 + 3" '
3^4
+ 5;2 = -4,
10.
11.
^^ + 5 = 3,
X y
?-? = l
X y
s r
^-? = 7.
s r
2r + 3t , ^+6
5 7
2r — bt , r + 7
3
h-2
3
2/^-7
4
= 2,
= 1.
3
13-J
= 0,
10.
104j
BIMULTANEOUS SIMPLE EQUATIONS
153
12.
13.
14.
15.
16.
17.
18.
19.
3x-{-2y-\-6_.
4:x-2y '
3-7.v_2
2ic + l •
8 15
y
20. .
21. .
llr 5^ ^o
|l2 8=^^-
r3 6 _17
2.T"^5.y 40'
7 4 11
2x 5 ;y 120
5x + 16y 3x—4:
Sy-2x = 7.
n
0,
2w-5 3m-7
2n-3 3m + l
r 3a;-2y + | _16
a; -2/
15 + y — 2a; _g
4a7— 5?/— 2
2a;H-i/-50 = 0,
? = ?l4-3.
4 3
5 + ? = 20,
a; y
^ + ^ = 10.
a; i/
3'
2v IV
= -3,
:|- + - = 23.
2^ w
22.
23
24.
25.
26.
-4s + i^-? = 0.
7« 7
|-i(2/-2)=K^-3),
^-i(2/-l)=i(^-2).
0,
^ + '
x-2 3-y
x-1 _2y + 11
6 5.5
.2y + .5 ^ .49a;-.7
1.5 4.2 '
.5 a;-. 2 ^41 1.5.^
1.6 16
11
27.
'5« + 6:
/ + 13_
3
42/-2
.'K + e
2'
13
1 x—y
_ 2i
x-y-
■3- .
r 1
2
1
4 ?,i + V
u — u (4 u
-f-v)'
-8
v 2
1^ 1
5
1^— V I
^(2^^-
-V)
154
HIGH SCHOOL ALGEBRA
[Cii. X
28.
29.
30.
31.
32.
7; + i (3 V - w - 1) = J- + f (to - 1),
^(4^ + 3^(;) = 3-V(7^o+24).
2^-x 2 '
^ - |f^ = ^^ + 3^-^ ^(.^ -g^ ^^^ p ^3^^
s o_5i + 2s,s — 3
2~^-~t::7'"^~2~^
2t-^s
«+l '
22,.
A or A-
a;
2
' 17-
-3a;
50 2'-
-1
+ ^ = 12--^
-2^
2
16 a; -f 19
3(^-2)
= 8?/ +
147 - 24 y
U2x I 3^ I ^^ + ^y -31 I 3a; + 4
8t/ + 7 6a;-3.i/ ^^ 4y-9
10 "^2(2/-4) "^ 5
J^J.gf., given
; to find X and y.
105. Literal equations. Literal equations may be solved
by the methods already employed in solving numerical
equations.
ax-\-by — c
hx -^ky= I
Solution. On multiplying the first of these equations by k
and the second by b, they become
aJcx + hky = cJc,
and bhx 4- bky = bl.
Subtracting member from member, we obtain
akx — bhx = ch — bl^
i.e. , (oik — bh)x = ck — bl\
ck — bl
whence
ak — bh
104-105] SIMULTANEOUS SIMPLE EQUATIONS
ir,5
If we multiply the first of the given equations by h, and the
second by a, and subtract, we eliminate x, and find
ch — al
y = •
hh — dk
Moreover, these values of x and y check, and are, therefore, a
solution of the given equations.
EXERCISE LXXII
Solve the following systems of equations and check the re-
sults ; eliminate without clearing of fractions where practicable :
I ax -f hy = m,
1.
2.
3.
4.
x + y = c,
x — y = d.
ax = by,
x-{-y = ab.
'y-\-az = 0,
by + z=^l.
X y
+ 1-2 = 0,
a b
' — ay = 0.
a b
^ + JL = a + b.
ab ab
[Hint. Let s = - and« = ^;
a
cf. Ex. 3, p. 152].
7.
X y
^4-^ = 1.
X y
HIGH 8CH. ALG.
10.
11.
12.
13.
14.
15.
I bx + ay = n.
x — y=a — b,
ax-\-by = a^ — b^.
0,
x-{-y x — y
a b
X -\-y _ x — y
a
1
c
1
b
b
a b
X y
c _a
X y
1.
ax by &
bx cy 0?
(a + &)aj4-(«+c)2/:
= a+6.
(a+c)a;4-(a+% =
= a-\-c.
a; + l a + & + l
^z + l^a-ft + l'
x-y=^2b.
'hx + lcy = 4.h\
1 . 1 _
h
x—k y — h k(y — h)
11
150 HIGH SCHOOL ALGEBllA [Cii. X
16. Under what circumstances has Ex. 8 above no finite solu-
tion ? Explain [cf. § 41 (iii)]. Answer this question with regard
to Ex. 9 also ; and with regard to Ex. 3.
II. THREE OR MORE UNKNOWN NUMBERS
106. Equations containing more than two unknown num-
bers. The methods already emploj^ed in the solution of
systems of equations containing two unknown numbers
(§§ 101-105) are easily extended to systems containing three
or more iinknoAvn numbers.
Thus, to solve the system of equations
r a^ + 32/-« = 5, (1)
3a^-j-62/ + 2^ = 3, (2)
[2x-3y-2z=^e>, (3)
we first eliminate some one of the unknown numbers, say 2;,
between (1) and (2), then eliminate the same unknown number
between (1.) and (3); in this way we obtain two new equations,
each containing the two unknown numbers x and y. On solving
these two equations we find x and y, and substituting their values
in (1) we find z, which completes the solution of the given system.
r x + ?.y- z = r,, (1)
Ex. 1. Given \Zx + (Sy -{-2z = ^, (2)
[2x-^y-3z=:Q', (3)
to find X, 2/, and z.
SOLUTION
Adding 2 times Eq. (1) to Eq. (2), member to member, gives
5x + 12y = l^, (4)
and subtracting Eq. (3) from 3 times Eq. (1) gives
a^-fl2y = 9. (5)
Now subtracting Eq. (5) from Eq. (4) gives
4 a! = 4,
whence x — 1.
On substituting this value of x in Eq. (5), we obtain 2/ = |; and,
with these values of x and y in Eq. (1), we obtain z = — 2.
Moreover, these values of x, y, and z check ; therefore they
constitute a solution of the given system of equations.
lOo-luej SIMULTANEOUS SIMPLE EQUATIONS 157
Note. Had the given system coiisisttd of four equations, containing four
unknown numbers, the same method of solution would still have sufficed.
For, by eliminating- some one of the unknown numbers, say x, between (1)
and (2), (1) and (3), and (1) and (4) in turn, we should have obtained a
system of three equations containing the remaining three unknown numbers,
which could then have been solved as in Ex. 1. And the vakies of these three
unknown numbers, being substituted in any one of the given equations, would
have determined the value of the remaining unknown number.
Similarly, a system consisting of five equations containing five unknown
numbers can, by eliminating some one of these, be made to depend upon a
system of four equations in four unknown numbers ; and so in general (see
also § 107).
(2x-3y-2z==-l, (1)
Ex. 2. Given 1 3 .« + 2; = 6, (2)
l-c-\-y + z = S; (3)
to find the values of x, y, and z.
SoLUTiox. Since the second of these equations is already free
from the unknown number y, therefore it is best to combine Eqs.
(1) and (3) so as to eliminate y, and thus obtain another equation
involving only x and z. On adding Eq. (1) to three times Eq. (3)
we obtain
6x + z='6, (4)
and on subtracting Eq. (2) from Eq. (4), we obtain
2a? = 2,
whence a; = l. (5)
On substituting this value of x in Eq. (2), we obtain
2^=3;
and on substituting these two values in Eq. (3), we obtain
Moreover, these values of x, y, and z, viz., 1,-1, and 3, check,
and therefore constitute a solution of the given equations.
EXERCISE LXXIH
Solve each of the following systems of equations:
(2x-\- 3 // + 4 :^ = 20, Ux-y-z = 5,
3. J3.r + 4//-f-r);^ = 26, 4. hx-Ay-\-16 = 6z,
[3x + oy-}-(Jz = 'Sl. [3y + 2(z — l) = x.
158
HIGH SCHOOL ALGEBRA
[Cii. X
5.
6.
7.
8.
9.
10.
11.
12.
13.
7x + 3y-2z = W,
2x-^5y-\-Sz = 3d,
5x — y-\-5z = 31.
5x — 6y-\-4:Z = lo,
7x-\-Ay-3z = 19,
2x-\-y-^6z = 4.6.
2x-{-Ay-\-5z = 19,
-3x-\-5y-{-7z = S,
Sx-3y-\-5z = 23.
5x-{-6y-12z = 5,
2x-2y-6z = -l,
4:X — 5y + 3z = 7^.
y^z-S6 = 72-5x,
93-ix-ly = ^^y-2z,
ix-^ly-\-lz = 5S.
ix-{-ly = 12-iz,
iy + lz==8-\-lx,
2x-5y-\-19 = 0,
3y-4:Z + 7 = 0,
2z-5x-2 = 0.
5 4"^5~ '
4 3^2
X y
1 + 1 = 8.
2 a;
10,
14.
15.
16.
17.
18.
.S 2 1
X y z
1+^=1.
x + z = 3a-\-h,
x-\-3y = ^c,
y-{-2z = x,
y-{-z=^x-2d.
s + e/i+l'j^o,
X yj
0,
u-3(!.l)=..
a^y ^1
x-\-y a
yz 1
5-2(1 + 1
y ^
19.
2/ + 2; 5'
xz _\
x-{-z c
[Hint. If ^^ = 1, then
x + y a
xy y X
(2v-{-Sx + y-z = 0,
3y-2x-\-z-4:V = 21,
2z — 3v — y-\-x = 6,
v+4:X-^2y-3z = 12.
IOC]
SIMULTANEOUS SIMPLE EQUATIONS
159
20.
v-}-y + z = 17,
V -\-x + z = 16.
Hint. Adding these equations
and dividing the sum by 3 gives
'y-\-z-\-v — x = 22,
z-\-v-^x — y = lS,
V -\- X -{- y — z =14:,
x + y-\-z — v = 10.
21.
22.
23.
y-\-z — Zx = 2a,
z-v-3y = 2b,
y-\-x — 3z = 2cy
l2v-{-2y =a-b.
(3u-\-5v-2x + Sz = 2,
2u-{-4:X-3y — z = 3,
u-{-v-\-z = 2,
(Sy-\-4iV + u = 2y
5 2; H- 4 a; — 7 v = 0.
PROBLEMS
[Leading to simultaneous equations in two or more unknown numbers.]
1. Find two numbers whose difference is 3^5 of their sum, and
such that 5 times the smaller minus 4 times the larger is 39.
SOLUTION
Let
and
Then, by the conditions of the problem
X = the larger number,
?/ = the smaller number.
x-y =
~35"'
and 5 y — 4 aj = 39.
Solving these equations, we obtain
X = 54: and y = 51 ;
and these numbers, which constitute a solution of the equations of the prob-
lem, also satisfy the problem itself, and are, therefore, the numbers sought.
2. Find two numbers snch that 3 times the greater exceeds
twice the less by 29, and twice the greater exceeds 3 times the
less by 1.
3. A lady purchased 20 yd. of gingham, and 50 yd. of linen,
for $ 29 ; she could have purchased 30 yd. of gingham, and 20 of
linen, for $ 16. What was the price of each material ?
4. If A's money were increased by $ 4000, he would have twice
as much as B. If B's money were increased by $5500, he would
have 3 times as much as A. How much money has each ?
ir>o
UIGIl SCHOOL ALGEBRA
[Ch. X
5. One eleventh of A's age is greater by 2 years than 1- of B's,
and twice B's age equals what A's age was 13 years ago. Find
the present age of each.
6. ABC represents a triangle whose perimeter is
82 inches. If AB = BC and 7 5(7= 17 AC, find the
length of each side of the triangle.
7. A man having $45 to distribute among a group
of children, finds that he lacks $1 of being able to
give $ 3 to each girl and $ 1 to each boy, but that he
has just enough to give $2.50 to each girl and $1.50
to each boy. How many boys and how many girls are
there in this group ?
8. John said to James, " Give me 8 cents and I shall have as
much as you have left." James said to John, " Give me 16 cents
and I shall have 4 times as much as you have left." How much
money had each ?
C 9. ABCD represents a flower bed in which
BC = ^ AB. If the perimeter of the bed is
40 feet, find the length of each of its sides.
10. A pound of tea and 6 lb. of sugar
together cost $ .96 ; if sugar were to advance
50%, and tea 10%, then 2 lb. of tea and
12 lb. of sugar would cost $2.28. Find the present price of tea,
and also of sugar.
11. A grain dealer sold to one customer 5 bushels of wheat,
2 of corn, and 3 of rye, for $6.60; to another, 2 of wheat, 3 of
corn, and 5 of rye, for $5.80; and to another, 3 of
wheat, 5 of corn, and 2 of rye, for $5.60. What was
the price per bushel of each kind of grain ?
12. The perimeter of the triangle CDE is 68 in. ;
four times CE equals CD increased by four times DE,
while twice CE equals DE increased by twice CD.
How long is each side of the triangle?
106] SIMULTANEOUS SIMPLE EQUATIONS 161
13. Divide 800 into three parts such that the first, plus -J- of
the second, plus |- of the third, shall equal the second, plus f of
the first, plus J of the third : each of these sums being 400.
14. Divide 90 into three parts such that ^ of the first, plus -|
of the second, plus i of the third, shall be 30 ; while the first part
increased by twice the second shall equal twice the third.
15. A boy spent $ 4.10 for oranges, buying some at the rate of
2 for 5 cents, some at 3 for 10 cents. Later he sold all at 4 cents
apiece, thereby clearing $ 1.58. How many of each kind did he
buy ?
16. If a certain rectangular floor were 2 ft. broader and 3 ft.
longer, its area would be increased by 64 sq. ft. ; but if it were
3 ft. broader and 2 ft. longer, its area would be increased by 68
sq. ft. Find its length and breadth.
17. Three rectangles are equal in area ; the second is 6 meters
longer and 4 meters narrower than the first, and the third is 2
meters longer and 1 meter narrower than the second. What are
the dimensions of each ?
18. The sum of the ages of a father and son will be doubled in
25 years ; the difference of their ages 20 years hence will just equal
^ of their sum at that time. Find the present age of each.
19. A merchant sold to Mrs. A. 2 yd. of cambric, 4 of silk, and
3 of flannel, for $5.05, and to Mrs. B., 4 yd. of cambric, 5 of
flannel, and 2 of silk, for $4.30. If 2 yd. of flannel cost 10 cents
more than 2 yd. of cambric and -J yd. of silk combined, find the
price of each per yard.
20. The tickets to a concert were 50 cents for adults and 35
cents for children. If the proceeds from the sale of 100 tickets
were $39.50, how many tickets of each kind were sold ?
Solve this problem also by using but one letter to represent an
unknown number.
21. Find three numbers such that the sum of the reciprocals of
the first and second is y\, the sum of the reciprocals of the first
and third is -f-g, and the sum of the reciprocals of the second
and third is ||.
162 HIGH SCHOOL ALGEBRA [Cii. X
22. The sum of the reciprocals of three numbers is 34 ; the re-
ciprocal of the second minus that of the third equals 4 ; the sum
of 3 times the reciprocal of the first and twice the reciprocal of
the second is less by 1 than 5 times the reciprocal of the third.
Find the three numbers.
23. In a certain two-digit number which equals 8 times the sum
of its digits, the tens' digit exceeds 3 times the units' digit by 1.
Find the number.
24. The sum of the digits of a two-digit number is 12, and if
the digits are interchanged, the number thus formed will lack 12
of being twice the original number. What is the number ?
25. The sum of the digits of a 3-digit number is 11 ; the double
of the second digit exceeds the sum of the first and third by 1,
and if the first and second digits are interchanged, the number
will be diminished by 90. What is the number ?
26. The third digit of a 3-digit number is as much larger than
the second as the second is larger than the first ; if the number
is divided by the sum of its digits, the quotient is 15 ; and the
number will be increased by 396 if the order of its digits is
reversed. What is the number ?
27. A capitalist invested ^4000, part at 5%, part at 4%, and
found that his annual income from this investment was ^175.
How much was invested at 5 %, and how much at 4 % ?
Solve this problem also by using only one unknown letter.
28. A capitalist invested A dollars, part at p %, part at q%,
and found that his annual income from this investment was B
dollars. How much was invested at p % ? at g % ?
Show that this problem includes Prob. 27 as a special case.
29. Divide the number N into two such parts that 1/m of the
first part, plus l/^i of the second, shall exceed the first part by M.
Specialize this problem, and find the solution of the special
problem by substituting in the general solution.
30. Three cities. A, B, and C, are situa,ted at the vertices of a
triangle ; the distance from A to C by way of B is 50 miles, from
A to B by way of C is 70 miles, and from B to C by way of A is
60 miles. How far apart are these cities ? (Make diagram.)
106] SIMULTANEOUS SIMPLE EQUATIONS 163
31. In the triangle ABC, AB = 12 inches, BC = 10
inches, BE^BF, FC=GG, AG = 4.iBF. If the
perimeter of the triangle is 42, find AG, AE, BE, FC.
32. A quantity of water which is just sufficient to
fill three jars of different sizes, will fill the smallest
jar exactly 4 times ; or the largest jar twice, with 4
gallons to spare; or the second jar 3 times, with 2
gallons to spare. Find the capacity of each jar.
33. Two men, A and B, rowed a certain distance,
alternating in the work ; A rowed at a rate sufficient to cover the
entire distance in 10 hours, while B's rate would require 14. If
the journey was completed in 12 hours, how long did each row ?
34. Two boys, A and B, run a race of 400 yards, A giving B a
start of 20 seconds and winning by 50 yards. On running this
race again. A, giving B a start of 125 yards, wins by 5 seconds.
What is the speed of each ? Generalize this problem.
35. If A and B can do a certain piece of work in 10 days, A
and C in 8 days, and B and C in 12 days, how long will it take
each to do the work alone ?
36. A and B together can build a wall in 5^^ days; being
unable to work at the same time, A works 5 days, then B takes up
the work, finishing it in 6 days more. In how many days could
each have built the wall alone ? Generalize this problem.
37. A man can row m miles downstream in c hours and m
miles upstream in d hours; what is his rate of rowing in still
water, and what is the rate of the current ?
38. From the solution of Prob. 37 find the solution of the
special problem in which m = 6, c = 1^, d = 4.
39. Two trains whose respective lengths are 1200 feet and 960
feet run on parallel tracks; when moving in opposite directions,
the trains pass each other in 24 seconds ; when moving in the
same direction, each at the same rate as before, the faster passes
the slower in 1^ minutes. Find the rate of each train.
164 HIGH SCHOOL ALGEBRA [Ch. X
40. Two trains are scheduled to leave the cities A and B, m
miles apart, at the same time, and to meet in h hours ; but, the
train from A being a hours late in starting, and running at its
regular rate, the trains met k hours later than the scheduled time.
What is the rate at which each train runs ?
41. From the solution of Prob. 40 iind the sohition of the
special problem in which m = 800, /i = 10, a = If , and k = Jq.
42. A train was scheduled to make a certain run at a uniform
speed. After traveling 2 hours it was delayed 1 hour by an
accident, after which it proceeded at -y- its usual rate and arrived
^ hour late. Had the accident occurred 36 miles farther on, the
train would have been 36 minutes late. Find the usual rate of
the train and the entire distance traveled.
43. Two boats which are d miles apart will meet in a hours if
they sail toward each other, and the second will overtake the first
in b hours if they sail in the same direction. Find the respective
rates at which these boats sail. Also discuss fully your solution,
i.e., interpret the results (cf. Prob. 4, p. 141).
44. Find an expression of the form ax^ -\-bx-\-c whose value is
6 when x = 2, 3 when x= —1, and 10 when a? = 4.
Hint. 4 a + 2b + c is the value of ax^ + bx -\-c when x = 2 ; therefore,
4a+26 + c = 6, etc.
45. Find an expression of the form ax^ + &x + c whose value is
7 when x = S,9 when x= —1, and 17 when x = — 5.
46. Find an expression of the form ax^ + bx^ -\- ex -\-d which
equals — 16 when x= —1,-4 when x = l, — 43 when x = — 2, and
— 100 when x= — 3.
47. Of three alloys, the first contains 35 parts of silver, to 5
of copper, to 4 of tin ; the second, 28 parts of silver, to 2 of
copper, to 3 of tin ; and the third, 25 parts of silver, to 4 of copper,
to 4 of tin. How many ounces of each of these alloys melted
together will form 600 oz. of an alloy consisting of 8 parts of
silver, to 1 of copper, to 1 of tin ?
10(5-107] SIMULTANEOUS SIMPLE EQUATIONS 1()5
48. If Prob. 47 demanded merely that the alloy should contain
8 parts of silver to 1 of copper (without specifying the amount of
tin), how many ounces of each of the given alloys would then be
required ? Why is this problem indeterminate (of. § 107) ?
107.* Determinate and indeterminate systems of equations.
As we have already seen, a system containing as many inde-
pendent equations as unknown numbers, can always be solved, i.e.,
the unknown numbers can be determined (§§ 101-106). Such a
system is, therefore, a determinate system.
On the other hand, a system in which there are fewer inde-
])eudent equations than unknown numbers is an indeterminate
system. It is easy to show that this statement — already seen
to be true in the case of a single equation containing two un-
known numbers (§ 99) — is true generally.
Thus, suppose we have three equations containing four unknown
numbers. By regarding one of these numbers temporarily as
kuown, we can solve the given equations for the other three ; i.e.,
we can express any three of the four unknown numbers in terms of
the fourth. To every assigned value, therefore, of this fourth un-
known number, there corresponds a set of values of the other
three (cf. § 99) ; hence the system is indeterminate.
Again, there can never be in a system more independent equa-
tions than there are unknown numbers.
For, if that were possible, suppose there are three independent
equations, viz.,
ax-{-hy = c, (1)
hx+jy^k, (2)
and Zfl? -f my = n. (3)
containing but two unknown numbers, x and y.
On solving (1) and (2) we obtain
cj — bk -, ch — ak
X = -^ and y = ,
aj — bh ' bh — aj
and on substituting these values for x and y in (3), we obtain
\aj — bh) \bh — qjj
* This article may be omitted till the subject is reviewed.
166 HIGH SCHOOL ALGEBRA ICu. X
i.e., the known numbers of these equations are not independent
(n, for example, is expressed in terms of a, b, c, Z, etc.), hence the
given equations are themselves not independent.
REVIEW EXERCISE-CHAPTERS VI-X
Find the H. C. F. and also the L. C. M. of:
1. 6a^ + 13a;-5 and3a^ + 2a.'2-f 2a;-l.
2. 12a^-29a; + 14and8a.-2-30 + 6a;*-f-lla^ + 33a;.
3. Chanere '^ ^^~ to an equal fraction whose denomi-
nator is 24 a? — 6 a^x^ ; also to an equal fraction whose numerator
is 1 — 10 ay — 5 a + 2 y.
Simplify :
go;'" — bx'^"^''- 1 + x l + o^
** a^ftic-dV * „ 1 + aJ- 1+a^
l-hx^ 1 + a^
5.
4a^
6xy-\-9y^ . Sa^-27f
r^«I-^)_yAp-l) ^ 4^
^4 _ ^ _ J ,^.2 _|_ ^, _j_ g 4 ic^ — 6 ict/
1^1 1
10.
(a-6)(6-c) (c-6)(c-d) {d-c)(b-a)
k-1 1-k
(]c - l)(k -m)(n- Tc) (I -k){n- k){k -p)
11. Why may a term be transposed from one member of an
equation to the other by merely changing its sign ?
12. When are equations conditional ? identical ? integral ?
fractional ? literal ? numerical ? indeterminate ? Illustrate each
of your answers.
Solve, and check as the teacher directs :
13. 3a;2_5i»-12 = 0. 2 -\- x ^ 19
14. 6m2-13m = -6. ' 2-a;~ 21
107]
SIMULTANEOUS SIMPLE EQUATIONS
167
16. 1---^
3^
2s
3s
X a
17
= a-\-b.
19. {x-2y+(x+oy=(x + 7y.
20. 7x + 5fl-^^ = a(x-a).
3 2
ic a
18. 6x'-\-7x^-20a^ = 0.
y — 4: ?/ — 6 _ y — 5 _ y_
21.
+
8v
2vH-5 2v-5 25-4^2
22.
2/-5 2/-7 2/-6 2/-8
23. (a? — a)(a — & + c) = (icH-a)(6 — a + c).
24. Show that while 2 is a root of the integral equation which
results from clearing — ^ + — rr- = 8 H of fractions,
x-^o {x + 5)(x-2) x-2
it is not a root of the given fractional equation. How could we
avoid introducing this extraneous root ?
25. Form the equations whose roots are: 2,-9; — 3|-, 4;
I, f ; -2a, -6a; 1,3,-7; l-c,c-l.
26. When are two equations equivalent ? inconsistent ? simul-
taneous ? independent ? Illustrate each of your answers.
27. Explain the term '^ elimination" as applied to simultaneous
equations, and outline three methods of elimination.
Solve the following systems ; check as the teacher directs :
28.
29.
7-2a; ^3
5-37/ 2'
y — x = 4:.
2x-
y
= 4,
31.
32.
30.
32, + ^ = 9.
1 + 1-1
X y 4
33.
1_
12
5 ahx -\-2y =
166,
3 a5a.' + 4 ?/ =
18 6.
a 6
3a 66 3
y — z x-\-z
1
2 4
'2'
x—y x—z
—
5 6
'-'>
?/ 4- 2; _ a) 4- ?/
4 2
-4.
168 HIGH SCHOOL ALGEBRA [Ch. X
34. If asi^ -\-hx-\-c becomes 8, 22, 42, respectively, when a;
becomes 2, 3, 4, what will it become when x becomes — \ ?
35. The sum of two numbers is 5760, and their difference is \
of the greater. Find the numbers.
36. What number added to its reciprocal gives 5.2 ?
37. It takes 2000 square tiles of a certain size to pave a hall,
or 3125 square tiles whose dimensions are 1 inch less. Find the
area of the hall floor. How many solutions has the equation of
this problem ? How many has the problem itself ?
38. Divide the number a into two parts such that the second
part shall equal n increased by m times the first part.
39. What number must be added to m and to n in order that
the first sum divided by the second shall equal p/q ? What does
your answer become when p = q"} What does this indicate
(1) when m = n, (2) when m and n are unequal ?
40. In order to build a new clubhouse, a country club assessed
each of its 200 members a certain sum ; later an increase of 50
in the membership reduced the individual assessments by ^10.
Find the cost of the proposed house.
41. At what time between 3 and 4 o'clock is the minute-hand
25 minute spaces ahead of the hour-hand ?
42. The freezing point of Avater is marked 0° on a Centigrade
thermometer, and 32° above zero on a Fahrenheit thermometer.
If 100° Centigrade = 180° Fahrenheit, find the reading on a Centi-
grade thermometer corresponding to 68° Fahrenheit. (Make a
diagram of each scale.)
43. State and solve the general problem of which Prob. 42 is
a particular case. By substitution in the formula thus obtained
express in the Centigrade scale the following Fahrenheit readings :
44°; 212°; -10°; 0°.
44. A man rows a boat with the tide 8 miles in If hr. and
returns against a tide 1 as strong in 4 hr. What is the rate of
the stronger tide ? At what rate does the man row in still water ?
107] REVIE]r EXEliCISE 169
45. A man selling eggs to a grocer counted them out of his
basket 4 at a time and had 1 e^^ left over ; the grocer counted
them into his box 5 at a time and there were 3 left over. If the
man had between 6 and 7 dozen eggs, how many must there have
been (cf. § 99) ?
46. Of two wheelmen, A and B, A starts c hours in advance
of B, and travels at the rate of a miles in h hours, while B follows
at the rate of p miles in q hours. How far will A travel before he
is overtaken by B ?
Under what conditions is this solution positive ? negative ?
zero ? infinite ? Interpret the result in each case.
CHAPTER XI
INVOLUTION AND EVOLUTION
I. INVOLUTION
108. Introductory. For the meaning of the words hase^
exponent^ and power ^ as used in algebra, see §§9, 30, and 36.
The process of raising a number or expression to any given
power is called involution.
In this chapter, as in the earlier treatment of powers,
we shall use only positive integers as exponents. Later on
(Chapter XVI), however, we shall find it advantageous to
employ such symbols as aP^ a~^^ and or also, and we shall
then assign suitable meanings to such symbols.
109. Even powers, odd powers, powers of fractions, etc. A
power of any given number is called even or odd according
as its exponent is even or odd.
From the law of signs given in § 18 it follows that :
(1) All integral powers of a positive number are positive.
(2) All even integral powers of a negative number are
positive.
(3) All odd integral powers of a negative number are
negative.
And from § 83 (i) [cf. also Ex. 30, p. 121], it follows that
a^ fmY w* .
J5' [nJ^V''-
Let pupils fully explain each of the above statements:
170
108-110] INVOLUTION AND EVOLUTION 171
EXERCISE LXXIV
1. Answer again questions 18-20 on p. 39.
2. Write that power whose base is k and whose exponent is
m — 3. Are there any limitations here on the value of A;? on
the value of m?
3. From the definition of an exponent show that a^ • ar^ = a^.
Also that 2^. 2. 22 = 2^
4. For what values of n between 1 and 10 is (—3)" • (— S)^"
positive ? Explain.
5. Show that an even power of a negative number is positive.
6. How is a fraction raised to a power (of. Ex. 30, p. 121) ?
Illustrate your answer.
Simplify each of the following expressions :
8.
(-!!)■■ - (-drj
110. Exponent laws. The following formulas state what
are known as the exponent laws. The bases (<2, 5, and c) stand
for any numbers or algebraic expressions whatever, but the
exponents are positive integers.
(i) First exponent law. a"" - a"" = a"*+«. [§ 30
For, just as a^ • a^ = (a • a • a) • (a • a)
= a^ i.e., «^+2.
so, too,
a^ ' a" =: (^a ' a ' a '" to m factors)(a - a • a ••• to n factors)
= a ' a • a '•• to (m + n) factors
Similarly, a"^ - a"" - a^ = o^^^'-^p.
HIGH SCU. ALG. — 12
172 BIGH SCHOOL ALGKhRA [Cn. XI
(ii) Second exponent law. (a'"y — a"***.
For, just as {aF)'^ = {a - a ■ a^
= (^a ' a ' a^ ' (^a ' a ' a)
= a^, i.e.^ a^'2;
so, too, (^a'^y = (a - a ' a ••' to m factors)**
= a • a • a • • • to mn factors
(iii) Third exponent law. a^ - b^ = (aby.
For, just as a^ • b^ = a • a ■ a - b - b -h
= ab • ab ' ab
^(aby;
so, too,
a"b^ = (^a • a • a ••' to n factors) - (b - b • b -" to n factors)
— ab ' ab • ab "• to n factors
= iaby.
Similarly, a^^c"" = (obey.
(iv) Fourth exponent law. a'" -r- a" = a""-". [§ 80
This law is an immediate consequence of (i) above, and
of the definition of division (§ 8), for since
therefore a"» ^ a" = «"*"".
EXERCISE LXXV
Simplify, and explain your work in each case :
7. (-5a)2.
9. (IT^S^)^
^ \2
11. ' ^
1.
a'h^ ' ah\
2.
3A-(-
2a?f).
3.
a ' 0? ' gC
■a'.
4.
5.
ccPe
6.
{x'zy.
m-
12.
X'^ ' oc^.
13.
X"" -T-Olf.
14.
(x-y.
15.
(2x'"'y,
16.
s« . s^.
17.
V^ ' V^ ' v^.
18.
c^r
110-111]
INVOLUTION AND EVOLUTION
- (-ir' - m.
- C-^J-
- (5T- - c^;
21. (H^)*.
22. (-c)2-
25. (,,,•.->)'. ^^ r.„_
26. (5 »•")». • H^a +
173
Write the following as powers of products [cf. law (iii) above] :
30. /i%2. 33. o?if. 36. a' • (2 6)»^.
31. r^s¥. 34. a^2/*- 37. 3* • ( - m)^ • (?i/.
32. c^d^ 35. — 2» . 3^ 38. ir-" • /« • 2^".
39. What does a represent in the proofs of § 110 ? May it rep-
resent a polynomial as well as a number ?
40. Translate the first exponent law (§ 110) into verbal lan-
guage (cf. § 30).
41. Translate the second, third, and fourth exponent laws into
verbal language.
42. Is (a • 6 • cf equal to a^ • 6^ • c^ ? Is (a + 6 -f c -h df equal to
a'- -f ^^ + c^ + c^^ ? Explain your answers.
43. Is [(-2)3]2 equal to [(-2)^^? Why? Is (a^)* equal to
{xy^ Why?
111. Powers of binomials. We have already seen (§§ 52
and 57) that
and (a + hy = a^-\-^a%^2>ah'^ + h^.
These powers (expansions) were obtained by direct multi-
plication, and the higher powers may, of course, be obtained
in the same way. Thus,
(a + 6)4 = ^4 4. 4 ^35 ^ 6 ^252 + 4 aJ3 4. 54^
(a -f 6)5 = ^5 + 5 a^i + 10 ^352 4. 10 aW 4- 5 aft* _|_ js^
(a 4- 6)6 = ^6 4. (3 ^56 4. 15 ^4^2+ 20 a%^+ 15 ^254 4. 5 ^554. je, etc.
174 HIGH SCHOOL ALGEBRA [Ch. XI
The following questions may serve to bring out the strik-
ing similarity of these expansions :
1. How does the exponent of the first term in each ex-
pansion compare with that of the corresponding binomial ?
2. How, in each expansion, does the exponent of a change
as we pass from term to term toward the right ?
3. In which term of each expansion does b first appear ?
How does the exponent of b change from term to term ?
4. How many terms in each expansion? What is the
sign of each term ?
5. What coefficient has the first term of each expansion ?
the second term ?
6. Multiply the coefficient of any term in any of the ex-
pansions by the exponent of a in that term, and divide this
product by the number of the term ; how does this quotient
compare with the coefficient of the next term ?
7. Assuming that the expansion of (a + by is similar in
form to the expansion of (a + ^)^ (a+i)^ etc., complete
the statement :
(a + 5)8 = a8 + 8 a^b + 28 a%^ + ....
112. Binomial theorem, (i) The answers to the first six
questioHS in § 111, when combined, may be expressed sym-
bolically thus :
(a + by = a--\-'^ a^-'b + ^(f -^) a"-2^,2
^ "^ 1 1.2
1 . ^ . o
This formula, which was discovered by the celebrated
English mathematician Sir Isaac Newton (1642-1727), is
called the binomial theorem; its correctness is proved in
§§ 206-207.
111-112] INVOLUTION AND EVOLUTION 175
(ii) Since a — 5 = aH-(— 5),
therefore
(a_5)3=[a+(-*)]3=«H3a2(-5) + 3a(-6)2 + (-6)3'
?'.e., (a — by differs from (a + b^ only in having the signs of
its even terms negative. So also for other powers of a — b, ^.e.,
(a -by=a^-^ a^-'b + ^i(!L:^^«-2j2
1.2.3 « ^ + •
EXERCISE LXXVI
Write down the expansions of the following binomials :
1. (a + xy. 5. {a-\-cy. 9. (m2 + 6)3.
2. (mH-f)^ 6. (i» + 2/)*. 10. {m-\-hy.
3. (1* + ?;)^ 7. (2/ + ;^)'. 11. (m^+fty.
4. (p + g)'. 8. {k-\-iy. 12. (m2 + 63)6^
13. Expand each of the following expressions : (x-\-yy, (^+2/)^
and (x + yy ; then multiply the first two expanded forms together
and thus verify that (x -\- yY • {x -\- yy = (a; + yy.
14. What terms in the expansion of {c — dy are negative?
Why?
15. Write the first five terms of (s — 2 ty and simplify your
result (cf. Ex. 3, p. 71).
16. How many terms are there in the expansion of (m-f-?iy?
How many in (a — by ? How many in (3 s — 2 ty ?
Write each of the following expressions in its expanded form :
17. (k-cy. 23. (4c + ^')'. 29. {v'-2y.
18. (r — sy. 24. (mii. — rsy. 30. (2 xy — ly.
19. (m — ny. 25. (ab + cdy. 31. (c + a)».
20. (c4-dy«. 26. (3a2 + c«(^)^ 32. (2 771+3)1
21. (x' + yy. 27. (A;4-l)'. 33. (2 -3 0-2^)5.
22. (2r-ay. 28. (^^-2)^ 34. (2-a'by.
176 HiGn SCHOOL algebea [Ch. xi
35. (.r-.9vy. gg_ fr-'lW 40- [.ci + (b + c)J.
36. (2xy^-\-x'yy. V ^^ 41. (c + d + e)^
37. fa-j-fl] . 39. - + - • ^ ^ ^
V a?; V« V 43. (2a^-m-l)^
44. Write the first four terms of (a + x)'-^ ; the first three terms
of (a; — y)^ ; the first three terms of (2 ax — 3 k^y.
II. EVOLUTION
113. Definitions. Here, as in arithmetic, by the square
root of any given number we mean a number whose square
equals the given number.
Thus, since T^ = 49, therefore 7 is a square root of 49.
Similarly, the third or cube root of a number is a number
whose third power equals tlie given number.
Roots are usually indicated by the radical sign ( V)i which
is a modification of the letter r, the initial letter of the Latin
word radix, meaning root. A small figure, called the index
of the root, is written in the opening of the radical sign to
indicate the particular root to be extracted. When no index
is written, the index is understood to be 2.
U.g., -yja indicates the second or square root of a, (1)
-^a indicates the third or cube root of a, (2)
and -y/a indicates the seventh root of a, (3)
and, in general, ya means the rith root of a,
t.e., i^ay = a. (4)
An indicated root is said to be an even root or an odd root
according as its index is an even or an odd number.
The process of finding a root of any given number is called
evolution ; it is the inverse of involution (cf. § 108).
Note. In practice the radical sign is usually combined with a vinculum
(§ 11) to indicate clearly just how much of the expression following the radi-
cal sign is to be affected by that sign ; thus \/9 + 16 means tlie square root
of the sum of 9 and 16, while VO + 16 indicates that 16 is to be added to the
square root of 9.
112-115] INVOLUTION AND EVOLUTION 177
114. Law of signs of roots. From the definition of root
(§ 113), and from § 109, it follows that :
1. An odd root of any number has the same sign as
the number itself. Thus, V8 = 2, and V— 8 = — 2, because
23=8 and (-2)3= -8.
2. An even root of a positive number has two opposite
values, i.e.^ one positive, the other negative. Th us, VSl = + 3
or— 3, since (+ 3)* = (— 3)*= 81. Instead of writing
V81 = -f 3 or — 3, we usually write V81 = ± 3 ; this expres-
sion is read, ''The fourth root of 81 equals plus or minus 3."
3. An even root of a negative number is neither a positive
nor a negative number. Thus, V— 9 is neither + 3 nor —3,
since ( -f 3)2 = ( - 3 )2 = + 9, and not - 9.
Note. Such indicated roots as V— 9 are called imaginary numbers
(cf. §§ 146, 164) ; all other numbers are, for distinction, called real numbers.
To provide for such roots as V— 9 we must again extend the number system,
just as we did when subtractions like 3 — 8 first presented themselves (cf .
Chap. II).
115. Roots of monomials. If a monomial is an exact
power, the corresponding root can usually be written down
by inspection.
E.g.y </S aV = 2 a% because (2 a^xf = 8 aV (§ 110) ;
■V'9^\>f=±3xy, because (+3 ay'y'y=(-Sxyy=9xY',
i/-32x'^ = - 2 x% because (-2x'y = -32 x'';
^ = 2^^ because f2^Y = 8™\
EXERCISE LXXVII
1. What is meant by the square root of a number ? Are both
5 and — 5 square roots of 25 ? Why ?
2. What are the square roots of 64 ? the fourth roots of 16 ?
Why ? If a is any ecen root of a number, then — a also is a root
(with the same index) of that number, — explain,
178 HIGH SCHOOL ALGEBRA [Ch. XI
3. What is the cube root of 27 ? of - 27 ? of 64 ? of -64 ?
Explain. How does -^32 compare with V— 32 ?
4. How does the sign of an odd root of a number compare with
the sign of the number itself ? Why ? Answer these questions
for an even root also.
5. Give the cube of each integer between 1 and 7. Name the
cube root of : - 8 ; 1000 ; - 1728 ; - f 7 . ii5 . _ 216 ; 8000.
6. What is the sign of any even power of a positive or negative
number? Can, then, an even root of a negative number be
positive ? negative ? Illustrate your answer.
7. Is — 13 a square root of 169 ? Why ? Is 5 as^ the cube
root of 125 aV ? Why ? How can you tell whether one given
number is a square root of another given number? a fifth root?
8. How do we find the exponents in the cube root of 8 o}^y?'}f ?
in the 4th root of a%^h' ? in the 6th root of m^^^^ ? in the nth
root of a^^n^in 9 Explain.
Find the following indicated roots, and check your answers.
Also, tell which are even and which are odd roots, and name the
index in each case :
9. Va'6V^ 16. V128a^^6i^. 3/ 21^ &d}''
22.
343 (c-df
10. Vl6aV2/^ 17 3 125<i/« ,
• A/ 112% ah^ 23 xl^m^hK
11. ^32^«. r^ -i, ^225
12. V-243aiV. ■ ^ 128 0^^^ ' 24.
1000 c^
125 d''
13. A/a^a^Y". 19. V J^ ^''^'"- 25 W ^''""'"^'"
14. ^'/-64^. 20 A^/ ^^^y . ^, x|aF5^^
2V
15.
5/--32_aV^ 3/ .O27 gV 4/ 8I (g-c)^^"^
\ 2432/2^ ' ' \. 064 6V' * \ 625 a*'"" »
28. Write a rule for the extraction of such roots as the above,
emphasizing particularly the matter of exponents and signs.
Does your rule apply to roots of polynomials also ?
116-116] INVOLUTION AND EVOLUTION 179
29. Is V9T16 equal to V9 • Vi6 ? Why ? Is V9Tl6 equal
to V9 + vTe ? Is VoM^ equal to Vo^^- Vfe' (cf. Ex. 42, p. 173)?
Is Va^ • h'^ equal to Va^ • V6^ ? State in words your conclusion
as to the square roots of sums and products.
116. Roots of polynomials extracted by inspection. If a
polynomial is an exact power of a binomial, or the square of
a polynomial, a little study usually reveals the corresponding
root.
Ex. 1. Find the square root of m^ + 4 m^n + 4 n^.
Solution. This expression is easily seen to be (m^H-2n)^;
therefore Vm*^ + 4 m^n -\- 4 n^ = ± (m^ + 2 n).
Ex. 2. Find the cube root of 8 a^ - 36 a^b -27b^ + 54 ab^.
Solution. This polynomial consisting of four terms, two of
which, viz., 8 a^ and — 27 b^, are exact cubes, may be the cube of
a binomial (§ 57) ; if so, that binomial must be 2 a — 3 6. (Why ?)
On cubing 2 a — 3 b, we see that
■\/Sa^-36a'b-27b^ + 54:ab^ = 2a-Sb.
Ex. 3. Find the square root of a^ + 6^ — 2 a6 — 4 6c + 4 c^ + 4 ac.
Solution. This polynomial consisting of six terms, three of
which are exact squares, and three of which are double products,
7nay be the square of a trinomial whose terms are the square roots
of the square terms (§ 56). A little further examination shows
that Va^ + 62 _ 2 a6 - 4 6c + 4 c^ + 4 ac = ± (a - 6 -f 2 c).
EXERCISE LXXVIII
By inspection find the following roots, and check results :
4. V4 aj- + 12 a; 4- 9. 6. V(m+^)^ — 4 (m + ?i) +4.
5. V25 2/^ -40 2/ + 16. 7. -Va^ -{-2xy + y^- 2 xz~2yz-\-z\
a V 8 u^ — 12 u^v — v^-\-6 uv^.
9. Va;'* — 4: oc^y -^ y^ — 4: xy^ + 6 icy.
10. v^8 /r - 84 h^k + 294 hk^ - 343 Ar^.
180 HIGH SCHOOL ALGEBRA [Ch. XI
11. VciP- b'-5a'b-^5 ab* + 10 a^b' - 10 a%\
12. Va' + 962_6a6 + 6(a;-2i/)(a-36)4-9(a^-4a;y + 4?/2).
13. ^x^ -6aba:^-\- 15 a^^x* - 20 a'b^x^-{- 15 a'b'x^- 6 a^6'x + a'6^
117. Square roots of polynomials.*
Sincev (A; + w)^ = /i:^ -f- 2 A:2* + w^^
therefore Vk^ -^ 2ku-tu^= k -\- u,
and we shall now try to find a method by which the root,
k -\- u, may be found from the power, k^ + 2ku + u^.
Manifestly the first term of the root (viz., k) is the square
root of the first term of the power.
And having subtracted k'^ (the square of this root term)
from the power, the next term of the root (viz., u} is found
by dividing the first term of the remainder (viz., 2ku-\-u^)
by twice the root term previously found (viz., 2 k^.
The actual work may be arranged thus :
k'^ + 2 ^•M + u^ \k -\-u
T^
Trial divisor f =2k
Complete divisor = 2 A; +
2 ku + w2
'Iku + u'^ = (2 k + u) u
The same method may be applied to other polynomials. ^
Ex. 1. Find the square root of 4 s- - 28 .s^ + 49 f.
Solution. Let k represent the first term of this root, and u
the next term ; then
4 s^ - 28 st^ + 49 t' contains (k + uf, i.e., k- + 2 few + u\
Now the first term of the square root of 4 s^ — 28 st^ + 49 f
is, manifestly, 2s, i.e., k = 2s; and the next root term may be
found as above, thus :
* For a more detailed discussion of this topic, see El. Alg. § 125.
t Twice the root already found at any stage of the work is usually called
the trial divisor (T. D.) and the trial divisor plus the next root term is called
the complete divisor (C. D.). % See Note 1, p. 181.
10-117] INVOLUTION AND EVOLUTION 181
4 .s2 - 28 .<?«3 + 49 «« I 2 s - 7 i-"^
k^ = (2 s)
•2-4 .V '
T.T>.=2k = 4s
C. D. =2^ + w=4s-7<5
- 2«.s'«' + 49 «6
- 28 st^ + 49 ^ = (2 A: + m) w
Checks : (1) Square 2 s — 7 f, or (2) substitute special values
for s and t (gL § 25).
Ex. 2. Find the square root of 9 x^ -\- 6 x^ — 11 x^ — 4: x -{- 4.
Solution. At any stage of the process of finding this root, let
k represent the term or terms already known, and let ii represent
the next term ; then
9x^-{-6x^ — lla^ — 4:X-\-4: contains k^-{-2ku-\- \C\ [§ h^
Here the first term of the root is 3 a^, i.e., k = 3x^, and the next
term (u) may be found as in Ex. 1, thus :
9x*+6a;3-llx2-4x+4 I 3 y.2 + X - 2
k^ = (3 a;2)2 = 9x4
T. D. = 2 ^• = 6 a;2
C.I).= 2k-{- u = 6x^ + x
T.B. = 2k* = 6x^ + 2x
C. D. = 2 A; + u = 6x^ + 2X-2
(5 x3 - 1 1 x2 - 4 X + 4
y3 + .7:2 = (2 A: + it) u
12x2 -4x + 4
12 x2 - 4 X + 4 = (2 k*-h n) u
Checks : (1) Square Sx'-{-x-2', or (2) use § 25.
Note 1. Before applying the process of Exs, 1 and 2 a polynomial should
be arranged according to ascending or descending powers of one of its
letters.
Note 2. Exs. 1 and 2 show how to find the square root of a polynomial
which is an exact square ; i.e., if the above process is continued until a zero
remainder is reached, then the square of the root thus found will be the given
polynomial. If, however, the same process is applied to a polynomial
which is not an exact square, then as many root terras as desired may be
found, and the square of this root, at any stage of the work, equals the
result of subtracting the corresponding remainder from the given polynomial ;
such a root is called an approximate root, and also the root to n terms.
* Here k represents 3 x2 + x, and u represents — 2. Observe also that the
first and second subtractions in this solution are together equivalent to the
subtraction of (3 x2 + x)2 from the given expression.
182 IIIGB SCHOOL ALGEBRA [Ch. XI
EXERCISE LXXIX
Find the square ropt of each of the following expressions, and
check your work :
3. 9 m^* - 66 m2 + 121. 5. 4: -\- S x - 4: a^ -\- x\
4. 16/ +104 7-^ + 169. 6. l + 2m-3m2-4m3 + 4m^
7. l-6y-{-5y^-\-12f-\-4:y\'
8. 9a* + 30a^a; + aV-40aa^ + 16a;*.
9. 4a^ + 17a^-22.r^ + 13a;*-24a;-4a^ + 16.
10. 4 a^ + 64 6* - 20 a^b + 57 a^b^ - 80 a¥.
11. 6a^2/ + 2a;«i/^-28aj.v^ + 9.T« + 42/« + 45a^/ + 43icy.
12. 3x''-2:f^-af-i-2x + l-\-x\
13. 48a* + 12a2+l-4a-32a3 + 64a«-64a^
14. 46x' + 25x'-Ua^-^0x + 4:xr'-\-25-12xi',
15. x'^ - 2 x'^y + 2 afz^ -2 yz^ + y^ -hz\
16. ^ + 16ay + 8a^v'. 18. 9 a;2_24a; + 28-^^ + i.
17. aj2_j.2a;_i_?+l :* 19. 4^2- 20« + 21 + — + i.
X x^ a a^
20. n^ + 4 71^ + -i + 2 7i + 4 + 4 n^.
21. a)'' + i + 4aj3 + i + 6a^ + -i- + 5 + 5a; + 5.
cc* ar 4 ar x
22. ^4_^_L..ii_.
r / ^ 4e^
23. (a; - 2/)2 -2{xy -i-xz- y^ - yz) + (y + zf.
25. 1 + », to three terms (cf. Note 2, p. 181).
* Observe that this expression is already arranged according to descending
powers of x.
117-118] INVOLUTION AND EVOLUTION 183
26. 1+2 m-, to four terms.
'27. a^ -f 1, to three terms.
28. 1 -f-.aj — a^, to four terms.
29. X* -\- 2 afy -{- y* -\- xy^ + ar^2/^ ^o four terms.
30. In Ex. 1 is not — 2 s, as well as -f 2 s, a square root of the
first term ? Solve Ex. 1, using — 2 s. Does your result check ?
31. Solve Ex. 2, using —Sx^ as the square root of the first
term, and compare your answer with that found in the text.
32. By extracting the square root until a numerical remainder is
reached, show that x*-\-4:a^-{-SQif-\-Sx—5 equals (ic-+2a; + 2)^—9,
and thus find the factors ofic'* + 4a^ + 8a:^ + 8ic — 5.
33. As in Ex. 31, find the factors of x'*'-\-6af-\-llx^-\-6x — S;
also of a« - 6 ci^ + 10 a^ + 9 a^ _ 30 a + 9.
118. Square roots of arithmetical numbers.* In order
to proceed systematically, and find the successive digits of
the root in their order from left to right, we first separate
the given number into periods of two figures each, toward the
right and left from the decimal point. The root may then
be extracted by virtually the same process as that used in
§117.
Note. The reason for the separation into periods lies in this : the square of
any number of tens ends in two ciphers, and hence the first two digits at the
left of the decimal point are useless when finding the tens' digit of the root ;
they are, therefore, set aside until needed to find the units' digit of the root.
So, too, the square of any number of hundreds ends in four ciphers, and hence,
for a like reason, two periods are set aside when the hundreds' digit of the
root is being found, and so on. Similarly for the periods at the right of the
decimal point.
Ex. 1. Eind the square root of 1156.
Solution. This number consists of two periods, hence its
square root consists of two digits. Again, since 9 is the greatest
* For a more complete discussion of this topic see El. Alg. § 126.
184
HIGH SCHOOL ALGEBRA
[Ch. XI
square in the left-hand period, therefore 3 is the first figure in the
root. Now, let k represent the known part of the root at any stage
of the work, and u the next root figure, then
1156 contains Tc^ + 2ku-\- u^,
and the work may be arranged as follows :
k^ = (30)'-^ =
T. D. = 2 A: = 60
C. D. = 2 ^' + w = 60 + 4
11'56 I 30 + 4 :
900
34
256
256
{2 k + u)u
Check: (34)2 = 1156.
Ex. 2. Find the square root of 315844.
Solution. Using k and u as in Ex. 1, we have
315844 contains A;^ + 2 ku + u%
and the work may be arranged as follows :
fc-^ =(500)2 =
31'58'44 1 500 + 60 + 2 =
250000
562
T.D.
CD.
= 2 A = 1000
= 2k + u = 1060
= 2k* =1120
= 2k + u = 1122
65844
63600 =(2 A; + u)u
T.D.
CD.
2244
22U =(2 k-\-u)u
Check: (562)=-' = 315844.
Note. When some familiarity with the above process has been gained,
the work may be abridged by omitting unnecessary ciphers, as shown below
in finding the square root of 315844 and of 10.5625.
31'58'44
25
10.'66'25
9
13.26
106
1122
658
636
2244
2244
62
645
156
124
3225
3225
* Here k = 560 ; compare footnote, p. 181.
118-119] INVOLUTION AND ^VOLUTION 185
EXERCISE LXXX
Extract the square root of each of the following numbers, and
check your results :
3. 1296. 6. 9216. 9. 667489. 12. 17424.
4. 841. 7. 12.96. 10. 26.2144. 13. 36.8449.
5. 2209. 8. 62.41. 11. 1664.64. 14. 101.0025.
15. How may the square root of a fraction be found?
Illustrate, using the fractions ^\ and f||. Is — 14 also a square
root of the latter fraction ? Why ?
16. A number contains one decimal place ; how many decimal
places in its square? How many, if the number contains two
decimal places ? if it contains three ? if it contains n ? Explain.
17. Show from Ex. 16 that if the right-hand period of a decimal
is incomplete, we must annex a cipher to complete it. Is this
true of the left-hand period of an integral number also ?
18. Extract the square root of 2 to two decimal places (cf.
p. 181, Note 2). How many periods of ciphers must be annexed to
2 for this purpose ? Why ?
Find the square root of each of the following numbers, correct
to two decimal places :
19. 13.5. 21. .017. 23. |. 25. 4|.
20. |. 22. 1.1105. 24. ■^. 26. .049.
27. Is V36 equal to V9- Vi? Is V27 (i.e., V9T3) equal to
3V3 where the roots are extracted to two decimal places? to
three decimal places ?
28. Is V450 (i.e., •\/225 • 2) equal to 15 V2 where the roots are
correct to two decimal places ? Show that V96 and 4 V6 are
equal, to at least two decimal places.
119.* Cube root of polynomials. The procedure here is like
that in § 117.
* Articles 119, 120, with Exercises LXXXI and LXXXII, may, if the
teacher prefers, be omitted till the subject is reviewed.
186 HIGir SCHOOL algebra [Ch. XI
Since Qc^uy = 7c^-{-^k^u-h3ku^-hu\
therefore VA;^ + 3 k'^u -\- 3 ku^ -{-u^ = k-{- u.
And this equation shows :
(1) that the j^rs^ tei^m of the cube root (viz., k) is the cube root
of the first term of the polynomial ;
(2) that the trial divisor for finding the next term of the root
is3A^;
(3) that the complete divisor is 3 A;^ + 3 ku -{-u^.
The actual work may be arranged thus :
F + 3 khi + P> ku^ + v^ \k-\-u
J^
T. D. = 3 k^
C. D. = 3 A:2 + 3 ku + u
3 A:2m + 3 kv!^ + u^
3 k'^u + ^ku^ + m3 = (3 k'^ + 3 A;?( + ifi) u
If now we let k represent the part of the root already known
at any stage of the work, and let u represent the next term, then
the above method will serve to extract the cube root of any
"arranged" polynomial (cf. § 117, Exs. 1 and 2).
Thus, the cube root of ofi -9x^ + 30 x^ - i^x^ + SOx'^ - 9x + 1 may be
found as follows :
I a;2 - 3 a; + 1
5c6 _ 9 a:^ + 30 0^4 - 45 a:3 + 30 ic2 _ 9 X + 1
(x^y = x6
T. D. =3(^2)2 = 3 X*
CD. =3x4-9x3 + 9^2
T.D. =3(x2-3x)2 = 3x*- 18x3 + 27x2
C. D. = 3 (x2 - 3 x)2 + 3 (x2 - 3 X) + 1
= 3x*- 18x3 + 30x2 -9x + 1
- 9x5 + 30x4 -45x3 + 30x2 -9x + l
-9x5 + 27x4-27x3
3x4- 18x3 + 30x2 -9x + 1
3x4 - 18x3 + 30x2 -9x + l
EXERCISE LXXXI
Find the cube root in Exs. 1-14, and check your results ;
1. Sa^-12x^ + 6x-l.
2. 27x^-189x'y-hUlxy^-34.Sf/
3. 125 n"" - 150 mn^-Sm^-\- 60 m'n.
4. 225uh + lS5uv^ + 125u''-\-27'i^.
liy-120] INVOLUTION AND EVOLUTION 187
5. a;^-20a^-6aj-f 15a;4-6x-^4-15ic2 + l.
6. 3aj^ + 9a;^-|-ic'' + 8 + 12a; + 13a^ + 18a;l
7. 342 x^ - 108 a; - 109 a^ + 216 + 171 x"" -21x^ + 27 x\
8. 156a;*-144aj«-99a^ + 64aj6 4-39aj2-9a; + l.
9. u X ■\- - -112 -^ + — + 0^-12 xUai. Ex. 17, p. 182).
10. 20 + i^ + 15c2 + c« + | + i + 6o^
11. 25 + ^ + 82/« + 302/-12 2/-^-25-i|.
2/ 2/ y
12. 6 aV-4aV -2 aV + 6 aV + 3 a^a; + «' + a^^ - 3 a«8.
13. 108 'ifz -21 f- 90 2/V + 8 2« - 80 y V ^ 50 yh^ + 48 2/2^
14. 'y3» -f- 9 ^Sn-S ^ 21 i;3"-2 — 42 'y3«-4_ 35 -y3n-5_9 ^3«-l_g ,y3«-6^
15. Find the first three terms of v 1 + x.
16. Find the first four terms of V 1 — 3 a? + aj^.
120* Cube root of numbers. The cube root of a number may be
found by virtually the same process as that used in § 119 for
finding the cube root of a polynomial (cf. §§ 118 and 117). The
number should be separated into periods of 3 figures each, begin-
ning at the decimal point (why ?), and the right-hand period, if
incomplete, should be completed by annexing ciphers.
Ex. 1. Find the cube root of 42875.
SOLUTION
k -\- u
42'875 1 30 + 5 = 35
27000
15875
T^ = (30)8 =
T.D. = 3 A:2 = 3 . (30)2 ^ 2700
CD. =Sk^ + 'Sku + u^ ,
= 3 (30)2 + 3 (30)5 + 52 = 3175 1 15875
* See footnote, p. 185.
HIGH SCH. ALG. — 13 •
188 HIGH SCHOOL ALGEliRA [Cii. Xi
Ex 2. Find the cube root of 9825.17, correct to tenths.
SOLUTION
F = (20)
9'825.'170 |20+ 1 + .4 = 21.4
8000
564.170
539.344
T.D. = 3 k-^ = 3 (20)2 ^ 1200 1825
CD. =3^•2 + 3^•« + M2
= 3 (20)2 + 3 (20) . 1 + 12 = 1201 1261
T.D. = 3A;2=: 3(21)2 =1323
CD. = 3 (21)-^ +3 (21) (.4) + (.4)2 = 1348.36
24.826
Check. (21.4)=^ = 9825.17 - 24.826 (cf. Note 2, p. 181).
EXERCISE LXXXII
Extract the cube root of each of the following numbers :
1. 1728. 3. 31855.013. 5. 39304.
2. 571787. 4. 148877. 6. 426.957777.
7. 75.686967. 9. .04, to two decimal places.
8. 34.7, to two decimal places. 10. 3^, to two decimal places.
121. Transformation of indicated roots.* From the defini-
tion of the symbol -Va (§ 113) it follows that, whatever the
values of h and k,
■Vm = hVk: (1)
for (h-Vky^JiVk-h^k
= hh--Vk^
= h^k, [since Vk Vk = k~\
i.e., (hVky=h%,
and h VJ is, therefore, a square root of hj^k.
Equation (1), read forward, tells how to simplify an indi-
cated square root of a number which contains a square factor ;
and read backward, it tells how to insert a coefficient under
* Omit § 121 if radicals are to be studied before quadratics : see Preface.
120-121]
INVOLUTION AND EVOLUTION
189
a square root 8i(/n. Let the pupil translate this eqiuttion
into verbal language, reading it both ways.
It also follows from the definition of -Va (§ 113) that
(V^^0^=-^- (2)
Equations (1) and (2) will be useful in Chapter XII.
EXERCISE LXXXIII
Simplify the following expressions [cf. Eq. (1), § 121]:
1. Vl2.
2. V50.
3. V48.
4. V63.
5. V4a26.
'■4- ■
"■ x/f-
20.
|V9.
23. —5cVb.
6. V54ajy.
Insert the following coefficients under the radical signs
15. 3V2. 19. -i-VS. 22. aV2^.
16. 5V7.
17. — 4V5.
18. 7ViO. 21. f Vii. 24. |aV6'-4ac.
Expand the following expressions, and unite like terms :
25. (3+V5)l 28. (6-V^2)l 31. [^(3-2V5)?.
26. (3 + V^^)'. 29. (&-V6'-4ac)l
27. (-H-V3)'. 30. (6-V4ac-62)^
32.
D
& + V62 - 4
ac
2a
CHAPTER XII*
QUADRATIC EQUATIONS (Elementary)
I. EQUATIONS IN ONE UNKNOWN NUMBER
122. Definitions. A quadratic equation has already been
defined (§ 93) as an equation wliicli, when simplified, is of
the second degree in the unknown number or numbers.
Thus, 3 §2 _ 4 = 7 s^ aoiP' + 5a; + c = 0, and dm = 4:m^ are
quadratic equations in s, x^ and 7n, respectively.
By transposing and uniting terms every quadratic in x^
say, may evidently be reduced to the standard form
ax^ -{- bx -^ c = 0,
wherein a, 6, and c represent known numbers, and are usu-
ally called the coefficients of the equation. The term free
from X, viz., c, is called the absolute (also constant) term.
U.g.^ by transposing, etc., 2x'^-^b — Sx = lx— S becomes
2:^2 — 10a;-|-13 = 0: hence, for this particular equation a= 2,
5 = -10, and c = U. Similarly, 6x^- 2x=Sx^- 4:-2x
becomes 3 a:^ -}- 4 = ; here a = 3, 5 = 0, and c = 4:.
An equation of the form ax^ + c = is often called a pure
quadratic, while one containing both the first and second
powers of the unknown number 'is called an affected quadratic.
123. Solution of pure quadratics. All pure quadratic equa-
tions may be solved like Exs. 1 and 2 below.
Ex. 1. Given 3 a;^ _ 12 = ; to find x.
* Chapter XII may, if the teacher prefers, be omitted until Chapters XIV
and XV have been studied: see Preface.
190
122-123] QUADBATIC EQUATIONS 191
Solution. On dividing through by 3, and transposing, the
given equation becomes
a;2 = 4,
whence x= ±2,*
i.e., a; = 2ora;=— 2;
and each of these values is found to check.
45 s^
Ex. 2. Solve the equation 5 -— - = 0.
16
Solution. On dividing the given equation through by 5, clear-
ing of fractions, and transposing, we obtain
whence 3 s = ± 4,
I.e., s=4ors = — I;
and each of these values is found to check.
EXERCISE LXXXIV
Solve and check :
3. 4a^ = 36. 11. 3cx'-10Sc^ = 0.
4. x' = ^. 12. (a + l)V = 4a2.
5. i_a^=_48. 13. (k-6y = 72-^12k.
6 — = 27 ■*■*• <^-^) = 2-^^'
4 * 15. a;(a;4-l)+3a^ = x + |.
7. ^-Sy = 0. 16. (r-Sf = 25.
^y 17. (u+iy-^%=o.
8. 4('y2 + 3)=32. 18. (a?-a)2 = 9&l
9. a^ + 3a; = 3(a;+ll)-ll. 19. 4a^-l = a2 + 2a.
^ (, + 3)(r-3), 20. ^(x-^)=(l-f
9 8 16^ ^ V 8.
* Using the double sign in each member here gives ± x= ±2,
i.e., either x = 2, (1) or - a; = 2, (3)
or a; = - 2, (2) or - ic = -2. (4)
But (3) and (4) give the same values of x as (2) and (1), respectively;
hence in solving such an equation as x^ = 4, the double sign need be used in
one member only.
192 HIGH SCHOOL ALGEBRA [Ch. XII
21. If X and y stand for unknown numbers, tell which of the
following equations are simple, and which quadratic (cf. § 93) :
aV -\-a^x-{-a = 0\ ~ =-; 6x—ly = ll; b x + xy — 1 y — 11',
Z X
y 2/ + 2
22. Reduce 5a:^H-2 — 8a; = 4(8 — x) to the " standard form."
What are its coefficients ? What is its absolute term ?
23. Are the equations in Exs. 3-14 pure or affected? Explain.
What is the absolute term in Ex. 9?
24. Show that the equation of Ex. 16 is a pure quadratic in
r — 3 but an affected quadratic in r.
Solve each of the following equations for each letter it contains :
25. S = \qt\ 26. 1 = ^. 27. E=^^ 28. R = K--
s V- 2 d-
A
29. The square on the hypotenuse (longest side) of a
right-angled triangle equals the sum of the squares on
the other two sides. In the right-angled triangle ABC,
the hypotenuse AO = 5, while -B(7=| AB; find AB
and BC.
30. The area of a square is 169 square inches; find the perime-
ter and the diagonal of the square.
31. How many rods of fence will inclose a square garden
whose area is 2^ acres ?
E B 32. From a rectangular field, ABCD, whose width
is f of its length, there is cut off a square field,
AEFD, whose area is 10 acres. Find the area of
the rectangular field.
33. The surface area of a cube is 150 square
inches. Find one edge and also the volume of the cube.
124. Quadratics solved by factoring. A quadratic equation
ma}^ often be easily solved by reducing it to standard form
(§ 122), and then factoring its first member (§ 72).
123-124] QUADRATIC EQUATIONS 198
Ex. 1. Solve the equation 3 ic^ + 4 = ar' — li a; + 16.
Solution. On transposing, uniting, and dividing through by
2, the given equation becomes
ar -h a; — 6 = 0,
i.e., (x-2){x-{-3) = 0.
Now, as in § 72, this last equation is satisfied when
a^-2 = or ic-|-3 = 0,
I.e., when x = 2 or when a; = — 3 ;
and each of these vahies of x, when substituted in the given
equation, is found to check.
Ex. 2. Solve the equation x (a; — 3) + a; + 2 = 2 (1 — .-c^).
Solution. On transposing, etc., the given equation becomes
3ar'-2a; = 0,
i.e., a;(3a;-2) = 0,
wlience x = or -;
3'
and these values of x are found to check.
EXERCISE LXXXV
Solve the following equations by factoring, and check the roots
in each case :
3. /-5?/-24 = 0.
4. m--16 = 0.
5. a^ + 5x = 21 + x.
6. 5 a;-=a^- 14.
7. 5s- = Ss.
8. 2v--30 = 9v-v-
9. 2ar'-x=3.
10. 4. 0^ = 9.
11. 2c^ = x(x-^r).
12. (4 x)- = 14 (4 x) —
13. 22x + Sx' = 4:x'-
14. ^-^"- = 38.
15.
.t'--4a; = 117.
16.
13y + 2f=.5y-^4.y\
17.
2iK2-20x = x- -51.
18.
3a(3a-l)=3a + 24.
19.
2/-72/ 4-3 = 0.
20.
ic (a; + 7n) =n{—x—m)
21.
~U0 + x^=^-23x.
22.
7-2_5r = 5r-25.
23.
Zaj^ — Ikx -}- A:w = ma;.
)•).
24.
5^2_3 =10^-3^1
48.
25.
ao(? -\-bx — ex.
26.
XT _X i» , -j
bo be
194 HIGH SCHOOL ALGEBRA [Ch. XII
27. If a quadratic equation in one unknown number has no
absolute term, show that one root of the equation must be zero.
125. Completing the square. What must be added to
x^+Qx to make it the square oi x+ S? What must be added
to m^—14:m to make it the square of m — 7 ?
Since Qx ±k')^ = x^ ±2kx -\- k^, therefore the expression
x^±2 kx, whatever the value of k, lacks only the term k^ of
being the square of x±k; hence, if the square of half the
coefficient of the first power of x he added to an expression of
the form x^ + bx, the result will be an exact square.
Such an addition is usually spoken of as completing the
square.
j^.^., if (f )2 is added to «/2 + 5 ?/ it becomes (?/ + 1)2.
126. Solution of quadratics by completing the square.*
There are many quadratic equations which cannot easily be
solved by the method of factoring given in § 124. All quad-
ratic equations, however, may be solved by the method of
completing the square, which is illustrated below.
Ex. 1. Solve the equation 2a;^ — 3— 5a; = 7a;+ll.
Solution. On transposing, etc., this equation becomes
Now, adding 9 to each member (§ 125, and Ax. 1), we obtain
a^-6x + 9 = 16,
i.e., (a; -3)2 = 16,
whence (§ 123) a; - 3 = ± 4,
i.e., a; — 3 = 4 or a; — 3 = — 4,
and therefore a; = 7 or — 1.
Moreover, these values check, and are, therefore, the required
roots.
*For the solution of quadratic equations by means of a formula, see § 178.
124-126] QUADRATIC EQUATIONS 195
Ex. 2. Solve the equation x^ + 11 a; + 1 = 8 a:.
Solution. On transposing, the given equation becomes
a^-|-3ic = — 1,
whence, adding (f )2, x'-\-Sx + (^f = - 1 + (f )', [§ 125
i.e., (^-\-iy = h
and hence . a; + f = ± V|= ± iV5, [§121
3^1 /- -3±V5
^ = -2^2^^ = ^
Moreover, these values otx, viz., ~'^^ ^ and ~^~ ^ , check (cf. Ex.
25, p. 189), and are, therefore, the required roots.
EXERCISE LXXXVI
3. Solve the equation ax^ -\-bx+ c =0.
Solution. On transposing and dividing by a, this equation becomes
whence
a a
5 \2 12 c b^-4ac
a \2al ^a^ a 4 a^ ' '■*
\ 2a/ 4 a^
therefore ^ + A = ± J^!^ ^ ± ^Z"' - 4 ac ^ [5121
2 a ^ 4 a^ 2 a
ig ^^ b ±Vb'^-4ac ^ -b±y/b'^-4:ac _
2a 2a 2a
Moreover, thesevaluesof a;, viz., -h + ^b^-4.ac ^^^ -b-Vb^-4a c^
2a 2a
check (cf. Ex. 29, p. 189) and are, therefore, roots of the given equation.
4. What must be added to each of the following expressions
in order to complete the square (cf. § 125): x^-\-Sx', P^ — 5P;
(x + yy-4.(x + y)?
5. How do we find the number which added to r^ + ar com-
pletes the square ? Explain.
6.
m^ -6 m =40.
7.
y'-10y = 75.
8.
W = 2x-hx'.
9.
-S = 2x^-h'^0x.
10.
a^ = x-\-l.
11.
Sx'-2x = l.
196 HIGH SCHOOL ALGEBRA [Ch. XII
Solve the following equations by the method of completing the
square, and check the roots in each case :
20. a!2-3a;-2 = 0.
21. ar^- 3 a; H- 4 = 0.
22. |c^— 7c = c(cH-l).
23. 6 + ot = et\
24. 12a^- x = 6,
25. «- = -j6' + 2.
12. 2a;2-f3 = 7a;. 26. r'-er=f.
13. 3a^-10 = 7a;. 27. 3 x^ -\-5x-7 =x^-2 x.
14. (2 2/-3)2 = 6 2/ + l. 28. 8m-10 = 3m^.
15. m(m + 4) = 7. 29. A-a;-|ar^+2=0.
16. a2-6a + 10 = 0. 30. (y -Sf -4.(y-3) = 117.
17. 3 ('y^-'?;) =2^2.^ 5 V + 4. 31. (2m-3)2-6(mH-l)4-8=0.
18. y^ — 2cy = l. 32. cV + 2(Za;=— e.
19. r2 + 2ar = d 33. (n + 1)2- 8 (n + 1) = 16.
34. Write a carefully worded rule for solving such quadratic
equations as those in Exs. 6-33 above.
35. Show that the rule asked for in Ex. 34 will serve to solve
such an equation asa^ + 6i« = 0. Is this equation more easily
solved by completing the square or by factoring ?
127. Avoiding fractions in completing the square. The
method employed in § 126 for completing the square often
introduces fractions into the work, and these sometimes be-
come troublesome (cf. Exs. 2 and 3, p. 195). A method
which avoids fractions is illustrated below.
Ex. 1. Solve the equation 5 x^ —6x= — 1.
Solution. On multiplying through by 5, we obtain
25aj2-30a;=-5,
i.e., (5aj)2-6(5a;) = -5,
whence, adding 9, (5 a;)^ — 6 (5 a;) + 9 = 4,
i.e., (5 X- 3)2 = 4;
12(i-127J QUADRATIC EQlATIOlSfS 197
therefore 5 ic — 3 = ± 2,
from which 5a; = 3±2=:5orl,
i.e.f x = \ or I ;
and each of these values is, on substitution, found to check.
Ex. 2. Solve the equation as? -{-hxz^ — c.
Solution. On multiplying through by 4 a, we obtain
4 a^ic- + 4 abx = — 4 «c,
i.e., (2 axf + 2 6 (2 ax) = - 4 ae,
whence, adding 6^, (2 aa;)^ + 2 6 (2 aa;) + 6^ = 6^ — 4 ac,
t.e., (2 ax -j- 6)- = 6^ _ 4 ^f^ .
therefore 2ax-\-h = ± V 6^ — 4 ac,
from which x = -l>±^h^-^ ^^^^ Ex. 3, p. 195.
• 2 a
Note. From the above solutions we see that, if an equation of the form
aofi + &a; + c = is multiplied through by a or 4 a, according as b is even or
odd, fractions can be avoided in the solution.
EXERCISE LXXXVII
3. What must be added to each of the following expres-
sions in order to complete the square : 4 x- -f 8 x; 9 m* + 12 m^;
25 cH"" - 10 cd; and 4 cV - 4 cm ?
In each of Exs. 4-11 belo^, (1) name the factor 'by which both
members of the equation must be multiplied if fractions are to be
avoided in completing the square; (2) solve the equation by the
method of § 127.
4. 5 0^4-6 x = 8. 8. 322 = 2 + 52.
5. 3/ + 4?/ = 95. 9. 7 = 2x + 3.^;^'.
6. 2m2-f3m = 27. 10. 6x^-x-^ = 0.
7. 2^2^7^ + 6 = 0. 11. 15a;2-7a!-2 = 0.
12. By the method of § 127, solve Exs. 12-14 and 20-25, p. 196.
198 HIGH SCHOOL ALGEBRA • [Ch. XII
Solve the following equations by the method of § 127 :
f^3t-{-5 ^ t + 1 u'-^u '^^ + ^ ^0
2 3 * 3 "^ 4
14. ma^ — 6 a; + 3 = 0. 18. 3/ — 4% + 2 = 0.
15. x^-\-px + q=:0. 19. mx^ -^ nx -\- p — 0.
16. mx^ = 2nx — k, 20. ax^ — 5ax = a — 11.
128. Fractional equations which lead to quadratics. As in
§ 97, so here, we first clear the given equation of fractions,
then solve the resulting integral equation, and finally check
the results so as to guard against the introduction of extra-
neous roots (§ 97, note).
Ex. 1. Solve the equation —^ — |- 1 = 3 a;.
^ x + 2
Solution. On clearing of fractions, etc., this equation becomes
3 a;2 + 4 .T - 7 = 0,
whence, solving as in § 127, we obtain
aj = l or — 1^;
and each of these values, when substituted in the given equation,
is found to check.
Ex. 2. Solve the equation -^— _^4a; + 3_ 2 x"
1 — x x-\-l x^ — 1
Solution. On clearing of fractions, etc., we obtain
a^_2x-3 = 0,
whence a; = 3 or - 1, [§ 126
of which 3 checks, but — 1 is extraneous (§ 97, note).
EXERCISE LXXXVIII
Solve the following fractional equations, being careful to ex-
clude all extraneous roots :
3. 15 a; + - = 11. 5. - = -•
X bx-{-b x-\-±
4. l-2 + .. = 2. 6. -1- + _!- = §.
X X 1— sl+s3
127-129] qUADRATlG EQUATIONS 199
7 __3 L_ = l. 11 2y + l 5^y-8
' 2(a^-l) 4(aj + l) 8 ■ 1-22/ 7 2
8 ^"^ I ^ + ^ = 2/^ ^"^^ Y 12 ^^ + ^ I ct-2a; ^22.
07 + 2 a;— 2 V^~^/ 2a — x a-{-2x
9. _l_ + ^_= J_. 13. -*^ + 6 = «J£+26).
a; — 1 a;— 2 3 — a; a — a; a-{-b
10. ■20.+ 40 ^_3M:7. ^^_ . __ ._.__,.
:+3 s'+is-\-3 S + 1 X— 1 2 + 1
15.^^ + ^ -=8+ ^
x + 5 (a; + 5)(a; — 2) a; — 2
129. Problems which lead to quadratics. As in § 50, so
here, the important steps in the solution of a problem are :
1. To translate the verbal language of the problem into
algebraic language, i.e., into equations.
2. To solve these equations.
3. To check, and interpret, the results.
Special emphasis should be laid upon testing and interpret-
ing results: a problem often contains restrictions upon its
numbers, expressed or implied, which are not translated into
the equations, hence the solutions of the equations may or
may not be solutions of the problem itself (cf. § 98).
Prob. 1. A farmer purchased some sheep for $ 168, and later
sold all but four of them for the same sum. If his profit on
each sheep sold was $ 1, how many sheep did he buy ?
SOLUTION
Let X = the number of sheep purchased.
Then — - = the number of dollars each sheep cost,
X
1f>8
and = the number of dollars received for each sheep.
A I. 168 168 ^
and hence = 1 ,
X — 4: X
therefore (§ 128) a; = 28 or -24.
"Profit on each sheep
being $1
200 BIGH SCHOOL ALGEBRA [Ch. XII
The first of these values, viz., 28, is found to be a solution of
the problem as well as of the equation, but while the second satis-
fies the equation it cannot satisfy the problem, since the number
of sheep purchased is necessarily a positive integer.
Prob. 2. At a certain dinner party it is found that 6 times the
number of guests exceeds the square of | their number by 8;
how many guests are there ?
SOLUTION . ^
Let X = the number of guests.
Then the expressed condition of the problem is
«-(¥)'=»■
i.e., 2x^-27 x + 36 = 0,
whence x = 12 or |.
Here, too, an implied condition of the problem is that the
answer must be a positive integer ; hence f , although it satisfies
the equation, it is not a solution of the problem.
Prob. 3. The sum of the ages of a father and his son is 100
years, and one tenth of the product of the numbers of years in
their ages, minus 180, equals the number of years in the father's
age ; what is the age of each ?
SOLUTION
Let X = the number of years in the father's age.
Then 100 — x = the number of years in the son's age,
and the condition of the problem states that
whence cc = 60 or 30.
Although both 60 and 30 are positive integers, yet 30 is not a
solution of the problem : it would make the son older than the
father. Hence the father is 60 years old, and the son 40.
If, in the above problem, " two persons " be substituted for " a
father and his son," etc., then both solutions are admissible, and
the ages are either 60 and 40 years, or 30 and 70 years.
1210 QUADliATlC EQUATIONS 201
EXERCISE LXXXIX
4. Divide 10 into two parts whose product is 22f .
5. Find two numbers whose difference is 11, and whose sum
multiplied by the greater is 513.
6. A man bought a flock of sheep for $ 75. If he had paid
the same sum for a flock containing 3 sheep more they would
have cost him $ 1.25 less per head. How many did he buy ?
Is each solution of the equation of this problem a solution of
the problem itself? Explain.
7. A clothier having bought some cloth for $30 found that if
he had received 3 yards more for the same money, the cloth would
have cost him 50 cents less per yard. How many yards did he
buy ? Has this problem more than one solution ?
8. Find two numbers whose sum is 10 and whose product
is 42. Can these numbers be real (see Note, § 114) ?
9. Find two consecutive integers the sum of whose squares is
61. How many solutions has the equation of this problem ?
Show that each of these is a solution of the problem also.
10. Are there two consecutive integers the sum of whose squares
is 118 ? Are there two numbers whose difference is 1, and the
sum of whose squares is 118 ? What are they? How does the
second of these questions differ from the first ?
11. Find three consecutive integers whose sum is equal to the
product of the first two.
12. Is it possible to find three consecutive integers whose sum
equals the product of the first and last ? How is the impossibility
of such a set of numbers shown ?
13. In selling a yard of silk at 75 cents, a merchant gains as
many per cent as there are cents in its cost. Find the cost.
14. A cow staked out to graze can graze over a circle 616 square
feet in area ; how long is the rope by which she is tied ? [The
area of a circle of radius r is tt • ?'^ ; 7r = 3|, approximately.]
202 HIGH SCHOOL ALGEBRA [Ch. XII
15. Two circles are such that the difference of their radii is
3 inches, and the sum of their areas 279|- sq. in. Find the radius
of each circle.
16. Find two numbers whose sum is |, and whose difference is
equal to their product. How many solutions has this problem ?
17. The product of three consecutive integers is divided by
each of them in turn, and the sum of the three quotients is 74.
What are these integers ? How many solutions has this prob-
lem ? Explain.
18. If the product of two numbers is 6, and the sum of their
reciprocals is ff, what are the numbers ? How many solutions
has the equation of this problem ? How many solutions has the
problem itself ? Explain.
19. A merchant who had purchased a quantity of flour for
found that if he had obtained 8 barrels more for the same money,
the price per barrel would have been f 2 less. How many barrels
did he buy ? How many solutions has this problem ? Explain.
20. Why is it that the solutions of the equation of a problem
are not always solutions of the problem itself ? (Cf. § 129.)
21. In a rectangle whose area is 55i sq. in., the sum of the
length and breadth is 15 in. ; find the length.
22. Find the length of a rectangle whose area is 464 sq. in.,
and the sum of whose length and breadth is 16 in.
Interpret the imaginary result in this problem (cf. § 98). Does
an imaginary result always show that the conditions of the problem
are impossible of fulfillment (cf. Prob. 8, p. 201) ?
23. The number of square inches in the surface area of a cube
exceeds the number of cubic inches in its volume by 8 times the
number of inches in one edge. Find the edge of the cube. How
many solutions has this problem? Explain.
24. In the trapezoid ABCD, whose area is
75 sq. in., the altitude BE equals BCy and AD
is 5 in. longer than BC \ find BC and AD.
A E
^ [The area of ABCD = \{BC-\- AD) • BE.-]
120 j QUADRATIC EQUATIONS 208
25. A triangle whose base is 2 in. longer than its altitude has
an area equal to that of a rectangle 10 in. by 4 in. Find the
base and altitude of the triangle. [The area of a triangle equals
half the product of its base and altitude.]
26. A boating club on returning from a short cruise found that
its expenses had been $90, and that the number of dollars each
member had to pay was less by 4i than the number of members in
the club. How many members were there in the club ?
27. If in Prob. 26 the expense of the cruise had been $145
the other conditions remaining unchanged, how many members
would the club contain ?
What is the significance of the fractional and negative results
in this problem ? Do such results always indicate that the con-
ditions of a problem are impossible of fulfillment ?
28. The number of miles in the distance between two cities is
such that its square root, plus its half, equals 12. What is this
distance ? Has this problem more than one solution ? Explain.
29. When a certain train has traveled 5 hours it is still 60
miles from its destination. If by traveling 5 miles faster per
hour, it could make the entire trip in 1 hour less than the sched-
uled time, find the entire distance ; also the actual speed.
30. The hypotenuse of a right-angled triangle is 10 inches,
and one of the sides is 2 inches longer than the other ; required
the length of the sides (cf. Ex. 29, p. 192).
31. It took a number of men as many days to dig a trench as
there were men. If there had been 6 more men, the work would
have been done in 8 days. How many men were there ?
32. A crew can row 5|- miles downstream and back again in 2
hours and 23 minutes ; if the rate of the current is 3-^ miles an
hour, find the rate at which the crew can row in still water.
33. From a thread whose length is equal to the perimeter of a
square, one yard is cut off, and the remainder is equal to the
perimeter of another square whose area is -| that of the first.
What was the length of the thread at first ?
HIGH SCH. ALG. — 14
204 HIGH SCHOOL ALGEBRA [Ch. XII
34. The diagonal and the longer side of a rectangle are to-
gether equal to five times the shorter side, and the longer side
exceeds the shorter by 35 yards. Find the area of the rectangle.
■ 35. A ladder 13 ft. long leans against a vertical wall. When
the distance from the base of the >vall to the foot of the ladder is
7 ft. less than the height of the wall, the ladder just reaches to
the top of the wall. How high is the wall (cf. Frob. 30) ?
36. If one train, by going 15 miles an hour faster than another,
requires 12 minutes less than the other to run 36 miles, what is
the speed of each train ?
37. A tank can be filled by one of its two feed-pipes in 2 hours
less time than by the other, and by both pipes together in 1|-
hours. In what time can each pipe separately fill the tank ?
38. The owner of a lot 56 rods long and 28 rods wide divided
it into 4 equal rectangular lots, by constructing through it two
streets of uniform width. If these streets decrease the available
area of the lot by 2 acres, what is their width ?
39. One of two casks contains twice as many gallons of water
as the other does of wine ; 6 gallons are drawn from each cask,
exchanged, and emptied into the other; it is then found that the
percentage of wine in each cask is the same. How many gallons
of water did the first cask originally contain ?
40. A and B together can do a given piece of work in a certain
time ; but if they each do one half of this work separately, A works
one day less, and B two days more, than when they work
together. In how many days can they do the work together ?
41. In going a mile, the hind wheel of a carriage makes 145
revolutions less than the front wheel, but if the hind wheel were
16 inches greater in circumference, it would then
make 200 revolutions less than the front wheel.
What is the circumference of the front wheel ?
42. In the figure, AB = BC= CD = DA = 10
inches, the diagonals AC and DB bisect each
other at right angles, and DB is 4 inches longer
than AC. Find the lengths of AC and DB, and
the area of the figure.
l-2i1-K30] (QUADRATIC KQrATIONS 205
130. Equations in quadratic form. Equations which con-
tain onl}^ two different powers of the unknown number, one
of these powers being the square of the other, are said to be
in quadratic form. Thus, a^+1 x^=S, av?"" + 6m" + c = 0,
and (2 s^ + 1)^ — 5 (2 s^ 4. 1) = 4 are in quadratic form.
Such equations may be solved as follows :
Ex. 1. Solve the equation 2y?{y? -\-X)=h — oi?.
Solution. When simplified, the given equation becomes
or, putting y for a^, 2 2/^ + 3 ?/ — = 0,
whence y=\ oy -\, [§126
Li
i.e. (since y — oc^), a^ = 1 or — -f ,
whence a; = ± 1 or ± V — |.
Moreover, each of these values (1, —1, V— f, and — V— f)*
checks (§ 121), and is therefore a root of the given equation.
Ex. 2. Solve the equation Vx' — 5 x + 10 = 2 .'e^ — 10 aj + 14.
Solution. This equation may be written thus :
Va;2-5a^ + 10 = 2(a.-2-5a; + 10)-6;
and, on putting y for Vaf — 5 ;r + 10, the given equation becomes
2/ = 2/-6,
whence y = 2 ov—l, [§ 126
i.e., Vaj^ — 5 a; + 10 = 2 or — -|,
and therefore aj'^ — 5 a; + 10 = 4 or f , [Squaring
whence a; = 2, 3, ^ ^ ~- ^ ^ or ^-V^ ^ ^^ ^26
all of which values check (cf. Ex. 28, p. 189), and are therefore the
required roots.
EXERCISE XC
Solve, and check as the teacher directs :
3. wt^ -16 = 0. 5. ?/' - 25 ?/2 + 144 = 0.
4. a;*_8x2 + 12 = 0. 6. n* = 18n2-32.
206 BIGH SCHOOL ALGEBRA CCh. XII
o r» /- Hint. Write equation thus :
8. x = 3 — 2-Vx. ^ ,
(2 k^ _ 1) - 6 \/2 A;-^ - 1 = 7.
9. 20x'-23x' = -6. ,
19. y?-x-\-^x^—x-?, = ^.
10. 4 — ?ii^ = 18 m^.
20. 5a;2 + 2V5a;2_a; = 8 + iK.
11. 13 V^! — 5 = 62;.
. 1 21. ?2^±2 + --A_=2.
13. (x' + iy + 4(a^ + l) = 45. Hmx. Let»: = <±2 j^^^i^y
15. x-2=Vi^+6. 23. '^±5 + ^^ = 7.
r r' + b
16. (m + l)-5Vm + l = 6. y + 2 2(/ + 4) ^51
17. 2s-3 = 7V2i^^-12. ^*' / + 4 2/ + 2 5*
25. 5Vm2-10m + 42 = m2-10m + 6.
26. ^2_7i_pVi2_7^^18^24.
27. a;4 + 4i»3-8x + 3 = 0.
Hint. By extracting the square root of the first member this equation may
be written in the form (x^ + 2 a; - 2)^ = 1. (Cf. Ex. 32, p. 183.)
28. 2/' + 22/^ + 52/' + 42/ = 60.
29. 16a;^-8aj3-31i»2 + 8a; + 15 = 0.
30. 9a;* + 6a^-83a;2_28x + 147 = 0.
II. SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS
[Two Unknown Numbers]
131. One equation simple and the other quadratic. Wlien
one equation is simple and the other quadratic, we may
alwa.ys eliminate by substitution [§ 103 (i)].
Ex. 1. Solve the following system of simultaneous equations:
(1)
(2)
I 3 a; - 2 2/ = 3, 1
l;jo-l3l]
QUADRATIC EQUATIONS
3 + 2j
»- 3 :
Solution. From Eq. (1),
whence, by substituting this value of x, Eq. (2) becomes
and, on expanding and simplifying, Eq. (4) becomes
102/^ + 32/-27 = 0,
whence (§ 126) ^ = f or - 1.
Substituting these values of y in Eq. (3) shows that
207
(3)
(4)
2/ = |or -f.
according as
ia? = 2, ix = — h
and \ g
satisfy the given system, and are therefore the solutions sought.
EXERCISE XCI
Solve the following systems of equations and check your results:
6.
'I
8.
x'-f = SO,
9.
x-\-y = 10.
x^-\.xy = 12,
x-y = 2.
10.
3 uv — v—lOuj
u-^2 = V.
4=x + Sy = 9,
11.
2x'-{-5xy=:3.
(a^ + 3)(2/-7) = 48,
12.
x-{-y = lS.
■st=A2,
13.
s-t = 19.
1 + ^ = 10.
14.
f-f=^-
2s + 3^=10,
8.
f2s + 3^ =
\t(s + t) =
x^y 15'
x—y = — 10.
faj + f 2/ = 15,
3ir- — = 24.
y
1.5 ic— .52/ = 6,
.1 x^ -\- .1 xy = 16.
16 + 4'y + 2w2 = 5i^v,
11 v — 5 w = 4.
-H-l^ = J + %
xy ar ^/^
2-1 = 7.
o; y
208 HIGH SCHOOL ALGEBRA [Ch. XII
132. Both equations quadratic : one homogeneous. An
equation is said to be homogeneous if all of its terms are of
the same degree in the unknown numbers (of. §§ 34, 93).
If one of two given quadratic equations is homogeneous,
the system may always be solved as follows :
Ex. 1. Solve the following system of equations :
6£c24-5a;2/-^6/ = 0,l (1)
2x2-2/2 + 5 0^ = 9. J (2)
Solution. On dividing Eq. (1) by y^, it becomes
whence (§126) ^^|or^^-|, (4)
i.e., x = lyov x = -^y. (5)
On substituting the first of these two values of x, viz., ^y, in
Eq. (2), we obtain
2(t2/)^-/ + 5(f.v)=9, •
i.e., /- 30 2/4-81 = 0,
whence (§ 124) 2/ = ^^ or' 2/ = 3,
and, since x = ^y, the coiTesponding values of x are 18 and 2.
'x = 2
Moreover, these pairs of values, viz., <j ' and
.y = *^' J
J are
^ = 3,
found to check, and are therefore solutions of the system.
Again, by substituting in Eq. (2) the second of the two values
of X in Eq. (5), we find two other solutions of the given system
VIZ. \ ^ ^ and \ ^'
Note. The above method may be somewhat simplified by substituting a
single letter, say v, for the fraction x/y in Eq. (3), i.e., by putting x — vym
the homogeneous equation. Thus, putting vy for x, Eq. (1) becomes
6 i?2y2 4. 5 W2/2 - 6 2/2 = 0^
and hence, dividing by ?/2^ 6 v^ + 5 u — 6 = 0,
whence (§ 126) w = f or t7= -f ;
and, since x = vy, therefore x = ^ y or x = — ^ y. From here on the work
is the same as that already given,
132-133]
QUADRATIC EQUATIONS
209
EXERCISE XCII
2. Which of the equations in Ex. 3-12 below are homogene-
ous ? Why ? Write a homogeneous cubic equation involving
the unknown numbers r and s.
By the method of Ex. 1 (or of the note) solve the following
systems of equations ; check your results as the teacher directs :
6.
xy-\-f=2S,
4:x'-27 xy + lSy^ = 0.
x^ -\-xy — 14: = y — Xj
2 a^ — 3y^ = xy.
^(s + Q=36,
b s" -1^ St + 12 f = 0.
^
i3y?-5uv=:2v\
\u{u — v) = 8.
10.
11.
12.
5a^-7a;2/-24 2/2 = 0,
xy-ir2y'^z=5.
aJ2/ + 3/-20 = 0,
5x^ = 13xy —^y^.
8aj2 + 2/2 = 36,
4a^-9a;2/ + 52/2 = 0.
2{:»? + f) = nxy,
x'-y' = 75.
5af + 4:xy = y%
x^-\-3x = 5-{-y.
Ex. 1. Solve
133. Both equations homogeneous except for the absolute
term. A system consisting of two quadratic equations each
of which is homogeneous in the terms containing the unknown
numbers can be solved by the method of § 132, Note.
x^ + 3xy + 2f = S, (1)
xy-4:f = 2. (2)
Solution. By substituting vy for x in Eqs. (1) and (2) we
obtain 3 vY + 3 vy^ -{- 2 y^ = S, (3)
and vY — vy^ — 4:y^ = 2, (4)
whence, from (3) and (4), respectively,
2
\x'
y
therefore
whence
3 -y^ + 3 v + 2
8
and 2/^
V'
4'
3v^ + 3v + 2 'y2_-y_4'
v = - 2 or 9.
(5)
(6)
210
HIGH SCHOOL ALGEBRA
[Ch. XII
Hence, from (5),
I.e.
if = l or/=3\, ^
2/ = 1 or — 1 or 2/ = + V 3^;
34
or
V,>j,
and substituting these values of y and v in
x=vy,
we obtain, as corresponding values,
a? = — 2 or 4-2, and also + 9 V^^^ or — 9 V^;
and, checking, we find the solutions of the given system to be :
Note. The success of the '■'•vy method" employed above is due to the
fact that by eliminating the absolute term from the given system of equations
we obtain a homogeneous equation (cf. § 132).
EXERCISE XCIII
Solve the following systems, and check your results:
2.
3.
6.
7.
ar^ + 2/2 = 29,
xy = 10.
m^ — mn = 8,
mn -\-n^ = 12.
af — Sxy = — 14,
xy-\-2f = S9.
f + 2xy = -%
x^ — xy=z70.
2x^-xy = 2S,
a^ + 22/' = 18.
?/2 4. 15 = 2 xy,
x' + y' = 21 + xy.
8.
10.
11.
12.
x^ + xy — 40,
27 + 2y'-3xy = 0.
2a?- + 3a!?/ + / = 20,
5a;2 + 42/' = 41.
xF — xy — y^= 20,
a;2_3(»2/ + 22/2 = 8.
i*^ H- 3 wv + 'y^ = 61,
u^-v^^Sl-2uv.
3f_5y2x_ ^
4 iC
~3"
'^ _ y^ + 2xy
l~2{l-x)'
13. Solve Ex. 1 by eliminating the absolute term and then
applying § 132.
By the method suggested in Ex. 13, solve :
x'-3xy + 4:f=:Sy, ^^ i3x'-5 xy -4.f = Sx,
x^-4:xy-^2y^ = 2y. ' [9 a^ + xy- 2 y^=6 x.
x'-4.xy-^3f = -3y, ^^ Ux' -^6xy -f = ^,y,
3x^ — 5xy = 6y. \6x-—9xy + 2y- = 2y.
14.
15.
138-134] QUADRATIC EQUATIONS 211
134. Special devices. The kinds of systems of equations
specified in §§ 131-133 occur frequently, and, although they
present themselves in a great variety of forms, they may
always be solved by the methods there given.
Special devices for elimination, however, often give sim-
pler and more elegant solutions; some of these devices are
illustrated below.
(i) Solving hy first finding x-\- y and x — y,
x-y = 5, (1)
Ex. 1. Solve the equations ,
[xy = -6. (2)
Solution. From (1), x" - 2xy -^y^ = 25, (3)
and from (2), 4 a;?/ = - 24 ; (4)
adding (4) to (3), x^-\-2xy-{- y' = 1, . (5)
whence x-\-y = ±lj (6)
and from (1) and (6), « = 3 or 2.
The corresponding values of y are y = —2ov — 3 ;
. • P 1 . (x = 3, (x = 2,
i.e., the solutions of the ffiven system are: J ^ and J
^ ^ \y = -2, l.y = -3.
(a;2 + 2/' = 5, (1)
Ex. 2. Solve the equations i „ , „
^ [x'-xy + f^^- (2)
Solution. Subtracting (2) from (1), xy = 2, (3)
whence, adding 2 • (3) to (1), x'' + 2 xy + 2/^ = 9, (4)
and subtracting 2 • (3) from (1), x^ — 2xy -{-y'^ = l\, (5)
therefore, from (4), a; + ?/ = ± 3, (6)
and from (5), x — y = ±l] (7)
from (6) and (7), we now easily find the following solutions :
\y = l, \y = 2, U = -l, U = -2,
all of which check.
212 HIGH SCHOOL ALGEBRA [Ch. XII
(ii) Solving hy dividing one equation hy the other.
a^^-/ = 3, (1)
Ex. 3. Solve the equations .
'^-2/=l. (2)
Solution. On dividing (1) by (2), member by member, we
obtain x-\-y = d, (3)
whence, from (2) and (3), a; = 2, and y = l.
ar^ + ^« = 26, • (1)
Ex. 4. Solve the equations ,
^ [x +y= 2. (2)
Solution. On dividing (1) by (2), member by member, we
obtain x^ — xy-^y^ = 13, (3)
and (2) and (3) may now be solved either like Exs. 1 and 2 above,
or by the method of § 131.
(iii) Solving by considerations of symmetry. An equation
is symmetric with regard to two of its letters if it is not
changed by interchanging those letters. Thus : x + y = S^
and §2 ^ st -\- 1^ = 5 (^s -{- f) are symmetric equations.
Two equations which are symmetric (or symmetric except
for the signs of one or more terms) may often be solved by
substituting u + v and u — v, respectively, for their unknown
numbers.
^ + 2/^=6, (1)
Ex. 5. Solve the equations
^xy = 2(x^y)-5. (2)
Solution. On putting x = it-{-v and y = u — v, the given equa-
tions become, respectively,
2u^ + 2v^ = 6, Sindu^-v^ = 4:U-5', (3)
therefore, eliminating v^ and simplifying,
u'-2u + l = 0,
whence u = l.
Substituting this value of u in either one of Eqs. (3), gives
v=±V2,
whence (since x = u-^ v, and y = u — v),
x=l± V2, and y~lT V2.
134] QUADRATIC EQUAIIONS 213
( oc^ -^ Tf = xy — 5f (1)
Ex. 6. Solve the equations { ^ ^
\x-^y + l = 0. (2)
Solution. On putting x = u-\-v and y =u — v, (1) and (2) be-
come, respectively,
2u^ + Quv'-u^ + v^ + b = 0, (3)
and 2 2^ + 1 = 0. (4)
From (4) u = — |,
and substituting this value in (3) gives
whence a; = 1 or — 2, and 2/ = — 2 or 1.
EXERCISE XCIV
By the method of 134 (i) solve the following systems :
ja^ + / = 13, |a^ + 2/' = l,
^' \xy = ^. ^^' |25a^y + 12 = 0.
1^2 + ^2^61,
'm4-n = 24.
[5mn-2 = 0.
mn Q |a;2 + 2/2 = a,
— y. 12.
9 [a;-f-y = 6.
By the method of 134 (ii) solve the following systems :
13- „ 15. ^
\r — s = l. [r— p = 7.
14. \ 16
r4-s= 7.
aj^ + 2/3 = a,
a: + 2/ = 6.
By the method of 134 (iii) solve the following systems :
17. |-^ + -"-* = 26. f2(. + ,) = -§f^,
I a; + 2/ = 6. 20. ] 5
a: -2/ = 7.
18. \ ■ -2/) = -^^2/' f2x2_ _^22/^ = 62,
2. 21. { _ ^.
[ a; — 2/ — o 0^2/ = ~~ "1-
^^- [ar^ + 2r'' = 37. ^^' 1^ + 2.9/4-^-22.
^ |3(a;-2/
[a; + 2/ =
214
HIGH SCHOOL ALGEBBA
[Ch. XII
Solve the following systems of equations, choosing for each
the method (§§ 131-134) which seems to you best:
23.
24.
25.
26.
27.
28.
29.
30.
31.
^_?/_16
y X lo'
a;-2/ = 2.
xF — xy = 6,
a;2-f2/2 = 61.
^+^. = 74,
a;- 2/ .
1-1 = 2.
rcf. Ex."!
13, §104. J
;i-^ y^
i+l=
= 91,
7.
•1 + 1-1
X y Z
.^2/ 18
a; + ?/ . X —y _ 10
a; — ?/ x-{-y 3'
a;^ + 2/^ = 45.
1 1
= 119,
-7 = i.
aJ^ + 2/^ + i» = 2/ + 14:,
a;?/ = 6.
= 96 — 4mn ,
6.
m -\-n
33.
34.
35.
36.
37.
38.
x-^y = 25,
V^+ ■Vy = 7.
a^ + 2/ + 2Va; + 2/ = 24,
a; — ^ -f- 3Va/' — 2/ = 10.
2(a^2 + /)=o.r2/,
x^-2xy = S y%
y{x + y)=4..
(2+^(2/ + l)=4,
V2 + aj-V// + l = -J-.
{a' + h' = n,
|a262 = 4.
39. I
40.
41.
42.
43.
44.
45.
a^ 4- 2/^ + 6 Va;^ + 2/^ = So^
a;2_2/2 = 7.
tv=- 26.
^' + sV + ^^ = 9,
s2 + 6-^ + 2/' = 3.
s^-t^ = 37,
st(s-t) =12.
x^ + 2/^ = 97,
a; + 2/ = — 1-
o , 3 771 f.
3 mn H = o,
n
3^n + — = 2.5.
m
2/ a; '
i;U] qUADHATIC EQUATIONS 215
PROBLEMS
1. The sum of two numbers is 14, and the difference of their
squares is 28. What are the numbers ?
2. Find two numbers whose difference is 15, and such that if
the greater is diminished by 12, and the smaller increased by 12,
the sum of the squares of the results will be 261,
3. Find two numbers whose difference is 80, and the sum of
whose square roots is 10.
4. Given that one root of a quadratic equation is 4 times the
other and that their product is ^, find the roots ; then form the
equation which has these roots (cf. § 72).
5. The sum of the roots of a quadratic equation is 12, and
their product is — 189. What is the equation ?
6. The sura of two numbers, their product, and also the dif-
ference of their squares are all equal ; find the numbers.
7. If the length of the diagonal of a rectangular field, contain-
ing 30 acres, is 100 rods, how many rods of fence will be required
to inclose the field ?
8. Find the dimensions of a rectangular field whose perimeter
is 188 rods and whose area "will remain unchanged if the length
is diminished by 4 rods and the width increased by 2 rods.
9. The sum of the circumferences of two circular flower beds
is 56|^ feet, and the sum of their areas is 141^ square feet. Find
the radius of each. (Cf. Ex. 14, p. 201.)
10. A circular table whose radius is 3J- feet has the same area
as a rectangular table whose length is 5 inches more than its
breadth. Find the dimensions of the rectangular table.
11. A sum of money lent at a certain rate of interest gives an
annual income of $ 450 ; if the sum were ^ 500 more and the
rate 1 % less, the annual income would be $ 50 less. Find the
principal and the rate.
12. A sum of money at interest for one year at a certain rate
amounted to $11,130. If the rate had been 1 % less and the
principal $100 more, the amount would have been the same.
What was the principal, and what the rate?
216 IHGU SCHOOL ALdKliRA [Ch. XII
13. A formal rectangular flower garden is to be enlarged by a
border whose uniform width is 10 % of the length of the garden.
If the area of the border is 900 sq. ft., and the width of the old
garden is 75 % of the width of the new one, find the dimensions
of the garden and the width of the border.
14. A certain kind of cloth loses 2 % in width and 5 % in
length by shrinking. Find the dimensions of a rectangular piece
of this cloth whose shrinkage in perimeter is 38 in., and in area
8.625 sq. ft.
15. The perimeter of a right-angled triangle is 24 ft., and its
area is 24 sq. ft. Find the length of each side in the triangle.
16. In the right-angled triangle ABC, BD is
drawn perpendicular to AC. If BC =12, AC =11,
and BD=^AD'DC, find BD, AD, and DC
17. The combined capacity of two cubical coal
bins is 2728 cu. ft., and the sum of their lengths
is 22 ft.; find the length of the diagonal of the
smaller bin.
18. Find two numbers whose product is 8 greater than twice
their sum, and 48 less than the sum of their squares.
19. Find two numbers such that the sum of their fourth powers
is 881 while the sum of their squares is 41.
20. The total area of the walls and ceiling in a room 9 ft.
high is 575 sq. ft. Find the length and breadth of the room
if their sum is 24 ft.
21. A farmer found that he could buy 16 more sheep than
cows for $ 100, and that the cost of 3 cows was $ 15 greater than
the cost of 12 sheep. What was the price of each ?
22. If 5 times the sum of the digits of a certain two-digit
number is subtracted from the number, its digits will be inter-
changed; and if the num.ber is multiplied by the sum of its digits,
the product will be 648. What is the number ?
23. Find two numbers such that the square of either of them
equals 112 diminished by 12 times the other.
1;)4-135J QUADRATIC EQUATIONS 217
24. If 5 is added to the numerator and subtracted from the
denominator of a given fraction, the result equals the reciprocal
of the fraction ; and if 2 is subtracted from the numerator, the
result equals J of the original fraction. Find the fraction.
25. The distance (s) in meters, through which a body falling
from a position of rest passes in the ^th second of its fall is
given by the formula s = \g (2t—l) ; and the total distance (S)
fallen in t seconds is S=^gt^. How long has a body been falling
when s = 44.1 meters and S = 122.5 meters? If g is less than 10,
what is its value?
26. Solve the problem of Ex. 25 if s and S are each expressed
in feet, and s = 112^7^ and S = 2571
27. In going 40 yards more than ^ of a mile the fore wheel of
a carriage revolves 24 times more than the hind wheel ; but if the
circumference of each wheel were 3 ft. greater, the fore wheel
would revolve 16 times more than the hind wheel. What is the
circumference of the hind wheel ?
28. A merchant paid $125 for an invoice of two grades of
sugar. By selling the first grade for $91, and the second for
$36, he gained as many per cent on the first grade as he lost on
the second. How much did he pay for each grade ?
29. Two trains start at the same time from stations A and
B, 320 miles apart, and travel toward each other. If it re-
quires 6 hr. and 40 min., from the time the trains meet, for the
first train to reach B, and 2 hr. and 24 min. for the second to
reach A, find the rate at which each train runs.
30. After traveling 2 hr., a train is detained 1 hr. by an acci-
dent ; it then proceeds at 60 % of its former rate, and arrives
7 hr. 40 min. late. Had the accident occurred 50 miles farther
on, the train would have been 6 hr. 20 min. late. Find the
distance traveled by the train. (Cf. Ex. 42, p. 164.)
135.* Simultaneous quadratics not always solvable by methods
already given. While many systems containing quadratics (and
* § 135 may, if the teacher prefers, be omitted till the subject is reviewed.
'21S HIGH SCHOOL ALGEBRA [Ch. XIl
some containing still higher equations) may be solved by the
methods of §§ 131-134, these methods do not always suffice for
the solution of such systems.
Thus, inspection shows that the system
a^-3a; + 82/ = 4,
3a;^-16/ + 20 2/ = 9,
cannot be solved by the methods of §§ 132-134; and elimination
by substitution (as in § 131) leads to an equation of the fourth
degree in one unknown number, viz., to
a,>4 _ 6 ar^-i»2- 6 a; +12 = 0,
which cannot be solved by the elementary methods already studied.
Such equations are discussed in higher algebra.
For all systems like the above, however, approximate solutions
may be obtained by means of graphs (cf. § 143).
136.* Systems containing three or more unknown numbers. Some
systems containing three or more simultaneous equations, some
of which are quadratic, may be solved by elementary methods.
E.g., if one equation of a given system is quadratic, and all the
others are of the first degree, then a slight modification of the
method of § 131 will provide a solution (cf. El. Alg. § 180).
The solution of such systems in general is, however, beyond
the limits of this book.
* This article may, if the teacher prefers, be omitted till the subject is
reviewed.
CHAPTER XIII
GRAPHIC REPRESENTATION OF EQUATIONS*
137. Introductory. Although an equation in two unknown
numbers has (§ 99) an infinitely large number of solutions,
and is in that sense indeterminate, yet by a beautiful device,
due to the celebrated mathematician and philosopher Des-
cartes (pronounced da-kart', born 1596, died 1650), a perfectly
definite picture of such an equation may be made (cf. § 139).
138. Axes. Coordinates. Let us draw (as Descartes did)
two perpendicular straight lines X'X
and Y' Z", cutting each other in the
point 0, and call these lines the
coordinate axes. If we now agree
that distances measured toward the
right from Y'Y^ or upward from
X'X^ shall be positive, while dis-
tances toward the left, or downward,
shall be negative, then any point in
the plane of this page can be located
as soon as we know its distances from the axes X' X and Y' Y.
Thus, to locate the point P, 3 inches from F'^and 2
inches from X' X^ we measure off 3 inches (represented in the
diagram by 3 spaces) toward the right from 0, to the point
M^ say, and then 2 inches upward from M.
The numbers which serve to locate a point (in this case
1
(
p
— -1
,
r
J
M
-
-
-
p?
/
1
*This chapter should he included in the course whenever possible
omission, however, will not break the continuity of the work.
HIGH SCH. ALG. — 15 219
its
220
HIGH SCHOOL ALGEBRA
[Ch. XIII
3 and 2) are called the coordinates of the point. The point
P may be represented by the symbol (3, 2).
Similarly the point ^ (— 3, 4) is located by measuring 3
spaces toward the left from 0, and then four spaces upward.
The point i2 (— 2, — 3), also, is represented in the figure.
Note. This plan of locating points in the figure somewhat resembles that
used to locate places on the earth's surface by their latitude and longitude.
The coordinate axes correspond to the equator and the prime meridian.
y
1
p
X
,
^
X
Q
s
,
1
EXERCISE XCV
1. Name the ic-coordinate (i.e., the distance from the axis
Y'Y) of each point located in the figure below. Also name the
^/-coordinates of these points.
Draw a pair of axes as in § 138 and
locate the following points :
2. (5,4); (3,7); (4,-2); (-3,1);
and (-4, -6).
3. (-i,l); (i,|); (H, -3); (4,1);
and ( — I, —5).
4. (3, 0); (-5, 0); (0, 8); (0, 0) ;
and (0, -2).
5. Where are the points whose ^/-coordinate is ? Where are
those whose ^-coordinate is ? those whose ^/-coordinate is 3^ ?
6. Locate five points each of which has its a7-coordinate equal
to its ^/-coordinate, and draw a line through these points. Does this
line contain any other points whose two coordinates are equal ?
7. Where are the points which have their ^/-coordinates oppo-
site in value to their respective ic-coordinates ?
8. Verify that the equation 2 x — y==3 is satisfied by each of
the following number-pairs : (0, — 3) ; (1, — 1) ; (2, 1) ; (3, 3) ;
(4, 5) ; then locate the point corresponding to each pair. On
what kind of a line do these points lie ?
9. Measure the coordinates of several other points on the line
mentioned in Ex. 8. Are these coordinates solutions of 2 x—y=3?
i;W-140] (lliAPlIJC liEPIiESKNTATION OF EQUATIONS 221
10. Locate the following points : (0, 5) ; (0, — 5) ; (5, 0) ;
(-5,0); (4,3); (-4,3); (4,-3); (-4,-3); (3,4); (-3,4);
(3, —4); ( — 3, —4). On what kind of a line do they lie?
139. The picture (graph) of an equation. Consider the
equation 2 a; — y = 3.
This equation is, manifestly, satisfied by the following
pairs of values of x and 7/ (§ 99) :
(-1, -5); (0, -3); (1, -1); (2, 1); (3, 3); (4,5); etc.
If we now locate (as in § 138) the points A, B, (7, etc.,
corresponding to these number-pairs, we find that they are
not scattered at random over the
page, but that the^ all lie upon the
straight line US in the figure (cf.
Ex. 8, p. 220). Moreover, the co-
ordinates of every point in the line
RS, and those of no other points
whatever, satisfy the given equa-
tion.*
For these reasons, the line MS may
be regarded as the picture of the
equation ; it is usually called the
graph, also the locus of the equation. That is, the graph
(or locus) of an equation is the line (or lines) containing
all the points (and no others) whose coordinates satisfy the
given equation.
140. Drawing of graphs. The method illustrated in § 139,
for finding the graph of an equation in x and «/, may be
stated thus :
(1) Solve the given equation for ?/, in terms of x.
V /a
7
t
7
T
X o~r^ zE.
/
M
t
i
f
^
t.
-hi v^
^ r
* Let the pupil test this statement by careful measurement on a large and
well-drawn figure. The proof of its correctness follows easily from the
theory of similar triangles in geometiy.
•>•^:>
Hid 1 1 S(1I(K)L ALGEBRA
[Ch. XIII
(2) Assign to rr a succession of values, such as 0, 1, 2,
3, ••• (also —1, —2, — 3, •••), and find the corresponding
values of y ; ^.e., find a succession of solutions of the given
equation.
(3) By means of a pair of axes locate the points corre-
sponding to these solutions, — use cross-section paper.
(4) Draw a line connecting these points in regular order ;
this line is (approximately) the graph of the given equation.
E.g., to find the graph of the equation 3y—x^ = 0, we solve
the equation for y in terms of x, and tabulate the corresponding
values of x and y, thus :
Locating the points 0, A, B, C,
etc., and connecting them in order,
we obtain the line NML •••£/, which
X-.
X
y
Points
1
i
A
2
1
B
3
3
C
-1
i
H
— 2
i
K
-3
3
L
•
:
'
II ITT
~E ji T
j-i^ 4
t " 7
\ t
\ I
-M^ J?
A i^ t
-> f-
X ~K-
% t
\ /
T ' k\ /(^ t.
^ Tt- -^ ^
± :£ ^
vr'
is a good approximation to the graph
of the given equation.
By assigning to x values between and 1, 1 and 2, etc., and
finding the corresponding values of y, we can locate points
between and A, A and B, etc., and thus draw a closer approxima-
tion to the required graph.
Note. The graph of a first decree equation in x and y is (cf. § l.SO) a
straight line (hence a first degree equation is often called a linear equation).
In this case, of course, only two points (i.e., two solutions of the equa-
tion) need be found in order to draw the complete graph.
UO-Ulj GRAPHIC REPRESENTATION OF EqUATlONti 228
EXERCISE XCVI
1. Find six solutions of 2x-{-y = 12, locate the points
determined by these solutions, and draw the graph of the equation.
Using the plan of Ex. 1, draw the graph of :
2. a; + 2^ = 8. 5. 2a;-3?/ = 0.
3. x-2y = l. 6, 'Sx + 2y = 12.
4c. 3x = y. 7. 2 2/ - ar = 0.
8. Draw the graph of 3 a; = 2 [i.e., 3x-\-0'y = 2, cf . Ex. 5,
p. 220]; of 22/ = 5; of x' = -l; of ar = 9.
9. What is the graph of x = ? of ?/ = ? Without making a
drawing, show that the graph of 2 x — 7 y = must pass through
the point in which the axes intersect. Is this true for the graph
of every equation of the form ax -\-by = 0?
10. In the equation Ax — 5 y = 10, when x = 0, y=? When
y = 0, x=? From these two solutions of the equation draw its
graph (cf. § 140, Note).
Draw the graph of :
11. x-y = 0. 17. 5x-2y = 20.
12. x-\-y = 0. 18. Sx + 5y = 7^.
13. 3 a; = — 11. 19. 7x — y = S^.
14. ?/2-16 = 0. 20. Dx-\-2y=21.
15. 2x-\-Sy = 6. 21. 4:X = y\
16. 2x-3y = 6. 22. 3a^-4i/ = 0.
Calling the coordinate axes S'S and TT instead of X'X and
Y' Y, draw the graph of :
23. 4:t = Ss. 24. s-t = 5. 25. 2t-3s^ = 0.
141. Drawing of graphs (continued). Thus far we have
considered only the simplest kind of graphs ; the method
employed will serve, however, for any equations whatever
in two unknown numbers.
224
HIGH SCHOOL ALGEBRA
[Ch. XIII
Ex. 1. Construct the graph of 2y — x^ = 0.
CoxsTRucTiON. Solving this equation for y in terms of x, and
tabulating the corresponding values of x and y, we obtain :
X
y
Points
1
2
4
A
B
-1
-2
-4
H
K
:
:
'
Yi-
T
L
it
-4^
jT
^ t V-
^^ - ,-i^ i
v^
iy
if
^
J
j
I
I
' ^
• X
On locating these points and connecting them in order, we ob-
tain the required graph, viz., : KHOA •••.
Ex. 2. Construct the graph of 4 a^ + 9 2/^ = 144.
Construction. Proceeding as in Ex. 1, we obtain :
a; = |Vl6-
2/^.
y
X
Points
6 or -6
^or^'
1
1 Vis or - 1 \/l5
BovB'
-1
1 Vis or - f Vl5
Hov H'
-2
3 V8 or - 3 \/3
Kov K'
.IjL
AJ^"--^
^^
^.
V ^/
S
^44-
-2.]"-^
4^
XaT
n\
t
#s.±
n' ^^
\7
I
On locating these points, using approximate values of the
square roots, and connecting them by a smooth curve, we obtain
the graph ABNAN'A.
141-142 ] GliAPHIC liEPRESENTA TION OF EQ UA TIONS 225
Note. The limitations of the graph in Ex. 2 are interesting. Thus, since
X = |\/16 — 2/2, X must be imaginary when y is greater than 4 ; hence, as our
graphic representation admits real values only, there are no points on the
curve whose ^/-coordinate is greater than 4. Similarly, it may be shown that
there are no points on the graph below y = — 4. And solving the given equa-
tion for y in terms of x shows that there are no points on the graph at the
right of ic = 6, or at the left of aj = — 6.
Ex. 3. Construct the graph of xy = 4.
Construction. Proceeding as in Exs. 1 and 2, we obtain:
4
X
X
y
Points
00
1
4
A
2
2
B
3
1
C
.
•
•
•
•
•
.
•
•
-1
-4
H
-2
-2
K
Y
1
1
\a
V
\
B
X
^,
N
'<
—
-
— .
^
o
X
N
K
\
\
fl\
-
On locating these points and connecting them by a smooth
curve, we obtain the graph AB • • • HK.
142. Intersection of graphs. Since (0, — 3) and (4, 0) are
solutions of the equation Z x — ^y
= 12, therefore its graph is the line
AB in the figure (of. § 140, Note).
If we now draw the graph of
3 rr + «/ = 2, using the same axes as
before, we obtain the line HK.
Moreover, since P, the point in
which AB and HK intersect (i.e,
cut) each other, lies on each of these
-L4-F-
kY^
r
X
^ B
, >^
-_U ^^ -
xT o^H 7^ -x
W-
^^
^'^ -t
^^ t
^^ \h
v'
X
226 IIIGa SCHOOL algebra [Ch. XIII
graphs, therefore its coordinates (§ 138) must satisfy each
of the given equations (cf. § 139).
Hence, we may find the coordinates of P by merely solving
the given equations as in § 101, and without even drawing
their graphs.
Approximate values of the coordinates of P may, of course,
be found by direct measurement of OM and MP ; this
measurement constitutes a graphical solution of the given
equations. Let pupils use both methods for finding these
coordinates, and compare results.
Remark. From what has just been said, and from the defini-
tions in § 100, it follows that (let pupils explain why) :
(1) The graphs of consistent equations intersect each other.
(2) The graphs of inconsistent linear equations are parallel
lines.
EXERCISE XCVII
Construct the graph
of
:
4. y' = Sx.
8. xy = 5.
5. y={x-iy.
9. 3x' + 4.f- = 12.
6. x' + y' = 25.
10. 3x'-4.y' = 12.
7. 16a^+/ = 64.
11. 4.f- = a^.
12. Show from the equation of Ex. 4 that no part of its graph
lies to the left of the ^/-axis (the line Y' Y) .
13. Show from the equation of Ex. 6 that no part of its
graph lies outside a certain square whose side is 5 ; similarly,
show that the graph of Ex. 7 is contained within a certain rec-
tangle whose dimensions are 16 and 4.
14. Show from the equation of Ex. 8 that its graph consists of
two infinitely long branches, one in the quarter XOY and one in
the quarter X'OY'.
15. If the graph of xy = — 5 were drawn, how would it differ
from that of xy = 5? Why ?
142-143J GEAPHW REPRESENTATION OF EQUATIONS 227
16. Draw the graph of 2x-\-y = dr -{-S and show that this
graph differs from that of Ex. 5 only in being moved two divi-
sions upward. Explain why this should be so.
17. Find, both by solving the equations, and by measurement,
the coordinates of the point in which the graphs of x-{-y = ^ and
2 a; — 2/ = 4 intersect ; compare your results.
In each of Exs. 18-23 below find, as in Ex. 17, the coordinates
of the point in which the graphs of the two equations intersect :
18. i ' 21. ' ^^ '
\2y-x=
6. 2a; + 2?/ = 8.
^^ 1 2.^^72/, . 22^ [^•+2/=3,
U-4-
\\x
y-x=5, ^ [^x + ^y=li.
2^ {Sy + 2x = 17, ^^ \2x-y = S, (cf. §§ 139,
2x-y = 5. [3y-x' = 0. 140.)
24. How are the graphs of two first degree equations in a; and
y related when the equations are inconsistent (cf. Ex. 21) ?
when they are simultaneous and independent (cf . Ex. 20) ? simul-
taneous and not independent (cf. Ex. 22) ?
143. Graphic solution of simultaneous equations. If the
graph of one of two simultaneous equations is drawn across
the graph of the other (^i.e., if the same axes are used in
both drawings), then the measured coordinates of each point
in which these graphs intersect constitute an approximate
solution of the given system (cf. § 142). The following
examples will illustrate this procedure.
Ex. 1. Solve graphically the simultaneous equations
Sx-4:y = 12,
3x-^y=2.
Solution. The graphs of these equations are the lines AB
and HK (figure, § 142) ; and the (measured) coordinates of P,
their point of intersection, are approximately .x- = | and y =—2,
which, by trial, are found to be a solution of the given system.
228
HIGH SCHOOL ALGEBRA
[Ch. XIII
Ex. 2. Solve graphically the system J ^ ^'
\y + 3 = 2x.
Solution. The graphs of
these equations are, respec-
tively, SPOQT and AB.
The coordinates of P, one of
their points of intersection,
are approximately a: =3.4 and
y = 3.7, which constitute an
approximate solution of the
given equations.
So, too, the coordinates of
Q (viz., X = .65 and y = —1.6)
constitute an approximate
solution of the system.
-y^ y
Ml ^-
l! ^-^"^
2^^
^^
VI
4^ ZJ- ~t
X "--jt X-
^^t
M
f\
/ ^s.
^ ^^
^ >- ~
^r
Ex. 3. Solve graphically the system
xy = 4:.
Solution. The graphs
of these equations are, re-
spectively, AB'A'B and
CPQC'P'Q'-, and the co-
ordinates of P, one of their
points of intersection, are
approximately x = 1.4 and
y= 2.9, which constitute an
approximate solution of
these equations.
So, too, the coordinates
of Q (x= 5.75 and y = .7),
P'(x = -1.4 and y= -2.9),
and Q' (x = — 5.75 and y =
— .7) are approximate solutions of the given equations.
Note to the Teacher. In the case of a pair of simple equations the
solution by the method of § 101 is usually easier than the graphic method,
and its results are exact instead of approximate. There are, however, many
other cases in which the graphic method is advantageous ; hence some prac-
tice with it, even on simple equations, is recommended.
Y J
1
c
[
\
\p
^*-'"*'^ "
"-k"**
■^s
t^
Si
X
X
Ar
^^ " I
)
n
A X
- ^;
s^^
y
_'^-^»:
-^
pTi
5^
- T-
7-
14^-144 J GRAPHIC liEPRE^EN TA TION OF EQ UA TIONS 229
EXERCISE XCVIll
Solve graphically the following systems of equations, and check
your results as the teacher directs :
^ {x-^y = 3, ^^ ja.'- + 2/' = 25,
5.
6.
8.
10.
11.
12.
x — y = 3.
12/ = ^-
2x-y = 5,
14.
■x' + f^ = 2o,
4.x = W-y.
x-\-y=^l.
4?/4-3a; = 5,
15.
y = 3x + 2,
0^=5.
x^ = 4.-y\
x + y = A,
y = 2-x,
5x-10y = S6,
2x-\-3y=-S.
16.
17.
{ . 4.x
x = l.
xy=- 10,
x+y = 2.
4a.- + |2/ = 6,
ix-^y=S.
18.
ix'+9==y,
\y = x^-5x-\-G.
2x^y\
19.
'x'-^y^ = 25,
2y=.x.
[xy=-^.
2a;-/-l = 0,
20.
'4.x'-9y' = S6,
2a; + 6v/ + T = 0.
y4-r = 25.
x-2y=-12,
21.
y = Bx—15,
y-x^=-2x-2.
x^-9x- + 23x-15 = 'y.
By referring to the graphs in the above exercises, find the
number of solutions of a system consisting of:
22. Two simple equations.
23. A simple and a quadratic equation.
24. Two quadratic equations.
144. Graphic solution of equations containing but one
unknown number. By slightly extending the method of
§ 143, we may find graphic solutions for quadratic equations
in one unknown number.
230
HIGH SCHOOL ALGEBRA
[Ch. XUI
IT
-\ \-
w jfe
M -j^
r- ^-4
X t
\ I
^ f
"^r lo j_ :S
-^^ m 7^
_^ L
^-4-
v/ ^
F -^
|2/ = ^=
12/ = 0.
Thus, the roots of x- — 2x — 2 = are manifestly the values of
X found b}^ solving the pair of simultaneous equations :
2x-2, (1)
(2)
Now the graph of (2), viz., the a>axis,
cuts the graph of (1), viz., the curve MQS,
in the points P and E, whose coordinates
are (approximately) ic = 2.75, y = 0, and
x= — .75, y=0. And since (§ 143) each
of these pairs of values constitutes an
approximate solution of (1) and (2), there-
fore 2.75 and — .75 are approximate roots
of a;2_2,^_2::=0.
EXERCISE XCIX
Find graphic solutions of :
1. x:'-6x + S = 0. 3. 6a;H-5x-4 = 0.
2. x''-Sx-\-5 = 0. 4. a^_3a;2-6a; + 8 = 0.
5. Show graphically that Ax^— 4:X-—llx-\-6 = has one
root between and 1, and a second root between — 1 and — 2.
What is the third root of this equation ?
6. Show that one root of cc^ — 7 x- + 9 .^• = 1 lies between 1 and
2. Between what integers do each of the other two roots lie?
7. Corresponding to any given value of x, how does the value
of y \\\ y = x' — 6x-\-6 compare with its value in y = x^ — 6 x-{-7 ?
Could, then, the graph of the second equation be obtained by
merely moving that of the first upward through one division ?
8. Compare the graphs of 2/=2a7^—10a;— Sand ?/=2ic^—10aj+l;
also those of y = 3-\-4:X — x''^ and y = 10-\-4:X — x"\
9. By first constructing the graphs of y = oi^ — 6x + 6,
y = x^ — 6 X -\- 7, etc., compare the roots of
X-
6 a; + 6 = 0,
X' - 6 X -\-7 = 0, x' - 6 X + S = 0, x^ - 6 X -{-9 = 0, X- - 6x -{-10 = 0,
anda;2-6a;4-ll =0.
144-14.-.] linAPlllr UEritESENTATION OF EQUATION t> 281
10. As in Ex. 9 compare the two smaller roots of
ic3 - 7 a;2 + 9 a; - 1 = with those of a;"^ - 7 a;^ _^ 9 ^ _ 3 ^ q ^^^^
jB3_7a;2_j_9^_5^()
Note. Exercises 9 and 10 illustrate how, by changing the absolute
term in an equation, a pair of unequal roots can be made gradually to become
equal and then imaginary.
11. Show that the roots of x'^-2x-2 = (§ 144) can be
found from the graphic solution of the system
y = ^, (1)
y-2x-2 = 0. (2)
12. Show that the graph of Eq. (1) in Ex. 11 may be used in
the solution of other quadratic equations {e.g., x^ -f 5 a? = 7) also.
13. Is the method given at the top of p. 230, or that suggested
in Exs. 11 and 12, to be preferred when we have several quadratic
equations to solve graphically ? Explain.
By the method of Ex. 11 solve :
14. a;^-2a^-2 = 0. 17. x' = x-\-^.
15. a;"^ — a; = 0. 18. 12 a; — 4 a;"' = 5.
16. x' + x-4. = Q. 19. 2a;^-a; = -3.
145. Use of graphs in physics, engineering, statistics, etc.
Descartes's plan for graphically representing equations has
now been adopted by practically all scientific men to repre-
sent simultaneous changes in related quantities. Physicists,
chemists, engineers, physicians, statisticians, etc., all find
that this graphic representation of related changes often
gives at a glance information which could be secured other-
wise only by considerable effort, and that it often brings out
facts of importance which might otherwise escape notice.
As a simple example of the use of graphs in this way let us
consider the following temperature readings, taken from the
U. S. Weather Bureau report, for 28 hours beginning at noon on
Feb. 5, 1906, at Ithaca, N.Y.
^32
UIGII SCHOOL ALGEnnA
[Ch. xm
IIR.
Tem.
-
-11
^°H
-
T
■V,
s
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
10°
9
8.5
7
6
5
3.5
3
2
1
0.5
-5
-6
-7
-6
-6.5
-6
-8
-9
-11
-3
2
3
3.5
5
3
Si
s
—
3
=1
s
\
^i
\
\
Z^
V
" z:^
\
z
-Ji
^,
\
I i£
^O 1 1
HOURS l_
- ^
O i
4
6
8
JO \ 1;2 1
2 4 6 8 10/
12 2 4
\
- T-
\
r
\
- ^ -
_^o
\
7
\
-^^^ / -
/
sy^'^s, 1 _
5 J
: si:
-10°
\t
yz .
/
1
the
mi
hi^
tal
sti
Her
3sa
ich,
;hes
^Ids
nila
idie
e
Ql
a
,t
t
Lt(
d
t
e
.n
a
h
id
a
d
n
is
\
nc
} t
les
he
a ]
ir
igi
Ic
ab
tic
)W
o^^
ifo
ire
om
ul
)n
r
re
rn
s
P
at
s
H
st
aa
y
ai
ed fig
as to i
)idly ]
point
Ltion £
iekl i
ed.
'ures and the graph answer
:he temperature — when, how
t rose or fell, what were its
s, etc. The graph, however,
it a mere glance, while the
b only after they have been
Note. For other interesting applications of graphs see Tanner and
Allen's Analytic Geometry, pp. 73-78. Also, and especially, " Graphic Meth-
ods in Elementary Algebra," by Prof. William Eetz, in /School Science and
Mathematics, vol. 6, pp. 683-687. This article gives many good suggestions
as well as valuable material and references.
EXERCISE C
By reference to the above temperature graph (usually called
thermograph)^ answer the following questions :
1. Between what hours was the temperature below 0° ? When
was it lowest ?
2. When was the temperature falling most rapidly ? Explain.
145 J GRAPHIC liEPBElsENTATlON OF EQUATION H 233
The following tables give the population (in millions) of the
countries named, for certain years between 1800 and 1900.
British Isles
Lands now in the
German Empire
France
Unitei
States
Year
Population
Year
Population
Year
Population
Year
Population
(millions)
(millions)
(millions)
(millions)
1801
15.9
1816
24.8
1801
27.3
1810
7.2
1811
17.9
1837
31.5
1821
30.4
1820
9.6
1821
20.9
1847
34.7
1841
34.2
1830
12.8
1831
24
1856
36.1
1861
37.3
1840
17
1841
26.7
1865
39.4
1866
38
1850
23.2
1851
27.8
1872
41
1872
36.1
1860
31.4
1861
28.9
1876
42.7
1876
36.9
1870
38.5
1871
31.4
1885
46.8
1881
37.6
1880
50.1
1881
34.9
1895
52.2
1891
38.3
1890
62.6
1891
37.7
1896
38.5
3. Taking the number of years after 1800 as the cc-coordinate
and the population (in millions) as the ^/-coordinate, locate the
several points represented by the above table for the British Isles,
and join these points by straight lines. Similarly, draw graphs
for the remaining tables.
4. By reference to your graphs, compare the population of the
countries named in 1830; in 1856; in 1880. By reference to the
tables compare the populations in 1871.
5. In making comparisons like those of Ex. 4, is it easier to
use the tables or to use the graphs ? Why (cf. § 145)?
6. By reference to the graphs answer the following questions :
(1) When did the population of the United States first exceed
that of the British Isles ? that of France ? that of Germany ?
(2) During what years has the population of the United States
increased most rapidly ?
CHAPTER XIV
IRRATIONAL NUMBERS - RADICALS
146. Preliminary remarks and definitions. A number
which may be expressed as the quotient of two integers,
positive or negative, is called a rational number.
J^.^., 3( = f); -7l( = ^); 2. 75( = |f|), etc., are rational
numbers.
Nearly all the numbers thus far used have been rational,
although we have met a few such forms as V2 and V — 5.
In this and the next chapter we shall examine more
closely such numbers as V2 and V— 5. These numbers are
particular cases of a/^, which is defined (§ 113) by the
equation (VaY^a; hence ( V2)2 = 2 and (V^=~5)2=-5.
The numbers V2 and V — 5 resemble each other in that
neither of them is rational (since no rational number squared
is 2 or — 5), but, as we shall soon see, they differ widely
in another regard.
By squaring 1 and 2, we find that V2 is greater than 1 and
less than 2 ; then by squaring 1.1, 1.2, 1.3, •••, we find that
V2 is greater than 1.4 and less than 1.5; similarly, V2 is
greater than 1.41 and less than 1.42 ; etc.
Since V2 lies between 1 and 2, 1.4 and 1.5, 1.41 and 1.42,
etc., therefore, if 1.4 (or 1.5) is taken for V2, the error is
less than 0.1 ; if 1.41 (or 1.42) is taken, the error is less than
0.01; and, by continuing this process, we can find rational
numbers which approximate V2 to any required degree of
accuracy.
234
140-147] IRRA TIONAL NVMBEIiS — liADTCA LS 235
On the other hand, since the square of any rational num-
ber is positive, therefore we cannot express V — 5, even
approximately, by means of rational numbers.
Numbers like V2, which are not rational, but which may
be expressed approximately to any required degree of accuracy
by means of rational numbers, are called irrational numbers.
^.g., V2, 5 — V7, and 10 4- v2 are irrational numbers.
Numbers like V — 5, wliich cannot be expressed, even
approximately, by means of rational numbers, are called
imaginary numbers (cf. § 114, Note). Rational and irra-
tional numbers taken together are called real numbers.
Note to the Teacher. Emphasis should be laid upon the fact that
although such numbers as V2 can be expressed only approximately by means
of rational numbers, they are, nevertheless, just as exact and definite as are
integers and fractions. n
Thus, let ABCD be a square whose side AB is 1 foot long,
and let x represent the number of feet in its diagonal AC,
then it is easily proved by geometry that
cc2 = 2, i.e., that x=y/2.
The numbers 1, 1.4, 1.41, 1.414, 1.4142, etc., are successive
approximations to the length of this diagonal, but its exact
length is V2 ; hence the necessity of including such numbers as y/2, in our
number system.
It will be worth while also to connect this latest extension with the exten-
sions previously made (see p. 16, footnote). Thus fractions arose from gen-
eralizing division ; negative numbers arose from generalizing subtraction ; and
in the present article it appears that generalizing evolution introduces two
further new kinds of numbers, viz., the irrational and the imaginary.
In other words : while the direct operations (viz., addition, multiplication,
and involution) with positive integers always produce results that are positive
integers, the inverse operations (viz., subtraction, division, and evolution)
lead respectively to negative, fractional, and irrational and imaginary num-
bers, and demand for their accommodation that the primitive idea of number
be so enlarged as to include these new kinds of numbers.
147. Further definitions. An indicated root is usually
called a radical; the number whose root is indicated is
called the radicand. If the root is irrational, but the radi-
cand rational, the expression is often called a surd. Thus,
HIGH SCH. ALG. — 16
236 HIGH SCHOOL ALGEBRA [Ch. XIV
V2, a/8, 6-v/45, V — 2, and V5 + VlO are radicals, whose
respective radicands are 2, 8, etc. ; of these radicals V2 and
6V45 alone are called surds.
The coefficient of a radical is the factor which multiplies
it, and the order of the radical is determined by the root
index. Two radicals which have the same root index are
said to be of the same order. Thus, the surds 12V5 ax^ and
m2V674 are of the same order, viz., the 7th, and their co-
efficients are 12 and m^ respectively.
Surds of the second and third orders are usually called
quadratic and cubic surds, respectively.
Radicals which, when simplified, are of the same order
and have their radicands exactly alike are called similar
(also like) radicals ; otherwise they are dissimilar (unlike).
Expressions which involve radicals, in any way whatever,
are called radical expressions ; they are monomial, binomial,
etc. (cf. § 20), depending upon the number of their terms.
Thus, V5 and 3V5 are similar, monomial, quadratic surds,
while b a -\- 3 V7 and 2 V9 + 3 Va: are binomial surds.
148. Principal roots. We have already seen that a number
has two square roots (e.^., V9 is + 3 or — 3), and we shall
see later that every number has three cube roots, four fourth
Yoot^^ five fifth roots, etc.
E.g., a/8 = 2, -1+V":r3, or - 1 -V^^, since the
cube of each of these numbers is 8 ; and VI6 = 2, — 2, 2V— 1,
or - 2V^^.
Although the number of roots always equals the order of
the radical, not more than two of these roots can be real; and
when there are two real roots, they are opposite numbers.
By the principal root of a number is meant its real root., if
there is but one real root, and its real positive root if there
are two real roots.
E.g.., if attention is confined to principal roots, V9 = 3
(and not - 3), -V^^ = - 2, ^125 = 5, a/T6 = 2, etc.
147-148] IRRATIONAL NUMBERS — RADICALS 237
EXERCISE CI
1. What is a rational number ? Use your answer to show-
that 7, f , — 8|-, and V36 are all rational.
2. What is an irrational number ? Is ^/8 an irrational num-
ber ? Why ?
3. By the method used in § 146 for V2, find two approximate
values for V3 (one larger and the other smaller than the true
value) which differ from V3 by less than 0.001.
4. Find two successive approximations to the value of V5.
Compare these approximations with the result of extracting the
square root of 5 by the method of § 118.
5. What is an imaginary number ? Give several illustrations.
For what values of n is-v^'— 5 imaginary ?
6. Is the number 2l4-Vl7 rational or irrational? Why?
What kind of number is 84 V5 - ^^-^ ? Why ?
7. Are both ^'21 and v 2 + V7 radicals ? Are they surds ?
Are all radicals surds ? Are all surds radicals ?
8. In Exs. 44-54, p. 242, point out the coefficient of each surd.
May the coefficient of a surd be fractional ? negative ?
9. Write a surd of each of the following orders : 2d, 5th,
3d, 7th.
10. Define similar surds, and illustrate your definition. May
the coefficients differ and the surds still be similar ?
11. What factor do two similar surds necessarily have in com-
mon? What kind of a number, then, is the quotient of two
similar surds ? Illustrate your answer.
12. Write a monomial cubic surd ; a binomial quadratic surd ;
a trinomial surd of the 5th order.
13. How many values has Vl6? What are they? What is
the principal square root of 16 ? What is the principal fifth root
of — 32 ? Define the principal root of a number.
238 HIGH SCHOOL ALGEBRA [Ch. XIV
149. Principles involved in operations with radicals. If
we exclude imaginary numbers, the principles employed
in operations with radicals may be symbolically stated
thus:
(i) Vlcy = ^Hc • Vy^
(ii) 4-=—^
(iii) 'Vx = ^'Vx = ^x,
wherein n and t are positive integers, while x and y may
have any values whatever, except that they cannot be nega-
tive when the root index is even.
150. Proof of the principles in § 149. For the sake of sim-
plicity, we shall (1) limit the proofs to principal roots, and
(2) assume that a change in the order of the factors of a
product, even when these factors are irrational numbers,
leaves the product unchanged (cf. § 42).
With these restrictions the correctness of (i), (ii), and
(iii), § 149, follows from the meaning of the symbol Va
(§ 113). Thus, to prove (i) we proceed as follows :
Qy/xVyY = VxVy • Vx Vy- •••to n factors [§ 9
= xy ; [since ( Va)" = a
whence Vx^y^^'xy, [§113
which was to be proved.
This principle may be translated into words thus :
The nth root of the product of two numbers equals the product
of the nth roots of these numbers.
The proofs of (ii) and (iii) are left as an exercise for the
pupil.
149-101] IHRATIONAL NUMBERS — liADICALS 239
EXERCISE Cll
Verify the following equations :
1. V9- V25 = V9.25. 3. Vi6T9 = Vl6. V9.
2. ^/38.^27=^-8.27. 4. ■v'lOOO a« = ^125 a' • -y/S.
5. Show that Exs. 1-4 are special cases of § 149 (i).
6. Find V5 • v3 correct to two decimal places (§ 118) ; then
find Vl5 (i.e., VS • 3) correct to two decimal places, and compare
results. Does this exercise illustrate any jwactical advantage in
knowing that -\/x • -s/y = -Vxy ? Explain.
7. By means of § 149 (ii), show that V35-^V7=V5, and
that VlOa^-^- V— 2a = V — 8a. State § 149 (ii) in words.
8. Find (correct to two decimal places, § 118) V7 -^ V5, also
Vl.4 (i.e., VT^S), and compare results. Does this exercise
n/ ni
illustrate any practical advantage in knowing that ^^ = -y- ?
-\/y ^y
9. Show that f^X^^ and thus prove that V*/^-^.
Verify that "Vx = yj</x = ^ VS [cf. § 149 (iii)] when
10. n = 2, p = 2, a; = 81. 12. n = 4, p = 3, a; = c-*d^l
11. n = 5, p = 2, x = m^. 13. n = 2, p=^S, a; = 64.
14. Use § 149 (iii) to find -^49 (i.e., \/49), correct to two
decimal places ; also find Vl44. How may you find the 9th
root of any given number? the 8th root?
15. Prove the correctness of § 149 (iii). What do n and t
represent in this principle ? May nt, then, equal 11 ? Explain.
151. Special cases of § 149. Besides the three principles
given in § 149, these four others are often useful :
(i) V^ = jrV/,
(ii) -^jr^... =Vx-V/- Vz--,
(iii) ^^=(^xy,
(iv) V]r=A/i^.
240 HIGH SCHOOL ALGEBRA [Ch. XIV
The correctness of these principles may be established by
the method used for the proof of (i), § 149 ; it is easier, how-
ever, to regard them as special cases of § 149, thus:
■v/i^ = V^V^, [§149(i)
which establishes (i) above.
So, too, ^xyz ••• = Vx ' ^yz ••• [§ 149 (i)
= ^x ' -\/y ' -Vz ••• = etc.,
which proves (ii) above ; and (iii) follows from (ii) by let-
ting x = y = z = '•', and supposing the number of these fac-
tors to be t.
Again, %/^=a/^ [§ 149 (iii)
, . , ,. . =V:c% [since </(xy = x'-
which proves (iv). *- ^ ^
EXERCISE cm
1. Is S VE equal to VS^Ts? Why? May 2-^ be written
as V2^ • 6 ? Why ? How may the coefficient of a radical be
inserted under the radical sign [cf. § 151 (i)]?
2. By means of § 151 (i) show that V20 = 2V5, and also that
V— 54=— 3V2. How may we simplify a square root which
contains a square factor? a 5th root which contains a factor
raised to the 5th power ?
3. By the method of § 150, show that x^y = V^.
4. Verify that V2 • V3 • V5 = V30 ; find at least two deci-
mal places in each member of the equation (cf. Ex. 6, p. 239).
How find the product of several radicals of the same order ?
5. By the method of § 150, show that ■\/x • -Vy • -\/z = -Vxyz.
6. Find, correct to two decimal places, (V6)^, also V6^, and
compare results. How would you raise Vl5 to the second
power? to the 5th power? Explain.
7. Show, using the method of § 150, that (^x) = -y/af.
151-152] IRRATIONAL NUMBERS — RADICALS 241
8. Verify that ■\/aP = -^a^, and that -\/a^^ = Va\ In each
of these equations compare the exponents of a, also the root-
indices.
9. Is a radical changed in value if we multiply its root-
index and also the exponent of its radicand by the same factor
[cf. § 151 (iv)]? if we divide both by any factor common to
them ? Illustrate.
10. Using § 151 (iv), show that -^9 (i.e./^3^) = V3 = ^8i.
Which is more easily computed, V32, or its equal, V2?
152. Reduction of radicals to their simplest forms. A
radical is said to be in its simplest form when the radicand
is integral, when the index of the root is as small as possible,
and when no factor of the radicand is a perfect power corre-
sponding in degree with the indicated root.
The following examples illustrate the application of the
foregoing principles in the reduction of radicals to their
simplest form.
Ex. 1. Reduce VSoV to its simplest form.
Solution. V8 aV = V4 a V . V2 ax [§ 149 (i)
= 2 ax^ V2 ax.
Ex. 2. Reduce V4 a^x^y^ to its simplest form.
Solution. -s/A.o'xY = </{2 aa^i/f = ^2^^. [§ 151 (iv)
Ex. 3. Reduce Vf to its simplest form.
Solution. -^ - = -J — 1
5_2
5.52
=^^\>m=Ujm, [§i5i(i)
EXERCISE CIV
Reduce each of the following to its simplest form :
4. V18 (^.e., V9^). 6. V45. 8. S^IG.
5 V24. 7. V75. 9. S^^^^.
242
HIGH SCHOOL ALGEBRA
[Ch. XIV
29.
45
/27 a?
10. 2\/54.
11. ^32.
12. VI 0-.e.,ViT7). 30. Va^ (cf. Ex. 2). 46. i^f.
13. VS
15. ^^ ^
16. VJ (cf. Ex. 3). 34. V216.
17. vi-
31. V^
32. Va;y.
33. a/25.
47. 3^125 a^a^.
48. y\^±y.
^ x-y
18. lOVf
19. </l
20. 2V|.
21. V^^
22. v=i:
35. V32m^7i'«.
36. 3^36 h-x'\
49. ■>/a2"a;"+\
50.
64 m«
37. V 32 a VI
39.
125
51. 6^320.
52. V- 486^2^
53. |V2|.
23. V27a;l
24. VSw?.
\9^
54. Vl8a-9.
25. Va«6^'.
26. VV^/*.
40.
41.
yV
55. Var+Wy^\
11,
2x
56. Va2''ft.""+^
4 25a«
57. </ - 40 a^^-^y\
27.
r
58. 4 Va^" — a*"aJ".
59.
28. J^
V
L 3a^^
-2 6a;
2a
60. V3a;2— 6«2/+32/^.
42. V162.
43. a/36.
44. 2V|V.
In the following, insert the coefficients under the radical
signs :
66. -4^1.
61. 3V7.
62. 5^4.
63. 2a/6.
64. -2 Vs.
65. y\V40,
70. (c4-l)V5c.
67. iVlW-.
71.
72.
73.
1 ,
O/. ^ V X24.
-VaMo^-i).
68. -V72a^l
4 w3vV4 wv^
69- ~^12a'x.
-2y'-2^<Jyh\
152-153] IRRATIONAL NUMBERS — RADICALS 243
153. Addition and subtraction of radicals. Similar radicals
(§ 147) may evidently be added and subtracted by regarding
the common radical factor as the unit of addition. The sum
or difference of dissimilar radicals can, of course, only be
indicated, and this is done by connecting the radicals with
the proper signs (cf. § 23).
The radicals to be added or subtracted should first be
reduced to their simplest forms ; the following examples
will illustrate the procedure.
Ex. 1. Find the sum of V75 and 3 Vl2.
Solution. Since V75 = V25T3 = ^25 • V3 = 5 V3, [§ 149 (i)
and 3 Vl2 = 3 V473 = 3 V4 . V3 = 6 V3, [§ 149 (i)
therefore V75 + 3 Vr2 = 11 V3.
Ex. 2. Find the sum of 5 Vl8 and — VOX
Solution. Since 5 Vl8 = 5 V9^ =5-3 V2 = 15 V2,
and -Vo:5^-VT=:--v/-^ = -|V2,
therefore 5 Vl8 + ( - Vo:5) = (15 - 1) V2 = 14i V2.
Ex. 3. Find the sum of V9 x - 18, 6 V4 a; + 8, V36 x - 72, and
-V25a; + 50.
Solution. V9 a; - 18 + 6 V4a;4-8 + V36a;-72 - V25a;-f 50
= 3 V^^^ + 12 V^T2 + 6 V^^^ - 5 a/^T2
=9 Vx'^ + 7 V^T2^
EXERCISE CV
Find the sum of :
4. V5, 7 V5, and - 3 V5. 8. 2 V20, -^ V45, and Vsa
5. Vl8, V50, and V98. 9. ^250, -v^, and -^U.
6. VT2, V75, and V27. 10. \/500, -y/WS, and ^^^^32:
7. V28, - 2 V63, and V700. 11. <^cdf*, 3 -\/cdf, V - 8 c'df.
244 HIGH SCHOOL ALGEBRA [Ch. XIV
12. Find (correct to two decimal places) the value of each of
the surds in Ex. 7 (cf. § 118) and add your results. How does
this sum compare with that previously found for Ex. 7 ? Is there,
then, any practical advantage in simplifying surds before adding
them?
13. Find the sum of a, 2 b, and c ; oi S x, 4: y, 2 x, and —5y.
14. What isjhe sum of 3V2 and 5^/7? of 3 V2, 5^/7]
- 2 V7, and -V2?
15. Write a rule for the addition and subtraction of radicals,
providing both for those cases in which the given radicals are
similar and for those in which they are dissimilar.
Simplify the following expressions as far as possible (cf . § 152),
and explain your work in each case :
16. ^/i35 + -^625-■^/320. 22. -^128^+ a/375^- -v^Sl^
17. -^40 + V28 4- V175 +^25: /«;,/«_ ja
19. V5 + V75-V12 + 2V3. 4»_ jW^ /£^
20. Vl47-Vi + iv/3+J^9: • ^bY \ bf ^^bf'
21. 6^|o+4^/I|-8^/|||: 25. V(a4-6)^c-^(a+6)V.
26. ^l92^-2-^3^-^/5^+v40^.
27. -y/abx -\- -^^^m^ - \^S~aW^.
28. V3a;3 + 30ar^ + 75a5-V3a^-6a^ + 3a;.
29. V5a^ + 30a* + 45a3-V5a^-40a^ + 80a^
30. V50 + -V/9-4 VJ + -v/24 + ^27-a/64.
31. V| + 6V|-iVl8 + ^36-^+^125-VS.
32. Va^ — a^x — Vaa?? — a^ — V(a + a;)(a^ — a?^.
154. Reduction of radicals to the same order. By (iv) of
§151,
^5 = ^-^5^ =^5^ =^625,
and ^7 = ^^P = -^^'P=-^^3"43,
153-154] IRRATIONAL NUMBERS — RADICALS 245
i.e., the radicals V5 and -^/l are equivalent, respectively, to
^625 and v'348 ; and these last two radicals are of the
same order, viz., the twelfth.
Moreover, it is evident that, by a similar application of
§151 (iv), any two or more radicals whatever may be
reduced to equivalent radicals of the same order. This new
order must, of course, be some common multiple (preferably
the L. C. M.) of the orders of the given radicals.
EXERCISE cvi
1. Is -^^ equal to Vx? Is ^3aV equal to ^9aV? Using
the method of § 150, prove the correctness of your answer to each
of these questions. Compare also § 151 (iv).
2. Eeduce ■y/25 mV and a/8 a^6V to equivalent radicals of
the 4th order, and explain (cf. Exs. 8-10, p. 241).
Reduce to equivalent radicals of the order indicated, and ex-
plain your work :
3. -s/ay^, 9th order. 7. 3 ax, 4th order.
4. -^SM, 6th order. 8. ^2 m\ 12th order.
5. -^2sf, 10th order. 9. ^AV, 12th order.
6. V2 sf, 10th order. 10. VciVa^, 8th order.
11. Express as equivalent radicals of the 6th order: -\/'
m,
■\/9 m^n\ and mn.
Eeduce the following to equivalent radicals of the same order:
12. V5 and ^/li. 19. Vl4, ^25, and ^95.
13. -v/T and V3. 20. 3V3, 5V2, and 2-\/20.
14. V27and-\/3^l 21. 2 V3, 3^2, and 2-v/39.
15. V6and3. 22. 3Va&, a^2^, and -^^^^
16. -Vp, V5j^ and -y/f^. 23. x'</2^, V^,and2^^^
17. i/W, V2, and ■^. 24. </x, 5 xVy, and V^.
18. ~^2¥y, \^3a? and xy. 25. ^d^ + 6"^, and aVa ~ b.
246 inan school algebra [Ch. xiv
26. Can the radicals in Ex. 22 be reduced, to equivalent radi-
cals of the 6th order ? of the 12th order ? of the 9th order ?
Give the reasons for your answer in each case.
27. What is the lowest common order to which you can reduce
the radicals in Ex. 23 ? those in Ex. 24 ? in Ex. 25 ?
28. By first reducing Vl5 and V6 to the same order, show
that -\/W is greater than V6»
29. Which is greater, V5 or Vl2 ? Explain your answer
(cf. Ex. 28). _
30. Which is greater, 3 VlO or 2 a/100 ?
Hint. First insert coefficients under radical signs (cf. Ex. 1, p. 240).
Arrange the following radicals in order of magnitude :
31. 3 V3, 5 V2, and 2 v'20. 32. 2 ^39, 3 -y/2, and 2 V3.
155. Product of monomial radicals. If two or more radi-
cals are of the same order, their product may be written
down immediately by § 151 (ii) ; while if they are of differ-
ent orders, they should first be reduced to equivalent radi-
cals of the same order (§ 154).
Ex. 1. Multiply a/5 by V2^
Solution. The L. C. M. of the orders of the given radicals is
6, and by §154 ^=V5^,
and V2=-v/2^;
therefore </5' V^ = </¥' ''^¥ = y/W^' [§ 149 (i)
= a/200.
Ex. 2. Find the product of 4 h-\/ax and V5 a^, and simplify
the result.
Solution. As in Ex. 1, 4 h^ax = 4 6 VaV,
and -\/5 a^ = V5^ a^ ;
therefore 4 b^ax • V5 a^ = 4 ft Va V . V 5^a*
= 4 6^52^V [§ 149 (i)
= 4a6A^25^^. [§151(i)
154-156] IRRATIONAL NUMBERS — RADICALS '247
EXERCISE evil
Find the following products, and simplify the results:
4. 2Vi5 . V5. '^2y^y'
6. -^ . ^^. 11. (8 - 2 Vl5) . 2 V6.
13. How may we find the product of two or more radicals
which are of the same order?
14. How may we find the product of two or more radicals of
different orders ? Illustrate, using the surds V3 and V2.
Find the following products, and simplify the results :
15. -Vab • V&c.
16. V3 . 3^.
17. V2 . </I.
18. 2\/5 . 7-^10. 26. V^' • Vl2^ . V75^^
23.
^f.
</2f.
24.
2V2
■ ^256.
25.
V2-
^1-^.
19. a/i . ^(if. 27. V2a& . ^abc • ^4:a^b\
20. \/^ . aP^
3/]^ J/W 28. V9(c-A;)=^.-lV3(c-A;).
29. (V72+^12-3 V98) • V2.
21. ^4a2 . WSa^
22. ^^ . V^.
156. Multiplication of polynomials containing radicals. The
product of two polynomials containing radicals is obtained
by multiplying each term of the multiplicand by each term
of the multiplier and adding the partial products, just as in
the case of rational polynomials.
248 HIGH SCHOOL ALGEBRA [Ch. XIV
Ex. 1. Multiply 5V2-2V3 by 3V2 + 4V3.
Solution. 5 V2 — 2 V3
3 V2 + 4 V3
30 - Q^Q_
+ 20V6-24
30 + 14 V6 - 24 = 6 + 14 V6.
Ex. 2. Expand (2V3— ■v2)^ by the binomial theorem.
Solution. (2V3-a/2)2==(2V3)2-2(2V'3)^2+(a/2)2
= 12-4^108 + ^4.
EXERCISE CVIII
Multiply, and express your results in the simplest form :
3. V5 — 5 by V5 + 1. fa S
/a 4b\ , fa 4/6\
4. 2V7-3by 2V7 + 4.
5. 5Vn-h4by 5Vil-4. 13. V2 + V3 + V6 by V2+4V3.
6. Vc + V2d by Vc - V2d. 14. x - Vxyz + 2/^ by Vx + V^.
7. (-Vac-VS^by, ^^ V2^-^3^by ^^-^^.
a (3Vi + 5yi)l _ ^^ ^^
9. (a - a6 V2 + 62)2. 16- 3 V3- V2 by 5 V4 4- V2.
10. ^3-3^9 by ^/3 + 3-^9. 17- 2 V3-^2 by 2 V3- ^4.
11. (</9 - 4 ^6 y. 18. ^ls-2^1 by V8 + 2V7.
19. VlO-3V5-4V3by 4V3-3V5.
20. Vic?/ + 2 ViC2! — V2/2! by ^xy — 3 ViC2!.
21. -\/m^v?- + Vm7i by -\JmT? -f- -^mhi.
22. 2-^/^-3 vi^s i^y a/16^2^^^.
23. Vaf" — VlO aj"+^ + V^ by V5^ + V2^«+^.
24. V(.'C+?/)*-V(a;+2/)'»+2by V(a:+2/)""+'+V(a;4-2/)'-
25. .+|-^Z^by.. + | + ^/Z±S". .
156-157] IRRATIONAL NUMBERS— RADICALS 249
157. Rationalizing factors.* Conjugate surds. The factor
by which ii given surd (radical) must be multiplied in order
to obtain a rational product is called its rationalizing factor.
Thus, of the surds V5 a^ and V 25 a, each is the rationalizing
factor of the other, since their product, 5 a, is rational ; the
same is true of the surds 2Va— V3 and 2 Va + VS. (Why ?)
Of two such binomial quadratic surds as 2 Va — V3 and
2Va + V3, which differ only in the sign of one term, each
is called the conjugate of the other. Moreover, since the
product of any two conjugate quadratic surds is rational
(§ 53), therefore each of them is the rationalizing factor
of the other.
EXERCISE CIX
1. Is V2a the rationalizing factor of V2a? Why? Show-
that VS kH and 2 V3 — V5 are the rationalizing factors of V'^ kH^
and 2 V3 + V5, respectively.
2. Is V5-2V3 the rationahzing factor of 2V3-hV5?
Explain. Are these surds conjugate to each other ?
Find the rationalizing factor of :
3. V3. 10. Vf. 16. 3a-V5^.
4 ^^7 2 6/"^ 17. ^5x — V2ay.
5. 2ViO. "' "'^^' ^«- Va» + 2^6.
6.
^■27/ X.. ^. - xi-vs-
'• ^4*^- 13. V2-V7. 20. ^Va-cJl.
3 _ b ^ d
8. V-i. 14. 4 + 5V3. ,14-3
9. 5^^^ 15. 2V3 + V8. ^^- \"r'^4^" •
22. How may we find the rationalizing factor of any binomial
quadratic surd? Why? Does the same method answer for a
binomial cubic surd ? Explain.
* See also § 177.
250 HIGH SCHOOL ALGEBRA [Ch. XIV
158. Division of monomial radicals. If the dividend and
divisor are of the same order, tlieir quotient may be written
down immediately by § 149 (ii), while if they are of differ-
ent orders, they should first be reduced to equivalent radicals
of the same order (§ 154).
E.g., to divide -^4 a:(?y^ by ■\/2 a^x, we proceed thus :
■V4 aoi?y- _ V16 a-^m^y^
-V2a'x V8a'
[§154
4
^ CH49(ii)
W2^^1^^^;^_ [§152
^
159. Division of polynomials containing radicals. If the
divisor is a monomial, then, manifestly, the quotient may be
obtained by dividing each term of the dividend by the
divisor.
Ex. 1. Divide 3V2 + 4V3 by V2.
Solution. ^V2 + 4V3 ^ 3 _^ ^^3 ^ 3 _^ 2V6. [§ 152
We may also solve Ex. 1 as follows :
3V2 + 4V3 ^ (3V2+4V3)V2 ^ 6 + 4V6 ^o ^ /^
V2 (V2)^ 2 -^ ^ '
i.e., we may, before dividing, multiply both dividend and divisor
by the rationalizing factor (§ 157) of the latter. This method
is known as " division by means of rationalizing the divisor " ; in
many examples it is easier than the one used in the first solution
above (cf. Exs. 28, 29, and 37 below).
EXERCISE CX
Eind the following indicated quotients and simplify each :
2. V20--V5. 4. 6V5-T-V4().
3. V2l6-f-Vl2. 5. V^-f-V3^'.
1 -is- lot) J inilATlONAL NUMBERS — RADICALS 251
6. V'M-^V^. 13. 6-r-3^4.
7. 2-v^54--v/216. 14. VI^TS^f.
8. Sv^lS-lS^J. 15. ^57i2^^-j- V2^.
9. </(a-6)---V(a + 6)l 4a; 3/^
10. 2V6-=-^/3. 2/ ^2/'
11. 2\/l2^V8. 1^- a^4ar*2/^-^2feA/2a;2/.
12. 10 -- V5. 18. 3 aV2~W'^^ -^ 2 6^3^^.
19. Find the value of 1 h- V2, correct to two decimal places :
(1) by finding V2 and dividing, and (2) by first rationalizing the
divisor. Which is the easier process ?
20. Find (correct to two decimal places) the value of :
-^ ; -^ ; ^ ; and -^. (Cf. Ex. 19.)
V3 ' V2 ' V2 ' V(5 ^ ^
Find the following indicated quotients and simplify each :
21. ( Vl5 — V3) -^ V3. 24. (x-Vy'z — ofyz^) -7- ^xyz.
22. (V6 + 2V3)--V2. 25. (-\/aFb--\/'b^ + -y/d^t)^^abi^.
23. (4-7V5)--V6. 26. (5-v/i2-2V6 + 4)-v--^4.
27. In w^hich of the above exercises is division by means of
rationalizing the divisor easier than direct division ?
Divide by means of rationalizing the divisor, and simplify:
28. -^ . 31. t 34. 3V2-4V5 .
V3-V2 Vc-Vd 2V3 + V7
29. ^ . 32. 8:t2V15. 35 g+V^M^- .
3^^-'^ ' V3+\/5 ' a_V^iM^
30. 2 + V3. 33 27-4V7 3^ V^+^^V^^
2-V3 ■ 2V7-1 ' V^+V^^i
37. If the result in Ex. 26 were wanted correct to 3 decimal
places, show in detail that it is far simpler first to rationalize the
divisor than to extract roots and divide by the ordinary arithmeti-
cal method. Is this true in Ex. 28 ? in Ex. 33 ?
HIGH. SCH. ALG. — 17
252 BIGH SCHOOL ALGEBRA [Ch. XIV
38. What is the product of (2 + V3) - V5 by (2 + V3) + V5 ?
of this result by 2 — 4 V3 ? What, then, is the rationalizing
factor of 2 + V3 - V5? Divide 2 by 2 + V3 + V5; also by
1 4. V3 - V2.
160. Powers of surds. Roots of monomial surds. Powers
of surds, being merely products of two or more equal surd
factors, may be found by the method of § 155 ; and roots of
monomial surds may be found by § 149 (iii).
Ex. 1. Find the fifth power of v'2 a^.
Solution. (^27')' = V(2^' [§ 151 (iii)
= ^2^^= 2 a^^4^ [§ 152
Ex. 2. Extract the square root of -v/4 a^x.
Solution. yj'^Ta^x= i/4r a^x. [§ 149 (iii)
This result may also be written in either of the following forms :
^j■V^o^ or '\J2 aVx. [§ 149 (iii)
EXERCISE CXI
9.
Simplify:
3. (V'r?)2. g (^FA\ 15. (V2^+V3^)^
4. (2-^aby, ' ^ ^J 16. (5V3-3V2)^.
5 (</6^m:ii?^' ^^ (-V2V^)3.
12
3,^,, 12. (^^F^^v^)\ ^^' (1-^V3).
7.
13. (9+V5)l 19. (Vm^-Vm7i)2.
8. (iViey. 14. (V7-V2)2. 20. (V2-3^3)l
Express each of the following by means of a single radical
sign, and simplify :
21. VV^. 24. ^-27V^. 27. VtV-V^^-
/ o , 3; 7= 28. -x/VS^
22. V\/3cdl 25. -^-^V^. ^
___^^ . 29 x\ 'fZHH
23. V^25mV. 26. yV(r + s)i«^ • A\8(e-/)
159-102] IRRATIONAL NUMBERS — RADICALS 253
161. An important property of quadratic surds. Neither
the sum nor the difference of two dissimihir quadratic surds
(§ 147) can be a rational number ; for, if possible, let
Vx±Vy = r^ (1)
-Vx and V^ being dissimilar surds, and r rational, and not
zero.
From Eq. (1) ±V^=r-Vi, (2)
whence, squaring, y = r^-~2r ^x -h a;, (3)
and, solving for V a:, ^x = — ^ ^ ;
2 r
i.e.^ if Eq. (1) were true, then the surd ^x would equal the
rational number — ^ ^, which is impossible; hence Eq. (1)
cannot be true.
From what has just been shown it at once follows that
if x-{-^/y = a-\-^^ where x and a are rational^ and V?/ and
V6 are quadratic surds^ then x = a and y—h.
For, if re + V?/ = a + VJ,
then V^ — V6 = a — x\
which, by the above proof, can be true only if each member
is zero, i.e.^ ii a = x and ■Vy=\^b. In other words, the
equation x + Vy = a-\- V3 is equivalent to the two equations
x= a and y = h.
162. Square roots of binomial surds. Some binomial quad-
ratic surds are exact squares; the following examples show
how to extract the square root of such surds.
Ex. 1. Extract the square root of 8 + V60.
Solution. If 8+V60 is the square of a binomial surd, let
Va;+ V2/ represent that surd, i.e., let
then, squaring, 8 + V60 = x-{-2 Vxy -{-y=x-\-y-\- 2^xy ;
therefore S = x-\-y and V60 = 2 ^/xy, [§ 161
254 HIGH SCHOOL ALGEBRA [Cii. XIV
and combining these last two equations (after squaring the second)
easily leads (§ 131) to the solution
x = 8 and 2/ = 5 ;
therefore ^js + V60 = V3 + V5,
as is easily verified by squaring the expression V3 + VB.
Note . This example might also have been solved by inspection ; for,
writing 8 + V60 in the form 8 + 2^V'l5, and then comparing it with
(\/x+ \/?/)2, i.e., with x -\- y -\- 2 y/xy, we see that we have only to find
two numbers whose sum is 8 and w^hose product is 15, and take the sum of
their square roots as the required root.
Ex. 2. Ky inspection, find the square root of 18 — Q-y/^.
Solution. Writing 18 — ^y/b in the form 18 — 2 •\/45, we see
that we need to find two numbers whose sum is 18 and whose
product is 45, and take the difference of their square roots. These
numbers are evidently 3 and 15, hence
Vl8-2V45 = V3-Vl5.
EXERCISE CXIl
Find (by inspection where practicable) the square root of each
of the following expressions, and check your results :
3. 4 + 2V3. a 30-20V2. 13. e4-4/-4Ve/:
4. 16 + 2j^l5. 9. 39-12V3. 14. 2 m + 9 n - 6 V2m?i.
5. 12 + 8V5. 10. 47-12VI1. 15. 146-56V6.
6. 17-12V2. 11. 63 + 24V5. 16. m + 2Vm,
7. 27-4V35. 12. Sxy-4.xyV^. 17. a — Ve.
18. In the first solution of Ex. 1 above, why does a; + 2/ = 8,
and2V^= VeO?
19. If the numerical value of '\21-f 8V5 is required, is it
easier to find first the binomial whose square is 21 + 8V5, or to
begin by extracting the square root of 5 ? Explain. Also answer
the question if 12 + 3V5 is substituted for 21 -f- 8 V5,
102-10:1] IRRATIONAL NUMBERS — RADICALS 255
163. Irrational equations. Equations which contain indi-
cated roots of the imkyioivn numbers are called irrational
equations (also radical equations) . Thus 6 ■\/x — 25 a; + 88 = 0,
ViTl+a; = 8,^^7r-4-l = 0, and 3 + iVi = ^a;2-l are ir-
-vx
rational equations, but such an equation -d^ x— V3 = 5 a; is
rational.
The solution of irrational equations is illustrated by the
following examples :
Ex. 1. Solve the equation V.^• + 1 + .t = 11.
Solution. On transposing, the given equation becomes
■\/x -h 1 = 11 — a?,
whence, squaring both members (Ax. 3),
fl; + l = 121-22a; + «2,
i.e., a;2_23a; + 120 = 0,
whence (§ 126), a; = 15 or 8 ;
and, on substitution, it is found that 15 satisfies the given equa-
tion if Vcc + 1 means the negative value of this root, while 8 satis-
fies it if the positive value of this root is intended.
Ex. 2. Solve the equation V5 x-\-l — wx + 2 = 3.
Solution. On transposing, the given equation becomes
V5 a; 4- 1 = 3 + Va? + 2,
whence, squaring both members (Ax. 3),
5 i» + 1 = 9 + 6 V^T2 + a; + 2,
i.e., ' 4a;-10 = 6V^T2;
whence, dividing through by 2, then squaring and simplifying,
4a^-29aj + 7 = 0,
from which (§ 126), a; = 7 or ^.
On substitution it is found that the given equation is satisfied
by x = 7 if each radical is regarded as positive, and by a; = ^ if
V5 a; + 1 is taken as positive and Va; + 2 as negative.
256 HIGH SCHOOL ALGEBRA [Ch. XIV
EXERCISE CXIII
Solve the following equations, and show what restrictions, if
any, must be made on the signs of the radicals in order that your
results shall be roots :
16. -\/3-\-x-\-Vx=—=' .
3.
V2a;-f 6 = 4.
4.
V4aj + 5 = 7.
5.
V3 a; - 8 = Va;.
6.
-y/x'-4x = 2^4:.
7.
i/x-\-oc' = -2c.
8.
9.
10.
^5 — x = x—5.
X 4- V^ =4:X — 4 Va;,
2/4- V^-20 = 0.
17. Vm + 5+ Vm— 8 = V3.
18. Vl+sVs2 + 12 = l+s.
19.
20.
^v — 8 Vv —4
V -v -}- 1 -Vv—2
2^ V3a; + l + V3^ ^o^
11. V 4y + 17 + Vy + 1=4. ^■^- V3¥+l-V3^-
12. V25-65=8-V25+66. ^^ V^-2 ^ V^4-1 .
13. cV^— d\/a?=c- + d2— 2cd. ' ^x-\-3 Vx-\-2
14. Va;+1+— 4^ = 2. 23. r+V^"^^
Va;+1 vr — c^
15. V^^^ ^ = 0. 24. a- Va^^-a^ ^_ V3-l
Vs + 7 a + Va^-a^ V3 + 1
^=-:^
25. V4 a; +1 — Vaj + 3 = Va; — 2.
26. Va; + a + Va: + ft = V2 a; + a + 6.
27. Va; + 3 + V4a.' + 1 = Vl0a; + 4.
28. ^^-^^^-^ = ^+V^^38.
a; - Va;^ - 8
29- y'-y- V2/'-2/ + 4 = 8 (cf. Ex. 18, p. 206).
30. a + 10 = 2 Va~+10 + 5.
31. How many roots has the equation in Ex. 27, if the radicals
are unrestricted in sign? How many, if each radical must be
taken positively ?
CHAPTER XV
IMAGINARY NUMBERS*
164. Definitions. In the solution of quadratic equations,
and elsewhere, such numbers as V— 5 and 6 — V— 10 fre-
quently present thenjselves; these numbers cannot be ex-
pressed, even approximately, as the quotient of two integers
(cf. § 146). _
Numbers of the form V— 5, where h represents a positive
number, are called pure imaginary numbers, while numbers
of the form a ± V — 6 are called complex imaginary num-
bers, or complex numbers. Two complex numbers are said to
be conjugate if they differ only in the signs of their imaginary
terms. Thus, V — 5, 2 V — 6, and V— |^ are pure imaginary
numbers, while 2— V— 3, 7 — 2V— 5, and 7-f-2V— 5 are
complex numbers, the last two being conjugates of each other.
From the definition of the symbol Va (§ 113) it follows
that (V^=T)2=-5; (1)
and by the method of proof used in § 150, it is easily shown
that V^=V6.V"^^. (2)
Remark. The second member of Eq. (2) may be regarded as
a standard form; and it will be found that operations with imagi-
nary numbers are usually much simplified by first reducing such
numbers to this standard form. The symbol V— 1 is often
called the imaginary unit, and is represented by /.
* Teachers who prefer less work in imaginaries than is here given may
omit §§ 166-168 ; the entire chapter should be read if time permits.
257
258 HIGH SCHOOL ALGEBRA [Ch. XV
165. Powers of V— 1, i.e., of i. As a particular case of
Eq. (1), § 164, we have
similarly (V- 1)3= (V- 1)2. V-l=-V-l, " i^= -i,
(V^:n)*=(V^^i)3. V^i:=l, " ^4=l,
(V^a)5 = (V^^)*- V^^ = V^, " ^5 = ^,
and so on for the higher powers of V — 1 ; any one of these
powers, when simplified, will be found to have one or another
of the four values : V — 1, —1, — V — 1, and 1.
EXERCISE CXIV
1. Define an imaginary number (of. §§114 and 146).
2. Which of the following are* imaginary numbers : V— 3,
V^, -V^^ V5, ^^^ 3-\/^ 4 aV^, and I + i V^=^ ?
3. Is V— a; imaginary when x represents a positive number ?
when X represents a negative number ? Answer the same ques-
tions for -^ — x.
4. Eeduce to the standard form (§ 164, Kemark) : V— 9 ;
V^^; V"=^l0"; V^^l3; V^^12; and V-i:5c^.
5. Show by the method of proof suggested in § 150 that
V^^T^ V7 . V^l.
6. Show that if ^ = V— 1, then
^2=-l, ^«=-l, ^l»=?
^3= -I, i'^=-i, ^"= ?
i*=l, i^ = l, i^'=?
7. Since any even number may be written in the form 2 n,
where n is an integer, and since a^"* = (ci^)"? show that every even
power of i is real.
8. When n is even, does i'-^" equal 1 or — 1 ? Why? Answer
the same questions if 7i is odd.
9. Give the values of the following even powers of i:
^8[ = (^2y]; P; |-32. ^-18. ^'64. ^100. ,^-6. ^^^
166-167]
IMAGINARY NUMBERS
269
10. Show that every odd power of i is either ^ or — i (cf. Ex. 7).
11. Find by inspection the value of the following odd powers
of i : i"^ ; f
i'\
12. Distinguish between pure and complex imaginary numbers,
and give three examples of each.
-B
A'
I I I.
B'
166. Graphical representation of imaginary numbers. Since
opposite numbers, such as + 3 and — 3,
are represented graphically by opposite
distances, such as OA and OA' ; and since
multiplying +3 by —1 gives —3; there-
fore we may regard the multiplier — 1 as
an operator which rotates OA through 180°
about 0, into the position OA' ,
Again, since i • ^= —1, therefore i may be regarded as an
operator which when applied twice in succession rotates OA
through 180°; hence using i as a multiplier once (instead of
twice) should rotate OA through 90° into the position OB.
In other words : 3v^ — 1 {i.e.^ OA • i) may be represented
graphically by OB. Similarly, any pure imaginary number
whatever may be laid off on the line B' OB, above the origin
if the number is positive., below the origin if it is negative.
E.g., if each division on the lines JT and RN in
the figure represents a unit, then OS=l, 0J= — 2,
OL = V^ ON = SV"^^, 0Q=- 2 V^ etc.
The lines JT and BN are often called
the axis of real numbers and the axis of
imaginaries, respectively.
J H
o
p
Q
R
S T
-i — H-
167. Graphical representation of complex
numbers. A complex number, such as 5 + 3V— 1, may be
graphically represented as follows : lay off OA, 5 units on
the axis of real numbers, and OB, 3 units on the axis of
imaginaries, tlien complete the parallelogram AOBR, and
draw its diagonal OR ; this diagonal is a graphical repre-
260
HIGH SCHOOL ALGEBRA
[Ch. XV
sentation of the complex number 5 + 3V— 1, i.e., of the sum
of 5 and 3V^=a.
Moreover, it is evident that any com-
plex number vrhatever may be graphic-
ally represented by the above method.
B
R
-/^i
a
O 1 1 1 J^
Note. The appropriateness of calling OR the
sum of OA and OB will be evident to pupils who
have an elementary knowledge of physics. Thus, if
two forces, represented in amount and direction by OA and OB, respectively,
act simultaneously on a body at 0, the result is the same as though a single
force represented in amount and direction by OB were acting on this body ;
i.e., the sum of the forces OA and OB is the force OB.
168. Graphical representation of the sum of complex num-
bers ; also of their difference. The sum of two complex
R numbers, such as 7 + 2V— 1 and
1 + 4v^— 1, may be graphically repre-
sented as follows : let OP and OQ
be the graphical representations of
7+2 V^^nr and 1 + 4V - 1, respec-
tively (§ 167) ; complete the par-
allelogram POQR, and draw its
diagonal OR ; then OB is the graph-
ical representation of the sum of 7 + 2V— 1 and 1 + 4V— 1
(cf. §167, Note).
Obviously the sum of any two complex numbers may be
represented by this method.
Again, to find the difference of two complex numbers
graphically, we have only to reverse the sign of the subtra-
hend, and proceed as in addition.
EXERCISE CXV
Represent graphically the following imaginary numbers :
1. 2V^^. 3. -7 1. 5. 2AL 1. V^5.
2. -5V^. 4. \i. 6. V^9. a V^.
167-160]
IMAGINARY NUMBERS
261
Perform the following additions and subtractions graphically
9. 4 + 2 [cf. § 4]. 13. 3-7.
10. 4.i + 2i. 14. 3^-10^.
11. Ai-2i. 15. 3-8-6.
12. 7
5. 16. V-25 4-2V-36-V-49.
Represent by drawing each of the following complex numbers :
17. 2 + 4aA3i. 21. 4 + 3l 25.
18. 4 + 2V^^. 22. -6-6i. 26.
2
V-i + 6.
V-2-7.
19
23. 3+V-36.
2 + V^^.
27. lO+V^^OS.
~8.
5V-1.
20. -7 + 1. 24. -:^ + V-9. 28. -i^-
Perform the following indicated operations graphically :
29. (l+4i) + (4 + 2i). 33. (5-V^~9) + (-3 + V^^).
30. (3 + 2i) + (5 + 3i). 34. -2^ + (-3 + V^^).
31. (2-50 + (8 + 0. 35. (4 + 3t)-(2 + i).
32. (3 + V^4) + (-2 + V^). 36. (8-20-(5-3i).
37. (10+V-9)-(-2+V-16).
38. (|_V38)_(7-V^.
169. Fundamental operations with pure and complex imagi-
nary numbers. If complex numbers are first reduced to the
standard form a + 5V— 1, and if we are careful to remember
that V — 1 • V — 1 = — 1 and not + 1, then the operations of
addition, subtraction, etc., with these numbers may be per-
formed exactly as are the corresponding operations with
real numbers. The following examples will illustrate these
operations.
Ex. 1. Find the sum of 2 + V- 9, 8 -V^i, and - V - 25.
Solution. Since 2 + V^9 = 2 + 3 V^^,
8-V=l = 8-2V"^,
and -V^r25= -5V^^ ,
10_4V^ i.e., 10-V^=36.
therefore the sum is
262 Jiiaii SCHOOL algebra [Ch. xv
Ex. 2. Multiply r>V^^ by W^^.
Solution. 5V^ = 5 V2 • V^,
and 4 V^^ = 4 V7 . V^=l,
hence the product is 20 V2 • V7 (V — 1)^ i.e., — 20Vl4.
Note. Observe that this product is not +20Vl4, as it would be if tlie
factors were i-eal numbers. Beginners should be especially careful to guard
against errors in the sign of a product of imaginary numbers.
Ex. 3. Multiply 3 + V^^ by 2 - V^^.
Solution. Writing these imaginary numbers in terms of the
imaginary unit, the work may be arranged thus :
3H-V5. V^
2-V3. yiTi
6 + 2V5.V^^
- 3 V3 ■ V^ - Vl5 (V3i)2
6+ (2 V5-3 V3) . V=3 + Vi5.
Ex.4. Divide 12 + V- 25 by 3- V- 4.
Solution. Such divisions are easily performed by first mul-
tiplying both dividend and divisor by the conjugate of the
divisor, thus :
12 + V- 25 ^ 12 + 5 V ^=n: ^ (12 + 5 V^I)(3 + 2 V^:i)
S-V"^^ 3-2V^=^ (3-2V^(3 + 2V^ri)
^ 36 + 39 V^i: + 10 ( V"^)'
9 _ 4 ( V^^)'
26 + 39V^^
9 + 4
2 + 3V-1.
170. Important property of complex numbers. By a method
altogether like that used in § 161 it may be shown that if
a + 5V— 1 = ^ + (^V— 1, then a = c and b = d. Moreover,
this fact may be used, as in § 162, to extract the square roet
of any complex number (cf. Ul. Alg, §§ 151, 182).
16l>-170j IMAGINARY NUMBERS 263
EXERCISE CXVI
5. Add 3 + 5 i and 7-|- v — 4 as in § 169; then add these
numbers graphically, and check your work by comparing results.
Simplify Exs. 6-13 below, and check as teacher directs :
6. 7-6^+2 + 3^. 8. (3 + 2 i) - (3 - 2 *).
7. (3 + 2*) + (3-2i). 9. (-4-A/^^=^)+(-4-V^.
10. V-4 + 4V-9 + V-25.
11. 3 + V-16 + V-4-5-V-9.
12. 3 + V-36-(l+2 V-25) + 3 V-16.
13. V-49 + 5 V-4-(6 + 2 V-9).
Simplify each of the following expressions (cf. § 162):
14. V^--(2V^ + o~3 V^24) + 3V^^18.
15. V-16 a^x" +VI-5 + 2V0-3O-V-9 a2«2 + V- aV.
16. a; V— 4+V — ar' — 2if — 1 — V— 32.
17. Solve the equation x^ — 1 = (cf. § 72) and find the sum of
the roots; check your addition graphically. Similarly find the
sum of the 3 cube roots of 8; of the 3 cube roots of —27.
Find the product of :
18. 3 V^=^ by 5 V-12. 20. 2 V^^ by V-4aV.
19. 5 V^^ by 2 V^^. 21. —1-6 r^ + i*^ by i\
22. V^^ + V^=^ by V"^ - V^=^
23. 3+2V^^by 5-4V^=n:.
24. V^r50_2V^^n^by V^^-5V^^.
25. 3P-4.i^hy2i^-3i^\
26. Show that the sum, and also the product, of a-\-bi and
a — pi is real (a and h being any real numbers). Show that the
same is true also for V— 4 — 3 and — V — 4 — 3.
27. Show that both the sum and also the product of any two
conjugate complex numbers is real.
28. Multiply V^a + V^^ + V"^ by V^^ — V^^ + V^^
264 HIGH SCHOOL ALGEBRA [Ch. XV
29.. (l-}-V^)'=? 30. (2-3if=? 31. (2a-3a;V^)'=?
32. Find the product of a V— ^ + 6 V— ot, a V— a + 6 V— 6,
and h V— h — a^ — a.
33. Show that — ^ + i V— 3 and — i — i V — 3 are conjugates
of each other, and also that each is the square of the other.
34. Reduce -^ ~ -\ — ^ to its simplest form.
3-V^4 3 + 2i
35. Simplify each of the following indicated quotients :
V^T5 . V^24. V^. Vc V^. V84 v:r6-i-2V^:r8~
V^^^' V"^^^' V^=^' V^^' V5 ' 2V^^' v^^
36. Show that ^ + ^^zl = «c + ?>^ + (^c-ad)V-l .
c+dV-1 ^^^
Perform the following indicated divisions (cf. Ex. 4), and check
your work by multiplying the quotient by the divisor :
37. -i-,. 40. ^-±^^.
l^i 5-2z
38 — 41 V2a;-3ai
^■' + ^' ' V2^ + 26^
39. 2W^. ^2 V^-lV6.
3 + V-2 ?:V6H-Va
43. If a and h are positive and unequal numbers, show that
V— a ± V— 6 cannot equal a real number (cf. § 161).
44. Show that if cc + V— 2/ = aH-V— 6, wherein y and h are
positive numbers, then x = a and ?/ = ft.
45. Find the square root of 5 —12 V— 1.
Hint. Let y/x-y/y. V^^l = Vs - 12 V^HT (cf. § 162).
Find the square root of :
46. 10-6V^^. 48. 3 + 2V^=l0.
47. 6V^^-17. 49. 5|-3}V^^.
CHAPTER XVI
THEORY OF EXPONENTS
ZERO, NEGATIVE, AND FRACTIONAL EXPONENTS
171. Introductory, (i) As originally defined (§9) an
exponent is necessarily a positive integer, and it is in this
sense only that we have thus far used it. Under this restric-
tion we have established the following exponent laws (§ 110),
wherein a is any real number except :
I dT" 'dP' = a"^+»,
II {^ory = a*»%
III {ahy = d^ . 5%
IV a"' : a» = a^-**.
(ii) We now propose to extend the meaning of an expo-
nent so as to include such symbols as a^ a~^, and a^, along
with our former exponent expressions.
In extending the meaning of any symbol already in use,
however, the extended meaning should be such as not to
disturb any rules of operation already established for the
symbol in question. Hence, we shall admit such symbols as
a^ a"^ etc., into our algebraic notation if, and only if, we can
assign to each of these symbols a meaning consistent with the
above exponent laws.
172. Meaning of such symbols as a^, a~^ and ai (1) If,
following the plan given § 171 (ii), wa let ?^ = in law I,
§ 171, we obtain
a"* ^0 = a"*, [since a^^^ = a^
hence a^ = 1 \
i.e., if law I is to admit the symbol a^ then this symbol must
have the value 1.
266
266 HIGH SCHOOL ALGEBRA [Ch. XVI
(2) Again, if in law I, § 171, we let m = 5 and n=—3,
we obtain a^ - a~^ = a^ = 1, [since a^~^ = a^
whence a~^ = — ;
aP
i.e., if law I is to admit the symbol a~^, then this symbol
must have the same meaning as — .
(3) And finally, if law I, § 171, is to remain valid for such
symbols as a^, then
a^ -a^ -a' = a\ [since a^^'^'^^ = or
and therefore a^ = Va\ [§11^
i.e., if law I is to admit the symbol a% then this symbol must
have the same meaning as ~va^.
p
173 Definitions of a^, a"^ and a^. Using as a basis the
special cases considered in § 172, we shall now define the
p
symbols a^ a~\ and a"" ?ls follows:
(1) aO=l,
(2) «-' = -,.
a''
and (3) a^-=</'^;
wherein a represents any number whatever, and k, p, and r
are positive integers.
Moreover, these definitions — since they are based upon
one exponent law only — must be regarded as tentative until
they are shown (§§ 174, 175) to satisf}^ all of the exponent
laws.
EXERCISE CXVII
Assuming the validity of § 173 (1) and (2), show that :
1. 3c-^=3.i=^^. 3. a%'c=c.
c c
172-174] TIIEOliY OF EXPOXIJXTS 267
Keduce the following to equivalent expressions free from zero
and negative exponents :
5. ««. 9. 5A-2. 13. Sx'^-r-y-*.
6. cc-3. 10. ifc-^. 14. f-^y-\
7. Qa\ 11. 7--r-3. 15. 2/'-2/"^
8. -4-!-a°. 12. Sa^V'- 16. 3 a^-^^ -- ic^/-^.
Assuming the validity of § 173 (3), translate the following into
equivalent radical expressions :
17. ai 20. t^. 23. (ia-^^s)!.
18. ai 21. {axy^. ^*' (^^"'>^-
19 a;l 22. (4c^d)i 25. (^^'y.
Write in the fractional exponent notation :
26. a/^. 30. -v/oV. 34. -V-^m^.
27. -v/m. 31. %V. 35^ &-^'l6^.
28. V 6. 36. 2\x y-^.
^\ 23 _i
29. V^24. 33. \-^,' 37. acv^a^c-^^.
38. Find the numerical value of: 5« ; 3"^; 4'^; 9* ; (i)^ ; 4-2-2^.
39. Show that if law II, § 171, is to admit the symbol a^, then
a^ must equal 1 (cf. § 172).
40. Show that if law IV, § 171, is to admit the symbol a"'*,
then a~^ must equal —•
a
41. Show that if law II, § 171, is to hold for a^, then a^ must
equal Va^.
174.* The symbols a^ and a~* obey all the exponent laws.
That a^ and a^^ as defined in § 173, satisfy all the exponent
laws may be shown by assigning zero and negative integral
*The proofs given in §§ 174 and 175 may, if the teacher prefers, be omitted
until the subject is reviewed.
HIGH 8CH. ALG. — 18
268 HIGH SCHOOL ALGEBRA [Ch. XVI
values to m and n^ both separately and together, in the equa-
tions which express those laws.
Thus, if we let w = in law II (n remaining a positive
integer), we obtain (a^y = a^ [since a" • « = a"
i.e., r = l,
which is correct ; hence a^ =1 is consistent with law II.
Again, let m = in law I V (ri remaining a positive integer),
and we obtain a^ : a"" = a~\ [since aO-"= a~"
i.e., l:a^ = a-%
which is a correct equation [§ 173 (2)] ; hence a^ = l and
a"" = — are consistent with law IV.
Once more, let m = — r and n= —s (where r and s are
positive integers) in law II, and we obtain
(a-0"* = «""""' = «'■*;
but this is consistent with § 173 (2), for
<-=(r=^
[§ 173 (2)
[§§ 109, 92
or
Moreover, if we similarly test the remaining combinations
of positive and negative integral and zero values of m and
71, in the four exponent laws, we find that definitions (1) and
(2) of § 173, and the exponent laws of § 171, are entirely
consistent. Hence we need no longer regard the definitions
of a^ and a~* as tentative (cf. § 173, last part).
The testing of some or all of these remaining cases may
be assigned as an exercise to the pupil.
p.
175. The symbol a' satisfies all the exponent laws. That
p
a^=:^aP is consistent with the exponent laws may be shown
as follows :
174-175] THEORY OF EXPONENTS 269
Let p, r, s, and t represent any positive integers, then
PI . _
= Va^^ • Va''^= VaP' • a'' [§§ 154, 149 (i)
pt+rs P,i
^.e., law I holds good for such symbols as a^,
p s^ s
Similarly, (</')' = ( V^)' = ^( Va^)^
= V^ [§ 149 (iii)
ps Pi.
P
i.e., law II holds good for such symbols as a^,
p .
Again, (a6)^= </(^ahy=<JaP • 6^
= 7^.V^ [§149(i)
p p
2
i.e., law III holds good for such symbols as a^.
The proof for law IV, being closely similar to that for law
I, is left as an exercise for the pupil.
Observe that the above proofs remain valid:
(1) if r (or t) takes the value 1, in which case ^ [or - j be-
comes an integer.
(2) if p (or 8) is negative or zero (cf. § 174), in which
^( or - ) becomes a negative fraction or integer, or zero.
r\ tj
Therefore these proofs, taken in connection with those
previously given, include all possible combinations of positive
and negative, integral, fractional, and zero values of m and n,
in the exponent laws of § 171. We need, therefore, no longer
p
regard the definition of a'^ [§ 173 (3)] as tentative.
p
Note. Observe also that a!" represents the principal rth root of a^, since
that is the meaning of the symbol y/(f (§ 150).
case
270 HIGH SCHOOL ALGEBRA [Cii. XVI
EXERCISE CXVIII
1. Does a^ equal x^ even when a and x are unequal ? Explain.
2. By means of § 173 (1) show that a"* -r- a" = a"*'" when n =
(cf. § 171, IV).
3. By means of § 173 (2) show that (a^)-^ = a-^ and that
a-'^a-' = a-' (cf. § 171, II and IV).
4. Show that 6a!^^ = 6aY ^^^^ ^^^^^ 3:Var^ ^ 2_:^^3 ^
y~* XT 2*xy~^ S^n~^x
5. By proceeding as in Ex. 4 show that, by changing the sign
of the exponent, di factor may be transferred from the numerator
to the denominator of a fraction, and vice versa.
6. Is ^^^-^ equal to ^L±A? Explain. Observe carefully
4 a? 4:Wx
that a factor but not a part may be transferred as in Ex. 5.
Free the following expressions from negative exponents, and
explain your work in each case :
7. ^.
8.
5-2
11.
52 . 12-1
10-1 . 3*
15.
(82_1)-1
2d'
12.
a-' + 2.
16.
9-V
6x-'
a-'x-' ,
13.
x-'-\-y-^
5
17.
9-V
h-'x-'
(-6)-i(a^2/)-^
3.2-2
8-2
14.
m~3 — n"^
m-3 . w-«
18.
3r-2-4s
10.
19. Is . ^ equal to 3 aa;-^ ? Why ?
a;2
20. As in Ex. 19, write the following fractions in integral form :
o^. 4:a-^c-\ 4-ia-V . m-i-3n-*
6^2/' ac ' 5a-id~i' m~^n
21. What is the diiference in meaning between mJ and m^?
between m-^ and m"'^ z.e., between m^ and m ^ ?
175-176] THEORY OF EXPONENTS 271
Write Exs. 22-26 below as radical expressions, and write Exs.
27-30 with positive fractional exponents :
22. a~T^6TV. ^^ 6~«/?. 28. -Vr-Ps^
23. 2^dh~'^. 1 n
26. a'^ — b^.
29. V-32r-V«.
24. — - . __ 3/ — . ._«,,_o, _. *
3^' 27. -v/-ic-«d-'e. 30. V(c+d)«-f--v^(c+d)*.
Find the numerical value of :
31. A 35. 6° -2-*. /27xY
32.4- 36. (.09)1. 'V.OOSr
40. (32-^^(32*)-^.
33. 9t. 37. (256).. 41. (-,W-(169)i
34. 8-^.25^ 38. (Hl)"^ 42. (64)-^ . (16-^)^
-(i)'-(r-(f )■'-"■»-■
176. Operations with negative, zero, and fractional expo-
nents. From the definitions of negative, zero, and fractional
exponents (§§ 173-175) it follows that in all operations
with such symbols as a^ a~^ and «s their exponents obey
the exponent laws of § 171 ; that is, these exponents behave
just as though they were positive integers.
When working with fractional-exponent expressions it is
frequently necessary to change the exponents to higher or
lower terms ; this does not change the value of such an
expression ;
for since V^= %^, [§ 151 (iv)
therefore a'' = a^'^ .
Operations with radicals may often be greatly simplified
by first converting the various radical expressions into their
fractional-exponent equivalents (cf. Exs. 29, 41, and 57 in
the following exercise).
272 - HIGH SCHOOL ALGEBRA [Ch. XVl
EXERCISE CXIX
.5
1. Simplify : V2 • ^4 • 2^ • 2-^
Solution. \/2 . ^ • 2' • 2"^ = 2^ • 2^ . 2^ . 2"^ [§ 173 (3)
= 2MH-|=2'3 = ^4. [§171, I
2. Simplify: {ah)^ - (p^)^.
Solution. (a&)^ • {h'^c)^ = (ab^c)^ = ahc^. [§ 171, III, II
6^r
3. Simplify: (-
s„.„„o.. (_) =(_) =(-) =_ f5„i,„
~ &2 - 52 •
Perform the following indicated operations, and express your
results in their simplest form :
4. 8t . 8^ . si 7. (^)f . A y . M\ 10. s-^i^-^s=^
8. 2x^ -h4: x^'
6. 2L8-i.27i 9.^?.^?. 12. (36a-tci)-i.
13. By means of fractional exponents, reduce Va^ and V^ to
equivalent radicals of the same order (cf. § 154).
2. 5
Solution. The given radicals are, respectively, equivalent to g^ and cc^'
and these expressions are, respectively, equivalent to g^ and o;^, i.e., to
\/g4 and v^, each of which is of order 6.
14. Solve Exs. 12-16, p. 245, by means of fractional exponents.
15. By means of fractional exponents, solve Exs. 19-22, p. 247,
Exs. 10-14, p. 251, and Exs. 9-12, 25-29, p. 252.
By means of fractional exponents, simplify :
16. V^-^Sv"^ . -y/x. 18. ("v/r^ . Vi . -v/O""
17. ("v/m^ . VOi 19. (8^ . 8-^)-2.
176]
THEOR
20.
21.
-^4a-2-^-8a-^.
22.
(^?)-^.(3s-V^-)l
23. ViC"^ • Vit*"^.
24. (Vo'^VO"'""-
i
THEORY OF EXPONENTS 273
26 (a"" + o^^)V^ .
27 m^ — m~^ + a/7 m^
28. !;!
o
25. f "" '\ ] • V ^ +
0,2
29. Eind the product of 3Va — BVy by 2Va + V^/.
Solution. Since SVa - 6\/y = S a^^ -^vK and 2 Va + S^y = 2a^+ ?/3,
therefore this product becomes
2a^ + y'^
qJ-^^ - 10 Jy^
+ 3 Jy^ - 5 y^'^^
Qa-7 a^y^ -6y\
If it is desired, this product may, of course, be written in either of the
following forms :6a- lVa\/y — 5\/«/2, or 6 a - iVcfiy^^ — 5v^.
Perform the following multiplications :
30. a^ + 6^ by a^ — &i 32. m^ — m^r^ -f- n^ by m^ + ?i^.
31. oj^ — Q^y^ + 2/^ ^y i»^ + 2/^- 33. m^ — m^n~^ + n" ^ by m"^ + n ^.
34. ia^^-TV^^/HTVa^^^Z-aV^/^byiaj^ + i^/^.
35. 81 yy^ - 27-s/'^</y + 9^/^Vf - 3 2/a/^4- 2/Hy 3 "V^ + y\
36. V« — 4-v/a^a; + 6^/ax — 4 A^aa;^ H- Vx by \/a — 2 Vax + V«.
37. '>n ^ + m"^ — 2 m^ + 4 m~^ by 1 + 2 m^ — -^— •
Vm
3 *_ 4 — i
38. ^r^ + q-^-^ — p-'^^q ^ by p~-^^ + Q" ^.
39. If Jx^/x + 2 n Vn + f a.'^« + 6 n^/^ by V7i - 3 a;^ + ^,«^-
40. 5 a-'x* + 3 a%"x-' - b'-V by x-^ - 3 ^"~%-' + ah
274 HIGH SCHOOL ALGEBRA [Ch. XVI
41. Divide x^ — y^hj Va? + Vy.
Solution. Since \/x + v^ = a;^ + y^, this solution may be put into the
following form :
x2 — ?/3
x3 + y^
x^ — x^y^ + xy - x'^y^ + x^y^ - y^
— x^y^ — y^
— x^y^ — x^y
x^y - y^
x% + xy^
- xy^ - y^
3 2.
— xy^ — x^y"^
1
X^y^ — y3
a; V + x^y^
- x^y^ ■
-x^y^
r
The above quotient may also be written thus :
VxP — Vx^ Vy -\- xy — y/o^Vy^ + Vx • y^ — y/y^.
Note. To appreciate one of the advantages of fractional exponents the
student has only to perform the division in Ex. 41, using the radical notation,
and compare his work with the above solution.
Perform the following divisions :
42. a + x^hy a • + x^. 43. m^ — n^ by m^ — n®.
44. x-^ + 3y~^ — 10 xy-'^ by x''^ Vy — 2.
45. J +2 ^ah-'^ + ^ by ^a + 6"^.
b
46. x^ + x^ -\/y — X -y/x y'^ — xy-{- -Vx y^ + y'^ by Va; 4- -y/y.
Simplify the following expressions :
47. / Va? + -v^.v Y. y/x-Vy ^^ x-y y^-yt
a;— +r'*'aj-"-r"** ' 2/ + V2/+I ' yt-1
176J
51.
THEORY OF EXPONENTS
1 . 1
275
a^
+
Write down, by inspection, the square root of each of the
following expressions:
52. 1 — 2i6^ + wi 54. p^ — 4 + 4p"i
53. x^ + 4 x^ + 4. 55. ax^ + 2 a^x^ -\- a^x,
56. 771 -\- n -{- p — 2 m^n^ + 2 n^p^ — 2 m^pK
57. Extract the square root of ^x* — 2 Va^ '+ 5 Va^ —4 Va? + 4.
Solution. This expression written in the equivalent fractional-exponent
form is cc^ — 2 a;'^ 4- 5 x^ — 4 x^ + 4, and in this form its square root may be
extracted just as though it were a rational expression (cf. § 117) ; thus :
2 x^ + 5 x5 -ix^ + A\x^ -xM- 2
x^ — x^
-2x^
+ 5x^
2x^ + x
2x^
2x^ + 2
4x^+4
4x^
4 x^ — 4 x^ 4- 4
hence the required root is xs — x? + 2, i.e., Vx^ _ vx + 2.
Extract the square root of each of the following expressions :
58. ar^ + 2 a;^ + 3 a; 4- 4 a;^ + 3 + 2 a."2- 4- x'K
59. a3 — 4 ot^ + 4 a 4- 2 a^ — 4 a^ + aK
60. n'^ — 2 wr^ii^' + 2 m^n^ + imT^n 5—2 mW 4- m^
Extract the cube root of the following expressions ; write the
results with all exponents positive, and then replace all fractional-
exponent forms by radical signs (cf. footnote, p. 185):
61. 8 4-12a;t4-6a;^4-aj2.
62. 8a;-i-12a;"%4-6aj-V-2/^.
63. r^_6ri4-15r^-204-15«^-6^4-^i
64. 8 a^h--^ 4- 9 a&^ 4- 13 a^ 4- 3 a^6 4- 18 a^h'^^ 4- ^' 4- 12 ah-\
276 HIGH SCHOOL ALGEBRA [Ch. XVI
Solve the following equations:
65. m^ = 4. 67. x~^ = 5. 69. a;"^ = — 27.
66. ^^ = 8. 68. \yi = 25. 70. yjm^ = 3\js.
71. 2r^ + 5r^ — 3 = 0. [Hint. Put?/ = ri].
72. £c"^ 4- 5 a;"^ 4- 4 = 0.
73. (2A;-3)-2 + 7(2A:-3)-i-8 = 0.
177. Rationalizing factors of binomial surds. Another
advantage of the fractional-exponent notation is that it fur-
nishes an easy method for finding the rationalizing factor of
any binomial surd whatever, — only quadratic binomial surds
were considered in § 157.
Ex. 1. Eind the rationalizing factor of x^ + 2/^-
Solution. Since (x'^y — (y^y is exactly divisible by x'^ -j- y^
when n is any positive even integer [§ 66 (ii)], and since 6 is the
smallest value of n for which both (x^y and (y'^y are rational,
therefore the rationalizing factor (cf. § 157) is
(X^y—(yh'^ X^—f 5 4 1 2 3 1, 6
-^^ — 7 1 — = ^ 7 = a?3 — x^y^ -\- xy — x^y^ -^ x^y^ — y^ '
.^3 -f ?/^ x^-\-y^
Ex. 2. Find the rationalizing factor of x^ + y'^.
Solution. As in Ex. 1 [cf. %66 (iv)], we find this factor to be
(X^y^ + (yh^^ X^ -f V^^ M 13 2 , J 2 4 11
x'5 _|_ 2/^ a;^ -f y^
EXERCISE CXX
Find the rationalizing factors of the following expressions :
3. s^+A 7. Sv^ — 2vK
11. ah'^-^3v\
4. s^ — t^. 8. s~^-\-t\
12. x~^-\-2y\
5. a^ — xK 9. 2m^ — ?ii
13. .'i;"* — Zi
6. m^ + ni 10. 2x^-3yk
14. 2r~h-'^-t'
CHAPTER XVII
QUADRATIC EQUATIONS
[Supplementary to Chapter XII]
178. Solution of quadratic equations by means of a formula.
Since every quadratic equation in one unknown number
may be reduced to the form ax^ -\-hx-{- c= ^ (§ 122), and
since the roots of tliis equation are ^whatever
the number 8 represented hy a, 6, and c (Ex. 3, p. 195), therefore
the roots of any particular quadratic equation may be found
by merely substituting for «, 5, and <?, in the roots of the
above general equation, those values which these coefficients
have in the particular equation under consideration.
E.g., since the roots of ax^ + hx+ c = {) are — ^ "^ —, therefore
the roots of 3 ic2 + 10 x - 8 = (in which a = 3, & = 10, and c = - 8) are
-10±VlO-^-4.3.(-8) ,,,.^-10^14 ,.,.,2,^^_4,
2.3 ' ' 6 ' 3
So, too, the roots of 6 y^ 4. 19 ^ _ 7 = are
19±Vl9^-4.6(-7) 1 ^^^ _ 7
2.6 ' ' 3 2
and the roots of a;2 - 3 x + 5 = are , -(-3)=tr V(- 3)2 — 4.1-5 ^
i& " 1
i.e., ^
Note. While the student should, of course, be able to solve quadratic
equations without the use of the formula, he is advised to commit this for-
mula to memory, and henceforth to employ it freely ; he will find this well
worth his while, because roots of quadratic equations are so often required
in mathematical investigations.
277
278 HIGH SCHOOL ALGEBRA [Ch. XVII
179. Character of the roots of ax" -\-bx + c = 0. Discriminant.
As we have already seen (§ 126), the roots of the equation
^ , ^ -b-h -VP-^ac , _5_ V62-4a(?
aar + oa; + <? = are ^ and t:
2a 2a
Hence it follows that, if a, 5, and c represent real and
rational numbers, the roots can be imaginary or irrational
only if V52 — 4ac is imaginary or irrational; ^.e.,
If h^ — 4tac is positive^ the roots are real and unequal;
ifb^~4:ac = 0, the roots are real and equal ;
if l^ — ^ ac is negative^ both roots are imaginary/ ;
and the roots are rational only when b^—iac is an exact square,
(Let the pupil fully explain each case.)
The expression b'^ — 4:ac^ which determines the character of
the roots, is usually called the discriminant of the equation.
Thus, without actually solving the equation 3 cc^ _ 5 aj — l = 0, we know
that its roots are real, irrational, and unequal because, for this equation,
\/&2 _ 4 ac = V37, and VST is real and irrational.
Similarly, we see that the roots of 2 r.^ + 5 a; — 8 = 4a; — 11 are imaginary,
because for this equation h"^ — Aac = — 23.
EXERCISE CXXI
1. By means of the formula of § 178, solve Exs. 6-17, p. 196.
Without first solving the following equations, tell whether
their roots are real, imaginary, rational, equal, etc., and explain :
2. a?-^x + Q> = 0. 5. 3^^4-ll^ + 17 = 0.
3. a;2-6a; + 9 = 0. 6. 1(3.^2+ 2) -i = i (a; -5).
4. ^^-1^-11=0. 7. 25?^2_20^^ + 7 = 0.
8. In each of Exs. 4-11, p. 197, determine the character of the
roots without solving the equation.
9. For what value of h will the roots of 3 .t^ — 10 ic + 2 A; =
be equal ?
Suggestion. The roots will be equal if (- 10)2 _ 4 • 3 . 2 A; = 0. Why ?
179-180] QUADBATIC EQUATIONS 279
For what values of m will Exs. 10-15 have equal roots ?
10. mic^— 6x4-3=0. 13. my^ — 5my-\-ll = m.
11. ^ + 3ma; + 7 = 0. 14. y\l - m) -\-7 y = 9 -3my.
12. 3a^-4mic-f 2 = 0. 15. -4i/2-32/-3 = m(2/-f 2/ + 4).
16. Translate into verbal language the conditions for the char-
acter of the roots of a^i? -\-hx-\-G — 0.
17. Show that if one root of a quadratic equation is imaginary,
then both are imaginary, and each is the conjugate of the other.
18. For what values of A; are the roots of 36 a?^ — 24 A;a; -f- 15 A;
= — 4 imaginary ?
Solution. Here the discriminant Ifi — ^ac= ( — 24 ky- — 4 • 36(15 Jc -\- i)
= 144(4 A;2 - 15 A; - 4) = 144(4 A; + 1) (k - 4) [cf. § 64]. Hence the roots are
imaginary for those values of k which make {4:k + l)(k — 4) negative, and
for those values only. Now (4A; + l)(^• — 4) is negative only when one of
its factors is positive and the other negativ^; hence the roots of the given
equation are imaginary when k lies between — ^ and 4. (Why ?)
19. For what values of k are the roots of 36 f — 24:kt-\-15k
= — 4 real ? How do the roots compare if k= — ^? k = 4:?
20. Without actually solving the equation, find the values of
m for which the roots of 4 mV + 12 m^x -\-10 = m are equal ; also
those values of m for which the roots are real ; also, those for
which the roots are imaginary.
180. Relation between roots and coefficients. If we let r
and / represent the roots of ax^ -^ bx -\- c — 0^ i.e,^ if
_54.V52-4a(? ■. , -b-Vh^-iac
2 a 2 a
then r + r' =
_5 + V^2_4^^ _5_V52_4ac_ 6,
2a 2a a
and r-r'^
_ J 4- V52 _ 4 ac -b- V62 -4:ac_c
2a 2a a
(2)
Let the pupil work out (1) and (2) in detail, and translate
each of these equations into verbal language.
280 HIGH SCHOOL ALGEBRA [Ch. XVII
EXERCISE CXXII
In the following equations, name the sum of the roots, also
their product ; then check your answers by solving the equations.
1. x' + 5x-2 = 0. 4. a;2_30a; + 25 = 0.
2. a^-10x = — 16. 5. a^-\-px = -q.
3. 4s2-6s=3. 6. ax^-{-2bx^c = 0.
7. In each of Exs. 4-11, p. 197, write down, without solving the
equation, the sum and the product of its roots; explain your
work in each case.
8. If one root of ic^ + 5 a; — 24 = is known to be 3, how may
the other root be found from the absolute term ? from the co-
efficient of the first power of a; ? Do the results agree ?
9. If one root of any given quadratic equation whatever is
known, how may the other root be most easily found ?
10. What is the sum of the roots of 3 mV -j-(Sm — l)x 4-5 = 0?
For what value of m is this sum 3?
11. If one root of2a^ — 3(2A: + l)a;-}-9A; = is the reciprocal
of the other, find the value of k.
Hint. Equate one root to the reciprocal of the other and solve for k.
12. For what value of k will one root of the equation in
Ex. 11 be zero ? With this value of Zc, what will be the value of
the other root?
13. Answer the questions of Exs. 11 and 12 for the equation
2(k+iyx'-S(2k + l)(k + l)x-{-9k=0.
14. Show that if one root of ax^ -\-bx-\-c = (whatever the
values of a, b, and c) is double the other, then 2 6^ = 9 ac.
181. Values of simple expressions containing the roots.
If r and / are the roots of a given quadratic equation,
§ 180 enables us to find the value of such expressions as
- + — , r^ + r'^, etc., without first solving the equation.
180-182] QUADRATIC EQUATIONS 281
E.g., the value of - + — , for the equation fl:;^ — 5 a; -f 3 = 0, may
be found thus :
1 l_r|-f-r.
r r rr
but, for this equation, r' + r = 5 and ri^' = 3,
therefore - + - = -.
r r' 3
Similarly, since r^ + r'^ = (r-{- r'y — 2 rr'j
therefore, for the above equation,
7^4-/2 = 25-6 = 19.
182. Formation of equations whose roots are given numbers.
(i) Sum and product method. If r and r' are the roots of
the equation ax^ -\- bx + e = 0, i.e., of a^-\--x-\--=0, then
a a
(§ 180) this equation may be written in the form :
2^ — (r + r'}x -\- rr' = 0.
And from this we learn how to write down a quadratic
equation whose roots are any two given numbers.
E.g., if the roots are to be 2 and — 5, we have
— (r + r') = 3, and rr' = — 10,
whence the equation is
a^4_3aj-10 = 0.
(ii) The factor method. An equation whose roots are any
given numbers may be written down as follows (cf . § 72) :
The roots of (^x — r}(x — r') =
are evidently r and / ; hence the equation whose roots are
2 and - 5 is (x - 2)(x + 5) = 0,
i.e. , as before, a^ -^Sx — lO = 0.
EXERCISE CXXIII
1. If r and r' denote the roots of x^ — lx + 12 = 0, find, with-
out solving the equation, the value of
J- J- -I- , -L 9 ■ 19
——,, —,, - + -, r^-\-r".
ly. _l_ y.1 n.iy.1 n. y>^
282 HIGH SCHOOL ALGEBliA [Cii. XVII
Find for each of tlie following equations the value of - + - ,
11 r r"
T^ + ?''^, r — r', and '-
r r
2. f~-12t = -m. 5. a;2 + 2a; = 4.
3. 6m2 — m = 2. 6. si? -{-px-\-q = 0.
4. 32/^-162/ + 5 = 0. 7. ax^ + hx-\-c==0.
8. Solve each of the above equations and thus verify the
results in Exs. 2-7.
By each ol the methods given in § 182, form the equations
whose roots are :
21. 1±V5.
16. r^, r'^. 2
17. V6±4. 22. 5, -^.
18. 5±2V3. 23. Vc±V^.
14. r, — r'. 19. 2 ± 5 ?:. 24. r±-,-
r
25. The roots of x' — ^x-\-2 — being ?♦ and ?•', form a new
equation whose roots are - and -. (Cf. Exs. 1 and 15 above.)
r r'
26. If r and r' are the roots of 3z^ — llz-20 = 0, form an
equation whose roots are - and _ ; also one whose roots are r^
r r'
and ?''^; also one whose roots are r -\ — andr' + -.
r' r
Write the equations whose roots are :
27. - 1, 2, - 5. 29. 1 ± V5, 5.
28. -a, -b, —c. 30. ± Vc, c + c?.
183. Factors of quadratic expressions. As in § 182, if r
and r' are the roots of the equation ax^ -{-bx-{-c = 0, then
a^ + - X + - = x^ — (r -{- r') X -\- rr' = (x — r^Cx — rO,
a a V yv ^
9.
5, -3.
10.
-4,4.
11.
-a, -6.
12.
i|.
13.
i-f
182-184] qUADUATlC EQUATIONS 288
^.e., multiplying by a^
ao^ -\- hx -\- c = a (x — r)(x — r^) ',
hence, if the roots of the equation aa^ + bx-i- c = are r and
r\ then the factors of the expression ax^ -{- bx -\- c are «, x — r^
and x — r'.
184. A quadratic equation has two roots, and only two.
By actually solving the equation ax^ -\- bx + c = (§ 126)
we find that it has two roots, say r and r^ That it can
have no other root, as r", is evident if we write the equation
in the form
a(x-r')(x-r^)=0 [§ 183
and observe that a (r" — r}(r" — r'} cannot be zero if r''
differs from both r and r',
EXERCISE CXXIV
1. Since 2 and 7 are the roots of x^ — 9 x-\- 14: = 0, what are
the factors of a^ — 9 ic + 14 ?
2. By first finding the roots of the equation 15 a^ — 4a?— 3 = 0,
find all the factors of the expression 15 a^ — 4 a; — 3. Check your
answer by finding the product of these factors.
3. Write a carefully worded rule for factoring quadratic
expressions by the method of § 183.
Find (in accordance with the rule just made) all the factors of
the following expressions, and check your results :
4.
5a;2-12a;-9.
11.
(2y-iy-5(y + l)+8.
5.
Sz'-4:5z-lS.
12.
a^+px-^q.
6.
a)2 + 9.
13.
ax^ + bx-\-c.
7.
s^+s+l.
14.
3m2 ^
-— m — 5.
8.
9.
4m2-24m-13.
8a^_2a;-3.
15.
10.
(^ + l)(2-a^)+9
HIGH 8CH. ALG.—
— X.
•19
16.
m^ + 6 m + 13.
284 HIGH SCHOOL ALGEBRA [Ch. XVII
17. Are the expressions in Exs. 4-16 equal to 0? What justi-
fication have we, then, for writing them so ?
18. How many roots has a quadratic equation ? Verify your
answer for the equations 28 x^ -f- 29 a; +6 = and m^— 10 m= —25.
19. Show that the cubic equation 27 2/^ — 1 = has three roots
and only three (cf. Ex. 17, p. 263).
REVIEW EXERCISE- CHAPTERS XJ-XVIi
1. Find the square root of x^ -\- 20 a^ -\- 16 — 4: oc^ -{- 16 x ; also
of a;'* — 2 x^y~^ — 4 cc^ + y~^ + 4 x^y + 4 xy^.
2. Find (correct to three decimal places) the value of Vll-7 ;
VA; V23561.
3. Expand by the binomial theorem : (a^ — 2/^" ; ( 9" ~^ ^
(l + c-y- (l-Sm^y-, ^-l
+ a;i
4. Use the binomial theorem to find (correct to five decimal
places) the value of (10.001)^ i.e., of (10 + .001)^
Simplify :
5. ^1 + -^^+^. 7. </ab^(ah-'c^)(a-h^c-').
prnj
\ z-l ^ z + 1 Xc** V'c^ V
9. (a3-62)^(-^^-V6).
^c-d^c-d ^c-{-d^c + d
11. (2V-3-3V^=^)(6V-2 + 4V-3).
13. By rationalizing the divisor perform the following indi-
cated divisions :
1 + r + Vl^^ . 1 V^-V^^3y . 3
l+r-Vl-r^' V6-V3-I' V^^-V^' 3*-f5"^
184]
q UADRA ric Eq ua tiojsi s
285
14. Express -^^89 — 28 VlO as the difference of two surds.
15. Find the sum of V- 25, - 1- V^ 5-lOV^, and
— 2 — 7 i, graphically ; also, by the same method, subtract the
fourth of these numbers from the third.
Solve the following equations and check as the teacher directs
20. m + 5 -h Vm -\-5 = 6.
16. f-| + 7| =
17. ^x-^l + x =
18. i-?l
= 1.
19 3 a^ 4- V4 a? — a?^ ^ 2
21. ^a-{-z-{--\/a — z = -y/b.
22. aj^ + 4 ic^ — 117 = 0.
23. 19a;*4-216a;^ = a;.
24. 144r2-l + 6V9r2-r=16r.
3 aj — V4 a; — x^
25. (7-4 V3)a:2+(2-V3)a;-2 = 0.
Solve the following systems
0^2 + 2/2 = 85,
, xy = 42.
26.
29.
(2
xy = 4:- y\
0^-2/2 = 17.
27.
r 0^-2^^ = 215,
I a^4-a^?/ + 2/' = 43.
28. ^
f 3r2-2rs = 15,
30.
31.
a? 2/
,a^ r
mo 4- Vv -h ty = 11,
t2r + 3s = 12.
2vw— Vv + w = 13.
32. Show that the difference of the roots of the equation
ic^ +pa; + g = is the same as that of a52 -|- ^px + 2 p^ + g = 0.
33. For what values of m will a^ - 2 (1 +3 m) «+ 7 (3 + 2 7^) =
have real roots ? equal roots ? imaginary roots ?
34. Form the equation whose roots are a, — a, and h ; also the
equation whose roots are the negative reciprocals of the roots of
ax^ — 6a; 4- c = 0.
35. If the roots of a^ — 2>a5 + g = are two consecutive integers,
show that 2>' — 4 g — 1 = 0.
286 HIGH SCHOOL ALGEBRA [Ch. XVII
36. A rectangular plot of ground contains 42 acres; find its
sides if its diagonal measures 1243 yards.
37. In a regiment drawn up in the form of a solid square,
the number of men in the outside five rows is -fj of the entire
regiment. Find the size of the regiment.
38. In a quarter of a mile drive, the fore wheel of a carriage
makes 22 revolutions more than the hind wheel ; if the circum-
ference of each wheel were 2 ft. less than it now is, the fore wheel
would make 33 revolutions more than the hind wheel. Find the
circumference of each.
39. A crew can row a certain course upstream in 8^ minutes,
and were there no current, they could row it in 7 minutes less
than the time it now takes them to drift downstream. How long
would it take them to row the course downstream ?
40. Two men, A and B, have a money box containing $ 210,
from which each draws a certain fixed sum daily, the two sums
being different. Find the sum drawn daily by each, knowing
that A alone would empty the box 5 weeks earlier than B alone,
while the two together empty it in 6 weeks.
CHAPTER XVIII
INEQUALITIES
185. Definitions. The symbols > and < stand for "is
greater than," and " is less than," respectively ; thus, the
expression a<b is read ''a is less than 5." One real number,
a, is said to be greater than another, 6, U a — b is positive ;
if a — 5 is negative then a is less than b.
E. g., 5 < 8, since 5 - 8 = - 3 ; and - 4 >- 9, since -4-(— 9) = +5.
A statement that one of two numbers is greater or less
than the other is called an inequality ; thus, 5a;— 3>2yis
an inequality, of which 5 a; — 3 is the first member, and 2 y
the second.
Two inequalities are said to be of the same species (or to
subsist in the same sense') if the first member is the greater in
each, or if the first member is the less in each ; otherwise
they are of opposite species.
Thus, the inequalities a > 6 and c + <? > e are of the same species, while
2-2 ^ y'2 -> 2;2 and m^<,n^ + mn are of opposite species.
186. General principles in inequalities. Before memoriz-
ing the following principles (1-7), the pupil should illus-
trate each by one or more numerical examples; he should
also try to invent a proof of his own for each principle before
reading the printed proof.
Principle 1. If the same number is added to, or sub-
tracted from, each member of an inequality, the result is an
inequality of the same species.
287
288 HIGH SCHOOL ALGEBRA [Ch. XVIII
For, if a<h, i.e.^ if a — h is negative, then a-{- c —(b -\- c^^
which equals a — b^ is negative, and therefore
a -f c<b-^ c,
and similarly a— c<b — c.
So, too, if « > 5, then a-\- c>b -\- c^ and a— c>b — c.
Principle 2. If each member of an inequality is multi-
plied or divided by the same positive number^ the result is an
inequality of the same species.
For, if a>b^ and n is any positive number, then an — bn,
i.e.^ (a — b^n^ is positive (why?), and therefore an>bn.
Similarly if we divide by n ; and so, too, \i a<b.
Principle 3. If each member of an inequality is multi
plied or divided by the same negative number^ the result is an
inequality of opposite species.
Hint. If « > 6, and n is negative, then an — hn is negative (why ?) and
therefore an < hn.
Principle 4. If several inequalities of the same species
are added^ member to member^ the result is an inequality of the
same species.
Hint. Let a<b, c<d, e</, ..., then (a - b)-\-(c - d) + (e -/)+ •••,
i.e., (a -f c + e + •••) — (6 + d +/+ •••)) ^^ negative (why ?), and therefore
Principle 5. If an inequality is subtracted from an equa-
tion^ or from an inequality of opposite species.^ member from
member., the result is an inequality whose species is opposite to
that of the subtrahend.
Hint. Let a<Cb, and c = d or c^d, then in either case, b — a -\- c — d,
which equals c — a —(d — b),is positive (why ?), and therefore c — a^d — b.
Principle 6. If the first of three numbers is greater than
the second., and the second is greater than the third., then the
first is greater than the third ; and conversely.
Hint. Let a > 6 and 6 > c, then (a — &) + (& — c), i.e. , a — c, is positive
(why ?), and therefore a>c.
186-187] INEQUALITIES 289
Pkinciple 7. If two or more inequalities of the same
species^ whose members are positive^ are multiplied together^
member by member^ the result is an inequality of the same species.
Hint. Let a > 6 and c> d, then, by Prin. 2, ac > he and he > hd^ whence,
by Prin. 6, aC^^hd -, and similarly for three or more such inequalities.
187. Conditional and unconditional inequalities. An iden-
tical or unconditional inequality is one which is true for all
values of its letters. Thus, a + 4 > a and (x — yY + 1 >
are unconditional inequalities.
A conditional inequality is one which is true only on con-
dition that certain restricted values are assigned to its letters.
Thus, a;-f4<3a;— 2 only on condition that x>'^.
A conditional inequality is solved by means of the princi-
ples of § 186, and in much the same way that an equation is
solved by means of the ordinary axioms.
Ex. 1. If 3 .'» — -2^ > -^ — X, find the possible values of x.
Solution. On multiplying each member of this inequality by
3, it becomes 9»=-25>ll -3x, [§186,2
whence 9 a; + 3 a; > 11 + 25, [§ 186, 1
I.e., 12 a; > 36,
whence ic> 3 ; [§ 186, 2
i.e., if the given inequality is true, x must be greater than 3.
By means of the principles established in § 186 the student may show that
each step in the above reasoning is reversible, and hence that the converse
is also true ; viz. , that if a; > 3, then 3 x — ^^ > ^^^ — ic.
rx- ^i_ . 1 .• (2a; + 32/>5, (1)
Ex. 2. Given the two relations \ . / /o\
I a; + 42/ = 6; (2)
to find those values of x and y that will satisfy them both.
Solution. On multiplying each member of (1) by 4, and each
member of (2) by 3, we obtain
8a;-}-122/>20,
and 3a;-|-122/ = 18;
whence, subtracting, 5 a? > 2, [§ 186, 1
and therefore a^ > f • [§ 186, 2
290 HIGH SCHOOL ALGEBRA [Ch. XVIII
Now substitute for x in (2) above, any number greater than |,
and find the corresponding value of y (this value of y will
always be less than J ; why ?) ; these values of x and y, taken
together, will satisfy both (1) and (2).
EXERCISE CXXV
3. Using the definitions of "greater" and "less" in § 185,
show that 5 > 2 ; that - 23 < - 12 ; and that 2 > - 9.
4. It x-^y > z — iv, show that x-\-w ^ z —y.
Hint. Apply Principle 1 twice.
5. May terms be transposed from one member of an inequality
to the other ? If so, how and why (cf . Ex. 4) ?
^ j^ m-n^ m-\-2n ^ ^^^^ ^^^^ 3(m -ii) < 2(m + 2 n).
How may an inequality be cleared of fractions ? Why ?
7. Show, from Principles 2 and 3, how to remove a common
factor from both members of an inequality.
8. What happens if the signs in each member of an inequality
are reversed ? AVhy ?
Hint. In the proof of Principle 3, put — 1 for n.
9. If a > 6 and c — d, show that G — a<d — h.
10. Illustrate by numerical examples that
(1) if a > & and c < d, then the sum of these inequalities may
be either a + c = 64-d, ora + c>&4-(?, or a + c<54-d;
(2) if a > 6 and c > d, then the difference of these inequalities
may be either a— c = h^d, a — c^h — d, ova — c<h—d.
11. Translate (1) and (2) of Ex. 10 into verbal language.
12. If a < 6 and c < d, and if d alone is positive, show that
ac > hd. Is this inconsistent with Principle 7 ?
13. If a, b, c, and d, are positive numbers, and if a > 6 while
c<d, which is the greater, ac or bd ? Why ? Illustrate your
answer numerically.
14. What operations Avith or upon inequalities lead to results,
one of which is certainly greater or less than the other ?
187] INEQUALITIES 291
15. Name and illustrate some operations with inequalities
which lead to results whose relations are uncertain.
16. Show that a^ + b->2 ah except when a — h.
Hint, (a — 6)2 is positive whether a> 6 or a < 6.
17. Distinguish between a conditional and an unconditional
inequality. To which of these classes does a^ + h^ -\-l>2 ah
belong ? Why ?
18. To which class of inequalities does 6 £c — 5 > 3 a? belong ?
AVhy ? Solve this inequality.
19. If 3 a; < 5 X — 9, show that x is greater than 4^ (cf. Ex. 1).
20. If x^ -\-2^ <C 11 Xj show that the range of values of x is be-
tween 3 and 8. i.e., that x must be greater than 3 and less than 8.
Hint. In order that (x - 3) (8 — x), i.e., \\x- x'^ - 24, may be positive,
both factors must be positive or both negative.
Find the range of values of x in each of Exs. 21-26 :
21. x" < 9.
22. a.-2-h24>lla;. 25.
Q^ 10 — a;>5.
23. 30>x-f4^>25. )
2 r3-4a^<7,
24. 28>3a; + x2. 26. |^_^2<4.
27. Show that no positive number plus its reciprocal is less
than 2 5 i.e., n being any positive number, that n 4- - < 2.*
n
28. Show that 4 ic- + 9 < 12 ic.
29. Show that 2 6 (6 a - 5 6) > (2 a + &)(2 a-h).
If a, &, and c are positive and unequal, show that
30. a' 4- ?>' > a%\ 32. a^ + 6^ > a^ft + ah\
g- a4-26 a4-3 ?> 33. a" -{■ h" -\- c^ > ah -\- he + ca.
' a-\-Sb a + 46 34. a^-{-]/-^cy^>3ahc(Gt'Ex.30,]).51).
35. If a=^ -f &2 = 1^ and (r' + cJ- == 1, prove that ah-\-Gd>l.
* Cf. Ex. 16. The symbol < stands for " is not less than," and > stands
for " is not greater than,"
4x-ll>|
292 HIGH SCHOOL ALGEBRA [Ch. XVIII
36. If both m and *i are positive, which is the greater,
m-i-n ^^ 2mn ^
2 m 4- ?i
Solve the following systems :
37 (2x-Sy<2, (y-x>9,
\2x-\-5y=6. 39. |7^_^j_^^
r3 a; + 2 2/ = 42, I 20 15
I 7 1 4 a; < 3 2/.
41. Find the smallest integer fulfilling the condition that ^ of
it decreased by 7 is greater than J of it increased by 6.
42. The sum of three times A's money and 4 times B's is $ 1
more than 6 times A's ; and if A gives $ 5 to B, then B will have
more than 6 times as much as A will have left. Find the range
of values of A's money and B's.
CHAPTER XIX
RATIO, PROPORTION, AND VARIATION
I. RATIO
188. Definitions. The ratio (direct ratio) of two numbers
is the 'quotient obtained by dividing the first of these num-
bers by the second. The numbers themselves are usually
called the terms of the ratio, the first being the antecedent,
and the second the consequent.
E.g.^ the ratio of 15 to 5 is 15 -^ 5, i.e., 3 ; the antecedent is 15, and the
consequent is 5.
The ratio of a to 5 may be vsrritten as « : 6, a -^ 5, or - ;
it is read "the ratio of a to 6," or "a divided by 6."
The inverse ratio of 6k to 5 is 6 -h a, i.e.^ it is the reciprocal
of the direct ratio of these numbers.
Two numbers are said to be commensurable or incommen-
surable with each other according as their ratio is rational or
irrational (cf. § 146), i.e.^ according as they have or have not
a common measure.
E.g.^ 1.5 and f are commensurable with each other; so also are 3\/2 and
5 v'2 ; but 3V^ and 5 are incommensurable.
189. Ratio of like quantities. The concrete quantities
with which algebra is concerned are expressed by means of
numbers, and the ratio of two like * quantities is therefore the
ratio of the numbers which represent these quantities.
E.g., the ratio $6 : $9 is the same as 6 : 9, i.e.., as 2 : 3.
* Unlike quantities can, of course, have no ratio to each other.
293
.294 TIIGU SCHOOL ALGEBRA [Ch. XIX
190. Properties of ratios. Since ratios are quotients, i.e.^
fractions, therefore they have all the properties of fractions.
Thus, a ratio is not changed if both antecedent and
consequent are multiplied or divided by any given number.
Again, if a, ^, and k are positive, and a<h, then since
a a + k . (a—b)k re oo
is negative (a — h being negative), therefore (§ 185), the ratio
a-\-k '. h -{-k is greater than the ratio a : h. [Translate this
important fact into words : (1) calling a ^ b ix proper fraction ;
and (2) calling a-i- b a ratio less than unity.]
EXERCISE CXXVI
What is the ratio of :
1. 6 to 8? 3. -10 to 24? 5. -I to 11?
2. 50 to 15 ? 4. 6.3 to .7 ? 6. 21 a;^ to 9 a;?
7. Which term of a ratio corresponds to the divisor ? What
is the other term called ? Illustrate from Exs. 1-6.
8. Form the inverse of each of the ratios in Exs. 1-6.
9. The ratio of x to 5 equals 2 ; find x, and check your work.
10. If the ratio of two numbers is f and the consequent is
6, find the antecedent.
In each of Exs. 11-14, find (and check) the value of x :
11. x^ :2 = ^. 13. 64 : a; = a;.
12. a; + 5: 2a; = -7. 14. 25:a;2 = 9.
15. What is the ratio oi x to y when 7(x~y)=S(x-\-y)?
when af -[-6y^ = 5xy?
16. A yard measure is divided into two parts whose lengths
are in the ratio 7 : 11 ; how many inches in each part ?
17. A and B divide $ 100 between them so that A receives $13
out of every $ 20. What is the ratio of A's share to B's share ?
How many dollars does each receive ?
190-191] llATIO, PROPORTION, AND VARIATION 295
18. What number must be added to each term of ^^ in order
that the resulting ratio shall be 2:3? Does this addition in-
crease or diminish the given ratio ?
19. Which is the greater ratio, 5 + 2 : 17 + 2 or 5 : 17 ?
21 + 8 : 11 -f 8 or 21 : 11 ? Does the addition of the same positive
number to both terms of a ratio always increase the latter's value
(cf. § 190)? Explain.
20. If a, b, and k represent positive numbers, translate (1) and
(2) below into words ; then give three numerical illustrations of
each:
(1) ^^^ < 7 when a<b',
— K
(2) ^'^>r when a>b.
b —k b
21. By a method similar to that used in § 190, show the cor-
rectness of (1) and (2), Ex. 20.
22. If X, y, and z are positive numbers, which is the greater
,. , -, i r>x 2ic4-5?/ ic-|-2?/o x — y-\-z x-\-y-\-Zo
ratio (and why ?), — — ^ or —^ — ^ ? ^L-L_ or ^ ^ ^ ?
2x-[-ly x-\-3y x-\-y — z x — y — z
23. Show that the following ratios are all equal: $12 : $9;
8 bu. oats : 6 bu. oats ; 4 T. of coal : 3 T. of coal ; 10 in. : 7^ in. ;
4:3; i:^^.
24. Eind the value. of each of the following ratios: 4V2 : V2;
4V2:2; 7V3 in. : 14V2 in.; $5.80 : 29 cents.
25. Eind two integers whose ratio equals 15| : 9f . Can the
ratio of any two numbers whatever be expressed as the ratio of
two integers (cf . Ex. 24, also § 188) ?
26. Which of the pairs of numbers (or quantities) in Ex. 24
are commensurable ? Which are incommensurable ? Why ?
II. PROPORTION
191. Definitions. If a, 5, c, and d are any four numbers
such that a : b — c : d, then these numbers are said to be
proportional, or to form a proportion ; i.e., a proportion is
a statement that two ratios are equal.
296 HIGH SCHOOL ALGEBRA [Ch. XIX
The proportion a :b = e : d (sometimes written a : b :: c : d^
is read : " the ratio oi a to b equals the ratio of c to c?," and
also "a is to 5 as <? is to c?." In this proportion a and d are
called the extremes, while b and e are called tlie means.
Ji a : b = e : d, then d is said to be the fourth proportional
to a, 5, and c ; while ii a : b = b : c, then c is called the third
proportional to a and 5, and 6 is called the mean proportional
between a and e.
A succession of equal ratios, in which the consequent of
each is also the antecedent of the next, is called a continued
proportion ; thus a : b = b : c = e : d = d : e = • • -^ is a continued
proportion.
EXERCISE CXXVII
1. Is it true that 8 : 12 = 10 : 15 ? Why ? How is this pro-
portion read ? What does it mean ?
2. In Ex. 1 name the means and the extremes of the propor-
tion, also the fourth proportional to 8, 12, and 10.
3. Is it true that 8 : 10 = 12 : 15 ? How does this proportion
compare with that in Ex. 1 ? Does a proportion remain true
after its means have been interchanged ? Try several numerical
examples and compare Principle 4, p. 298.
4. By arranging the numbers 3, 4, 6, and 8 in different ways,
make three different proportions.
5. Is 6 a mean proportional between 4 and 9? between 18
and 2 ? Is the same thing true of — 6 ? Name the third
proportional in each case.
6. Show that 2 : 6 = 6 : 18 = 18 : 54 = 54 : 162 is a continued
proportion in which each ratio equals ^. Write a continued
proportion of five ratios each of which equals f.
192. Important principles of proportion. Since a propor-
tion is merely an equation whose members are fractions,
therefore the principles of proportion may be derived from
those governing equations and fractions.
101-192] RATIO, PROPORTION, AND VARIATION 297
Note. Before memorizing the following principles (1-8) the pupil should
illustrate each by one or more numerical examples ; he should also try to
invent a proof of his own for each principle, before reading the printed proof.
Principle 1. If four numbers are in proportion, then the
product of the extremes equals the product of the means.
For, let a, h, c, and d be any four numbers which are in
proportion, then a :h = c : d\
a c
whence ad = he, [clearing of fractions
which was to be proved.
Principle 2. If the product of two numbers equals the
product of two others, then either pair may be made the
extremes of a proportion in which the other pair are the means.
For, if ad = be,
a o
then 7 ~ ;j' [dividing by hd
ct
i.e., a '.b = c : d.
In the same way it may be shown that, if ad = he, then
h : a = d : c, c : a = d : h, etc.
Eemark. From the proof just given it follows that the correct-
ness of a proportion is established when it is shown that the product
of the 7neans equals the product of the extremes ; this test is very
useful.
Principle 3. i/" four numbers are in proportion, then
they are in proportion by inversion ; i.e., the second is to the
first as the fourth is to the third.
For, ii a : h = c : d, then ad = he (why ?) ; hence h : a = d : c
(cf. Principle 2, Remark).
Suggestion. Let the pupil state Principles 4-7 below entirely in verbal
language, and prove each in detail (cf. statement and proof of Principle 2).
298 IIJGH SCHOOL ALGEBRA [Ch. XIX
Principle 4. If four numhers are in proportion^ then they
are in proportion hy alternation; ^.e., if a:h — c:d^ then
a : c = b : d.
Principle 5. If four rmmbers are in proportion^ then they
are in proportion by composition ; i.e.^ ii a : b = c : d, then
(a + ^) : a= (^c -{- d') : e ; [also, (a + 6) : 5 = (c + 6?) : c?].
Principle 6. If four numbers are in proportion^ then they
are in proportion by division or separation ; i.e.,if a : b = c : d,
then (a — b^ : a= (^c — d} : c; [also, (^a — b^ : b= (^c — d^ : dj^.
Principle 7. If four numbers are in proportion, then
they are in proportion by composition and division ; i.e.^ if
a : b = e : d, then (^a + 5) : (a — 6) = ((? + t?) : (c — c?).
Principle 8. In a series of equal ratios the sum of the
antecedents is to the sum of the consequents as any antecedent
is to its own consequent.
Thus, if a'.b — c:d=e:f=g:h = --' = x'.y.,
then <ia + c + e+g-\--" + x): (^b -j- d-hf+ h+ ■-■ -^ y} = e :f.
To prove this theorem, let each of the given equal ratios
be represented by a single letter, say r ;
, •, a c e a -, x
then -=r, -=r, - = r, ^ =r, •••, and - =r,
b d f h y
hence a = h\ c = dr^ e =fr^ g=hr., •••, and x = yr^
and, adding these equations, member to member,
a^c^e-\-g-\-"-^x = Q)^d^f-\-h-\--'^y)r,
and therefore ^ + ^ + ^ +.^+ •>• +^ ^^^e
b+d+f-\-h^-- + y f
which proves the principle.
Note. As in the proof just given, so it will often be found advantageous
to represent a ratio by a single letter.
192] RATIO, PROPORTION, AND VARIATION 299
EXERCISE CXXVIII
Using Prin(3iple 1, find .*; in each of Kxs. 1-4 :
1. 14 : 3 = 56 :x. 3. - 16 : .^• = 18 : 7.
2. aj : - 5 = 20 : - 2. 4. J : ic = x : g^^ .
Find a mean proportional between each of the following pairs
of numbers (cf. Ex. 4) :
5. 3, 27. 7. 25, — 4. - 9. ain'^, a^m.
6. -2,-5. 8. .25, .09. 10. a + h,a-b.
11. How many answers has each of Exs. 5-10 ? Why ?
Show that the mean proportional between any two numbers
equals the square root of their product.
12. Find the third proportional to 1 and 4 ; to — 25 and — 40 ;
also the fourth proportional to m — n, m^ — rr, and m + n.
13. Using Principles 2-7, make seven different proportions
from the equation cd = mn.
14. Add 1 to each member of the equation a:b = c: d; write
the result as a proportion and thus prove Principle 5.
15. Prove that like powers (also like roots) of proportional
numbers are proportional, i.e., prove that if a:b = G: d, then
a" : ft** = c" : d\
16. If a: b = c: d and e:f = g:h, show that ae :bf= eg: dh ;
also translate this principle into verbal language.
17. li a: b = c:d and e : /= a : h, show that - : - = - : -•
e f g h
Hint. Use a single letter to represent a ratio (cf. proof of Principle 8).
li p : q = r : s, Siud m and n are any numbers whatever, show
that the proportions in Exs. 18-25 are true.
18. mp : 7iq=mr : ??s(cf. § 190). 21. s^: q^ = r'^ : p^
19. op:r = 5q:s. 22. p -\-q :2 p = r + s : 2 r.
20 r:s=-: ' 23. pr : qs= r^ : s\
q p
24. j9'^-4r-:^"-4s'»=-|": -^•
HIGH ^ClI. ALG. — 20
300 HIGH SCHOOL ALGEBRA [Ch. XIX
25. 2) + q: r+ s = Vp~ -f g^ ■ V?^T^.
26. li a:b = c: d = e :f=g : h= •", and I, m, 7i, p, -" are any
numbers whatever, show that
(ma -\- Ic — ne -i- pg -^ •••) -.{mb -\- Id — nf -\- ph -{ ) = a:h.
Hint. Compare § 190, also Principle 8.
27. If {p-\-q^-r+s){p—q—r-{-s) = {p-q^r-s){p + q—r—S),
show that p : q = r : s.
28. If ax + cy ^ay + cz^ az + cx ^ ^^^^ ^^^^. ^^^,^ ^j jj^^^^
?>.?/ -|- dz bz +dx bx+ dy
ratios equals ^LjL^ (cf. Principle 8).
b -\-d
By the principles of proportion solve the equations :
29. a;:15 = aj-l:12. 33. a: : 27 = 2/ : 9 = 2 : a; -?/.
30. a;-:32 = a; + 2:12. a;+Va;-l 13
34. ^=iiiz = —
31. {\cx+V)rdx'=\c\ x--Jx-l 7
'x-y=2,
Hint. Apply Principle 7.
32. \ Q^-^f ^5 32 Va; + 7 + V^ ^ 4 + Va? ^
(a? + 2/f 9 ■ Va; + 7 - V^ 4 — Vx
36. AVhat number must be added to each of the numbers 7, 9,
11, and 21 in order that the four sums may be proportional ?
37. In the triangle ABC^ ^/^ divides BC into two parts, 5^ and
KC^ respectively proportional to AB and AC. If ^J5 = 10 in.,
AC = 16 in., and 5(7= 20 in., find BK and KC (Draw a figure
to illustrate.)
38. Find two different numbers, m and w, such that
m-\-n\m — n\ m^ + n^ = 5:3: 51.*
39. The perimeter of a triangle whose sides are in the ratio
5 : 6 : 8, is 57 meters ; find the lengths of the sides.
40. How may $10 be divided among three boys so that for
every dollar the first receives, the second shall receive 15 cents
and the third 10 cents ?
* The expression a:h:c = x:y'.z, means that a : h ■= x : y, a : c = x : z^
and b :c = y : z; and also the equivalent statement a : x = b : y = c : z.
192-194] RATIO, PROPORTION, AND VARIATION 301
41. Two rectangles are equal in area. If their widths are as
2 : 3, find the ratio of their lengths.
42. The sides of a certain rectangle are in the ratio 7 : 3.
Compare the area of the rectangle with that of a square which
has the same perimeter.
43. If a:b, c : d, e\f, g : h, ••• are unequal ratios, in which
a, b, c, '" are positive numbers, and if a: b is the greatest and
e : / the least among these ratios, show that
is less than a : b, but greater than e : / (cf. proof of Principle 8).
III. VARIATION
193. Variables, constants, and limits. Many questions in
mathematics are concerned with numbers whose values are
changing ; such numbers are usually called variables, while
numbers whose values do not change are called constants.
If the difference between a variable (in the course of its
changes) and a constant may become and remain less than
any assigned number however small, then this constant is
said to be the limit of the variable.
Thus, your own age, the height of the mercury column in a thermometer
tube, the length of the shadow cast by a given flagstaff, etc., are variables ;
while the difference between the ages of two given men, the weight of the
mercury in a given thermometer, the length of a certain flagstaff, etc., are
constants.
Again, the decimal .3333 ••• (i.e., .3 + .03 + .003 H — ) is a variable whose
limit is I ; this decimal grows larger and larger as more and more places are
included, and may thus be made to differ from i by less than any assigned
number however small.
So, too, 1 + J + I + I + Jg ••• is a variable whose limit is 2 (cf. § 202).
n n
194. Interpretation of the forms -, -, and -• Two of these
oo
forms were first hiet with in § 41, and were there inter-
preted by assuming that the definition of division, given in
§ 8, remains valid for infinitely large numbers and for zero.
302 HIGH SCHOOL ALGEBRA [Ch. XIX
It is better, however, to interpret these forms from the
standpoint of variables and limits.
(i) If the values 1, -jIq-, y^^, -f^foo^, •••, are successively
assigned to x^ what are the corresponding values of -?
of -, where a is any finite constant whatever ? Answer the
same questions when x takes the successive values 1, 10,
102, 103, ....
These examples illustrate two important facts, viz. :
(1) ^8 the divisor grows smaller and smaller^ approaching
zero as a limit (the dividend beirig a finite constarit^^ the quotient
increases without limit.
(2) As the divisor increases without limit (the dividend being
a finite constant^ ^ the quotient approaches zero as a limit.
For the sake of brevity, (1) and (2) are often expressed
by the equations
- = 00 and — = 0,
CX)
respectively ; but the interpretation of these equations is as
stated in (1) and (2) above.
(ii) In the fraction -— , as x takes the successive
x—1
values 1.1, 1.01, 1.001, 1.0001, what limit is approached by
the numerator? by the denominator? by the value of the
fraction? Answer the same questions for — -, as x
approaches 2 as a limit.
These examples illustrate the fact that as a dividend and
its divisor each approach zero as a limit, the quotient may
approach any value whatever ; this is often expressed by
saying that
- is indeterminate.
194] RATIO, PROPORTION, AND VARIATION 303
EXERCISE CXXIX
1. Which of the following quantities are constants and which
are variables : (1) the circumference of a growing orange ? (2) the
length of the shadow cast by a certain church steeple between
sunrise and sunset? (3) the length of the steeple itself? (4) the
time since the discovery of America ? (5) the interest earned by
an outstanding note ? (G) the principal of the note ?
2. A point P moves through half the distance AB {i.e., to P'),
then through half the remaining distance p pn pu
(i.e., to P"), then through half the remain- i i i
ing distance (i.e., to P"), and so on. Show ^
that the distance from A to P is a variable whose limit is AB.
3. In Ex. 2, is the distance from P to J5 a constant or a
variable ? What is its limit ? Explain both answers.
4. If X takes in succession the values .6, .06, .006, .0006, •••,
what is its limit ? Why ?
5. What is the limit of the variable sum .6 + .06 -f- .006 -\
(i.e., of the decimal .6666 ••-) ? Explain.
6. If r is any finite constant, trace the changes in the quotient
r/s as s passes through the values, 3, 1, ^, ^, -^j, •••; also as s
passes through the values 3, 9, 27, 81, •••. Is there a limit to
the quotient in the first case ? in the second ? Explain.
7. Translate into verbal language [cf . § 194 (i)] :
(1) ^ = 00; (2) ^=0.
8. As X approaches the limit 1, what limit does approach
X —1
in form f in value f Answer the same questions for the fractions
a; — 1 x^ — x 1 3 a^ — 2a; — 1
a^-1' a;-l'
ar
9. By means of your answers to the questions in Ex. 8, illus-
trate the fact that - is indeterminate in value.
304 IlIGIl SCHOOL ALGEBRA [Cii. XIX
195. Direct and inverse variation; etc. Of two variables
which are so rehited that, during all their changes, their ratio
remains constant, each is said to vary (also to vary directly)
as the other. The symbol employed to denote variation is ^ ;
it stands for the words " varies as," and the expression acch
is read '•^ a varies as 5."
If a Qc ^, ^.g., if a : 6 = A;, a constant, then a = kb (why ?) ;
hence a variation statement may be converted into an
equation.
E.g., if a tank contains v cii. ft. of water, each cubic foot weighing 02.5 lb.,
and if the total weight of the water is to lb., then :
(1) When V changes (as it must, for example, while the tank is filling),
w changes also.
(2) Since, no matter how the quantity of water changes, to = 62.5 -w, or
w : V = 62.d, therefore wcav; i.e., the weight of watisr varies as its volume.
One of two numbers is said to vary inversely as the other
if the ratio of the first to the reciprocal of the second is
constant. If a varies inversely as 5, then a ' h = k: let pupils
fully explain why.
Again, if x, y, and z are variables such that x = ki/z^ where
k is Si constant, then x is said to vary jointly as ^ and z ; and
if a; = -^, then x is said to vary directly as y and inversely as z.
z
E.g., the time required for a railway train to travel a given distance varies
inversely as the speed ; for if t, r, and d represent, respectively, the time, rate,
and distance, then t 'r = d, i.e., t: ~ =d, where d is constant.
r
Again, the cost of a railway journey varies jointly as its length and its
cost per mile ; while the number of posts required to build a certain fence
varies directly as the length of the fence, and inversely as the distance be-
tween the posts.
Note. It should be remarked in passing that such an expression as wccv
above (i.e., the weight of water varies as its volume) is merely an abbreviated
form of the proportion
w:w' =v:v',
wherein w and w' stand for the respective weights, and v and v' for the
volumes, of any two quantities of water.
195] RATIO, PROPORTION, AND VARIATION 305
The theory of variation is, therefore, substantially included in that of
ratio and proportion, and the only reason for even defining the expressions
"varies as," "varies inversely as," etc., here, is that this convenient
phraseology is so well established in physics, chemistry, etc.
EXERCISE CXXX
1. Explain and illustrate the following statements :
(1) The interest earned by a certain principal varies as the time.
(2) The circumference of a circle varies as its radius.
2. State (1) and (2) of Ex. 1 as equations (cf. § 195), also as
proportions (cf. § 195, Note).
3. It xccy and if a? = 12 when y = 3, find the equation connect-
ing X and y ; also find x when y==7.
Solution. Since xccy, therefore x = ky where k is constant (why?) ;
moreover, when x = 12 and ?/ = 3, the equation x = ky gives k = 4. There-
fore, under the given conditions, x = 4y; hence, when y = 7, x = 28.
4. li aocb and if a = 89 when b = —3, find a when 6 = 2; also
when & = I ; also find b when a = — 65.
5. If ^ X -B and BccC, show that AccC.
Hint. Show that A = kC\ where k is some constant.
6. If m cc n and pccriy prove that m±pccn.
7. If 3 m^— 18 oc 2n + 1, and if m = 4 when n = 2, find m when
n = 23.5.
8. The area of a circle varies as the square of its radius. If
a circle whose radius is 10 ft. contains 314.16 sq. ft., find the
area of a circle whose radius is 5 ft. ; of one whose radius is 12 ft.
9. Find the radius of a circle whose area is twice that of a
circle 10 ft. in radius (cf. Ex. 8).
10. If X varies inversely as y, how is the value of x aifected if
y is doubled ? if ?/ is multiplied by 10 ? ii y is divided by — 6 ?
Explain.
11. Give three numerical examples of inverse variation.
306 HIGH SCHOOL ALGEBRA [Cii. XIX
12. If X varies inversely as y, show that :
(1) xy = k (where k is constant).
(2) x' :x" = y" : y' (where x' and y', x" and y" are correspond-
ing vahies of the variables).
13. If x varies inversely as y, and if x = 4 when y = 2, find y
when x = — 8; when x = l^ ; when x = 2.5.
14. If x varies directly as y and inversely as z, and if a; = — 12
when 2/ = 2 and z = 7, find y when x = 2 and 2 = 3.
15. Solve Ex. 13 by drawing the graph of the equation con-
necting X and y (cf. § 141), and then measuring the ^/-coordinates
of the points whose respective .^-coordinates are — 8, 1^, and 2.5.
Also show, from the graph, that any change in x makes an oppo-
site change in y.
16. If the volume of a pyramid varies jointly as its base and
altitude, and if the volume is 20 cu. in. when the base is 12 sq. in.
and the altitude is 5 in., what is the altitude of the pyramid
whose base is 48 sq. in. and whose volume is 76 cu. in. ?
17. The distance (in feet) through which a body falls from a
position of rest, varies as the square of the time (in seconds)
during which it falls. If a body falls 257^ ft. in 4 sec, how far
will it fall in 5 sec. ? how far during the 5th second ? how far
during the 7th second ?
18. If the intensity of light varies inversely as the square of
the distance from the source of light, how much farther from a
lamp must a book, which is now 2 ft. away, be removed so as to
receive just one third as much light ?
19. The weight of a body comparatively near the earth's sur-
face varies inversely as the square of its distance from the earth's
center. Assuming that the radius of the earth is 4000 mi., find
the weight of a 4-lb. brick 2000 mi. from the earth's surface.
(Two solutions.)
20. The number of oscillations made by a pendulum in a given
time varies inversely as the square root of its length. If a pen-
dulum 39.1 inches long oscillates once a second, what is the
length of a pendulum that oscillates twice a second?
CHAPTER XX
SERIES — THE PROGRESSIONS
196. Definitions. A series is a succession of numbers
which proceed according to some definite law. The num-
bers which constitute the series are called its terms.
E.g.^ in the series 1, 2, 3, 5, 8, 13, eacli term after the second is the sum
of the two preceding terms ; in the series 2, 6, 18, 54, 162, each term after
the first is 3 times the preceding term ; and in the series 1, 4, 9, 16, ••., 81,
each term is the square of the number of its place in the series.
A series which consists of an unlimited number of terms
is called an infinite series ; otherwise it is a finite series.
The present chapter considers only the simplest kinds of
series — the so-called "progressions."
I. ARITHMETICAL PROGRESSION
197. Definitions and notation. An arithmetical series, or
arithmetical progression (designated by A. P.), is a series
in which the difference found by subtracting a term from
the next following term is the same throughout the series.
This constant difference, whether positive or negative, inte-
gral or fractional, is known as the common difference of the
series.
E.g.^ the series 2, 5, 8, 11, 14, ••. is an A. P. whose common difference is
3. So, too, the series 18, 11, 4, — 3, — 10, is an A. P. whose common differ-
ence is — 7.
The elements of any given A. P. are the first term (desig-
nated by a), the last term (Z), the common difference (c?),
the number of terms (n), and the sum of all the terms (s).
Thus, in the series 2, 5, 8, •••, 32, the elements are « = 2, Z = 32, (?= 3,
11 = 11, s= 187.
307
308 HIGH SCHOOL ALGEBRA [Ch. XX
EXERCISE CXXXI
1. Does a row of numbers written down at random constitute
a series ? Explain.
2. Show that 1, 7, 13, 19, 25 is an A. P. What are its ele-
ments ?
3. What is d in the A. P. 7, 11, 15, 19 ? Extend this series
four terms to the right ; also three terms to the left.
4. If the 1st, 3d, and 5th terms of an A. P. are 18, 24, and 30,
respectively, find d and write 8 consecutive terms of the series.
5. Write 10 consecutive terms of the series in which 19, 9,
and 4 are the 1st, 5th, and 7th terms, respectively.
6. What are the elements of the A. P. 5, 5 + 3, 5 + 6, 5 + 9,
5 + 12 ? How is any term of this series formed from the pre-
ceding term ?
7. Show that x,x+y, x-\-2y, x-{-3y, ••• is an A. P. What
is d in this series ? How many times must d be added to the
first term to make the 2d term ? to make the 3d term ? the 7th
term ? the 10th term ? the nth term ?
8. Show from the definition of an A. P. that such a series
may be written in the form
a, a+d, a-j-2d, a + 3d, '•-, I — 2d, l—d, I,
wherein a, d, and I represent, respectively, the first term, com-
mon difference, and last term.
198. Formulas. The elements of an A. P. are connected
by the two fundamental equations (formulas) numbered (1)
and (2) below.
Since each term of an A. P. may be derived by adding d
to the preceding term (cf. Exs. 6-8, above), therefore, if
I stands for the nth term
l = a-}-(n-l)d. (1)
107-198] SERIES — THE PPiOGnESSIONS 309
Again, since the sum of the terms of an A. P. may be
written in each of the two following forms,
s = a-^ (a-\-d)-^(a-\-2d)-h--' + (I- 2 d) -h (I- d) -^ I
and s = I i- {I - d) -{- {I - 2 d) -\- •" + (a -^ 2 d) -{- {a -\- d) -^ a,
therefore, by adding these equations, term by term,
i.e.^ 2s = n(^a-\- Z), [n terms
whence * = -^-^ — ^ ; (2)
or, substituting the value of I from (1),
n [Za + (n-l)<n ,
"-^ 2
Note. If any three of the five elements of an A. P. are given, the other
two can always be found from (1) and (2) above, because, in that case, the
two unknown elements will be connected by two independent equations.
Ex. 1. Find the sum of 8 terms of the A. P. — 3, — 1, 1, 3 •••.
Solution. Here a=— 3, d = 2, w = 8;
whence, from (1), Z = -3 + (8 - 1) 2 = 11,
and from (2), s = ^(-^ + ^^) = 32 ;
Li
i.e., the sum of 8 terms of the A. P. is 32.
EXERCISE CXXXII
Verify formulas (1) and (2) above for the following series :
2. 10, 13, 16, 19, 22, 25, 28.
3. 26, 19, 12, 5, -2,-9, -16, -23, -30.
4. _8,-5|, -3i, -1,H,3|,6.
5. By means of § 198 (1), find the 17th term of 7, 11, 15, ... ;
then by § 198 (2), and without writing all the terms, find the sum
of the first 17 terms of this series.
6. As in Ex. 5, find the 12th term of 1, 3.5, 6, 8.5, •..; also
the sum of the first 8 terms.
310 • HIGH SCHOOL ALGEBRA [Ch. XX
Find the sum of :
7. Ten terms of 4, 11, 18, ....
8. Thirty terms of - 2, - 0.5, 1, 2.5, ....
9. Nineteen terms of 2, 5, 8, •.-.
10. k terms of 2, 5, 8, •••.
11. n terms of 5, 5 -f A', 5 + 2 A:, 5 -f 3 A;, • ...
12. t terms of li, 2 h, 3h, •••.
13. Find the sum of the even numbers from 2 to 100 inclusive.
Compare your result with that in Ex. 12 when h = 2 and t = 50.
14. How many strokes are made in a day (24 hours) by a clock
which strikes the hours only ?
15. Suppose that 50 eggs are placed in a row, each 2 yd. from
the next^ and a basket 2 yd. beyond the last egg ; how far would
a boy, starting at the basket, walk in picking up these eggs and
carrying them, one at a time, to the basket ?
16. If a body falls 16.1 feet during the first second, 3 times as
far during the next second, 5 times as far during the third second,
etc., how far will it fall daring the 8th second? how far during
the first 8 seconds ?
17. By means of § 198 (1) find n when a = 2, cZ = 4, 1 = 66',
also find s for this series.
18. If a = -10,d = 3, s = 35, findZandn.
Hint. Substitute in § 198 (1) and (2) and solve the resulting equations
for I and n (cf. § 198, Note).
19. If a = l, d = -|, 71 = 9, findZand.9.
20. If 1 = -^, n = lS, s = -45i, find a and d.
21. How many consecutive odd integers (beginning with 1)
must be added to give the sum 225 ? 441 ? (Cf. Ex. 18.)
22. If s = 112 and n = 7, determine the unknown elements in
the series ..«, 10, 13, 16, .-., and write the series.
23. If s, n, and d are given, find a and I; i.e., find a and I in
terms of s, n, and d (cf. Ex. 22).
198-191)] SERIES—- THE PEOGliESSIONS 311
24. Find a and ?i in terms of d, I, and s. Make up and solve
eight other examples of this kind.
25. Show that an A. P. is fully determined when any three of
its elements are given.
26. If the 6tli and 11th terms of an A. P. are 17 and 32,
respectively, find the common difference, and also the sum of
the first 11 terms.
Hint. Since the 6tli term is 17, therefore 17 = a + 5 (Z. Similarly,
32 = a + 10 d
27. Show that if each term of an A. P. is multiplied (or
divided) by any given number, the resulting products (or
quotients) are themselves in arithmetical progression.
28. If each term of an A. P. is increased or diminished by any
given number, will the results be in arithmetical progression ?
Explain.
199. Arithmetical means. The two end terms of an A. P.
are called its extremes, while all the other terms are called
arithmetical means between these two.
E.g., in the A. P. 5, 9, 13, 17, 21, between the extremes 5 and 21 there
are 3 arithmetical means (viz., 9, 13, and 17).
In an A. P. of 3 terms the (one) arithmetical mean
between the extremes equals half their sum ; for if A is
the arithmetical mean between a and 5, then
A — a = h — A^ [definition of an A. P.
whence 4 =?-LA. %
Ex. 1. Find the arithmetical mean between 3 and 27.
Solution. The arithmetical mean between 3 and 27
= 3 + 27^ 15_
2
Ex. 2. Insert 5 arithmetical means between 3 and 27.
Solution. In this series a = 3, 1 = 27, and (since there are to
be 5 means) n = 5-}-2 = 7; whence, from § 198 (1), d = 4, and
the required series is 3, 7, 11, 15, 19, 23, 27.
312 HIGH SCHOOL ALGEBRA [Ch. XX
EXERCISE CXXXIII
3. Find the arithmetical mean between 14 and 9; between
-5 and 17; -3 and -4; fandf; - 2.75 and 11.4.
4. Insert 4 arithmetical means between 12 and 27.
5. Insert 15 arithmetical means between 19 and 131.
6. Insert 20 arithmetical means between 16 and — 40.
7. Insert 12 arithmetical means between — f and f .
8. Insert 7 arithmetical means between — .08 and — .0032.
9. If m arithmetical means are inserted between two given
numbers, such as a and b, show that the common difference of
the series thus formed is (6— a) -j- (m -f- 1).
10. What does the formula of Ex. 9 become when m = 1 ? Is
this consistent with the formula for A obtained in § 199 ?
11. Without actually finding the means asked for in Ex. 2,
find the sum of the series formed by inserting them.
12. Find three numbers in A. P. whose sum is 15, and the sum
of whose squares is 107.
Hint. Let x — y, x, and x -{- y represent the required numbers.
13. Find three numbers in A*. P. whose sum is 18 and whose
product is 202i.
14. The sum of the first seven terms of an A. P. is 105, and the
sum of the third and fifth terms is 10 times the first term. Find
the series.
15. The product of the extremes of an A. P. of three terms is 4
less than the square of the mean, and the sum of the series is 24.
Find the series.
16. The sum of four numbers in A. P. is 14, and the product
of the means is 12. What are the numbers ?
Hint. Let x — 3y,x—y,x + y,x-\-Sy represent the series.
17. The sum of an A. P. of five terms is 15, and the product of
the extremes is 3 less than the product of the second and fourth
terms. Find the series.
18. How many arithmetical means must be inserted between 4
and 25 so that the sum of the series may be 116 ?
199-200] SERIES— THE PROGRESSIONS 813
19. A number, expressed by three digits in A. P., equals 30.4
times the siim of its digits ; but if 9 is added to the number, the
units' and tens' digits will be interchanged. Find the number.
20. In the series 1, 3, 5, •• •, what is the n\h. term ? Prove that
the sum of the first n odd numbers, beginning with 1, is nl
II. GEOMETRIC PROGRESSION
200. Definitions and notation. A geometric series, or geo-
metrical progression (designated by G. P.), is a series in
which the quotient of each term (after the first) divided by
the next preceding term is the same throughout the series.
This constant quotient is called the common ratio, or simply
the ratio, of the series.
E.g.^ the numbers 2, 6, 18, 54, ••• form a geometric series whose ratio is 3 ;
while |, — 1, f , - I, V", ••• is a G. P. whose ratio is - |.
The elements of any given G. P. are the first term (a),
the last term (Z), the number of terms (n)^ the ratio (r),
and the sum of all terms (s).
E.g., in the G. P. 2, -6, 18, -54, 162, -486, 1458, a = 2, Z = 1458,
n = 7, r = - 3, and s = 1094.
EXERCISE CXXXIV
Which of the following are geometric series ? Explain.
1. 7, 21, 63, 189, 567.
2. 1, 4, 16, 64, 192, 576.
3. -6, 12, -24, 48, -96, 192, -384.
4. What are the elements of the G. P. in Ex. 3 ? Extend this
series two terms to the right, also five terms to the left.
5. If a is the first term of a G. P., and r the ratio, what is the
second term? the third? the fourth? the fifth? the fourteenth?
the twenty-third ? the nth ? Explain.
6. Are x, xy, xy\ xf, -■, and p'q-', 2A PV, Pg\ Q% P~V geo-
metric series ? If so, name a and r in each.
314 RIQH SCHOOL ALGP.nnA tCn. XX
7. What is r in the G. P. 2, |, |, ••• ? in the G. P. 21, 7, J, ••• ?
Are these two series merely parts of the same series ? Explain.
8. If the 1st, 3d, and 6th terms of a G. P. are 12, 3, and |,
respectively, find r and then write down the first 8 terms of the
series (cf. Ex. 26, p. 311).
201. Formulas. The elements of a G. P. are connected by
the two fundamental equations shown below (cf. § 198).
Since each term of a G. P. may be obtained by multiply-
ing the preceding term by r (cf. Exs. 5 and 6, p. 313),
therefore, if I represents the n{\\ term of such a series, then
l = ar-\ (1)
Again, if s represents the sum of a G. P. of n terms, then
sz=a-{- ar -\- ar^ -\- ai^ + • • • + «r"~^ + ar^~^ ;
whence, multiplying by r, we obtain
sr = ar -^ ar^ + a7^ -\- h ar^~^ + ar",
and, therefore, by subtracting the second of these equations
from the first, member from member,
s — sr = a— ar" ;
hence ^~~/ * (^)
EXERCISE CXXXV
1. By means of § 201 (1) write down the 6th term of the geo-
metric series 4, 12, 36, • • • .
2. As in Ex. 1, write the 7th term of the G. P. 3, 6, 12, .•• ;
then by § 201 (2) find the sum of the first 7 terms of the series.
Check results by writing down and adding these first 7 terms.
3. Find the 8th term of 24, 12, 6, .••; also the 11th term of
TTJinr? ~ Tu^y ttj"? *"•
Find the sum of the following series:
4. 1, 2, 4, . • • , to 10 terms. 7. 1, — 2 a;, 4 a^, • • • , to 7 terms.
5. 1, 1.5, 2.25, •••, to 6 terms. 8. -5, -2, -.8, •••, to A: terms.
6. 2, — I, I, ••• , to 7 terms. 9. x, x~^, x~^j ••• ,to n terms.
200-201] SERIES — THE PIWGEESSIONS 315
10. Find the G. P. whose 3d term is 18, and whose 8th term
is 4374.
Hint. By § 201 (1), 18 = ar^, and 4374 = ar'' ; hence, dividing the second
of these equations by the first, 243 = r^, i.e., r = 3.
11. Find the G. P. whose oth term is |- and whose 9th term is
i||. Also find the sum of the first 9 terms of this series.
12. Show that § 201 (2) may be written in each of the follow-
ing forms :
g (1 — r") a — rl rl — a ar" — a -, __a ar''
1 — r ' 1 — r' r— 1 ' r — 1 ' 1—r 1 — r
13. By actually dividing a(l — r") by 1 — r, verify the correct-
ness of § 201 (2).
14. If a = 4, I — 972, and n = C, find r, and write the series.
15. If 71 = 12, r = - 2, and 6- = - 1365, find a and I
16. Find the sum of a G. P. of 6 terms whose ratio is f and
whose last term is 32.
17. If r, n, and I are given, find a and s ; that is, find a and s
in terms of r, n, and I (cf. Ex. 16).
18. Find a and I in terms of n, r, and s; also r and s in terms
of a, n, and /.
19. Is a G. P. fully determined when any three of its elements
are given (cf. § 198, Note)?
20. Three numbers whose product is 216 form a G. P., and the
sum of their squares is 189. What are the numbers ?
Hint. Let -, a, and ar represent the required numbers,
r
21. Divide 38 into three parts which are in G. P., and such
that when 1, 2, and 1 are added to these parts, respectively, the
results shall be in A. P.
22. Find an A. P whose first term is 3, and such that its 2d,
4th, and 8th terms shall be in G. P.
23. If the population of the United States was 76,000,000 in
1900, and if it doubles itself every 25 years, what will it be in
the year 2000?
HIGH SCH. ALG. — 21
316 HIGH SCHOOL ALGEBRA [Cn. XX
24. Thinking $1 per bushel too high a price to pay for wheat,
a man bought 10 bu., paying 3 cents for the first bushel, 6 cents
for the second, 12 cents for the third, and so on. What did the
tenth bushel cost him, and what was the average price per
bushel ?
25. Show that the amount of $ ^ for n years at a given rate
{R) of compound interest is the nth term of a G. P. whose first
term is A and whose ratio is (1 + i?).
26. A gentleman loaned a friend $250 at the beginning of
each year for 4 years. If money is worth 5 % compound interest,
how much should be paid back to him at the end of the fourth
year to discharge the obligation ?
27. The president of a charity organization starts a " letter
chain " by writing three letters, each numbered 1, requesting each
recipient to remit 10 cents to the society, and. also to send out
3 other letters, each numbered 2, with a similar request, the chain
to close with the letters numbered 20. Should every recipient
comply, how much money would be realized ?
202. Infinite decreasing geometric series. A decreasing
G. P. is one in which r < 1, numerically, and in which,
therefore, the terms grow smaller and smaller as we pass
from left to right in the series. Thus, 6, 3, |, f, f, ••• is a
decreasing G. P.
If we let s„ represent the sum of the first n terms of this
series, then
6 _ 6(i)^ |-g^ j2, p. 315
l-\ 1-1
= 12-6(1)-^
and, since 6(J)"~^ approaches zero as a limit when n increases
without limit [§ 194 (i)], therefore s„ approaches 12 as a
limit when n becomes infinite ; this is often expressed thus :
8 =12.
201-202] SEBIES — THE PROGBESSIONS 317
Similarly, whenever r < 1, numerically, and 7^ = oo, the
formula
_ a _ gy"
becomes s = ;
* 1-r
since r" approaches the limit zero as n becomes infinite.
EXERCISE CXXXVI
1. If in Ex. 2, p. 303, AB = 2 ft., show that the successive
distances traversed, when expressed in inches, form the G. P.
12fi.S3 3 3 3 3 ...
±^, D, O, 2"? 4? 8^? T6J 3T' '
2. By § 201 (2), find s^ for the series in Ex. 1 ; also find Sg, Sg,
^10, and s„.
3. From Ex. 2, p. 303, show that in the series of Ex. 1 above
s„ < 24, no matter how large 7i may be. How near to 24 will s„
approach as n is made larger and larger ? Explain. Also find
s^ by § 202.
4. Find s^ for the series 0.6, 0.06, 0.006, •••, and thus show
that 0.6 {i.e., 0.666 •••) equals f ; similarly, show that O.iS (i.e.,
0.151515...) equals 3%.
Find s^ for each of the following series :
5. 1, -hh'-- 9- ^•^- 13. l,k,k',^"
6. l,i,i,.... 10. 0.12. (wherein A: <1).
7. 1,-1, A,-. 11- 1.362. 14. ^,-,-,,
1 1^
8. V2, 1, V(X5, .... 12. 4.7523. (wherein a; > 1).
15. If, in a G. P., r is positive and less than 0.5, show that any
given term of the series is greater than the sum of all the terms
that follow it.
16. A point traversing a straight line moves in any given
second 75 % as far as in the preceding second ; if it moves 24 ft.
in the first second, how far will it move before coming to rest ?
318 HIGH SCHOOL ALGEBRA [Ch. XX
17. If a sled runs 80 ft. during the first second after reaching
the bottom of a hill, and if its distance decreases 20 % during
each second thereafter, how far will it run on the level before
coming to rest?
18. If a ball, on being dropped from a tower window 100 ft.
above the pavement, rebounds 40 ft., then falls and rebounds
16 ft., and so on, how far will it move before coming to rest ?
19. Although s^ for the series ^, i, ^, ••• is 1, show that for the
series -J, J, J, ^, •••, 6\ grows larger beyond all bounds, by suffi-
ciently increasing 7i.
Suggestion. Write the series thus : s« = I +(i + i) + (^ + i + 7 + i) -i ,
putting 8 terms in the next group, 16 in the next, and so on, and show that
each group is greater than |.
203. Geometric means. The two end terms of a finite G. P.
are called its extremes, while all the other terms are called
geometric means between these two.
E.g., in the series f, i, ^, |, and ^, the extremes are | and ^4^, and |, ^,
and I are geometric means between them.
In a G. P. of three terms, the (one) geometric mean be-
tween the extremes equals the square root of their product ;
for if Gr is the geometric mean between a and 5, then
— = -77, [definition of a G. P.
a Gr
whence G- = ± Vab,
Ex. 1. Find the geometric mean between 6 and 24; also between
10 and 8.
Solution. The geometric mean between 6 and 24 is ± V6 • 24,
i.e. ±12; and the geometric mean between 10 and 8 is ± VlO • 8,
i.e. ±4V5.
Ex. 2. Insert four geometric means between f and — -^.
Solution. In this series, a = i Z = — y, and (since four means
are to be inserted) n = 4 + 2 = 6 ; hence by § 201 (1), - -V- = | • ?-^,
whence ?*^ = — 2^^- and r = — |. Therefore the required series is
4 _2 1 _4. 9 __2
3?
1. -I. I. -¥•
202-204] SERIES— THE PROGRESSIONS 319
EXERCISE CXXXVII
Find the geometric mean between the following number-pairs :
3. 18,8. 5. }, -ff 7. (a + b),(a-by,
4. 5,20. 6. 0.5,3.5. 8. 2x-3, {x-{-:iy.
9. Insert 4 geometric means between 3 and 96.
10. Insert 3 geometric means between 2 and ^ (two answers).
11. Insert 6 geometric means between — 3125 x^^ and ^^y •
12. If m geometric means are inserted between a and b, show
that r for the series thus formed is "''^b -^ a.
13. What does the formula of Ex. 12 become when m = l?
Is this consistent with the formula for G obtained in § 203 ?
14. Two numbers differ by 24, and their arithmetical mean
exceeds their geometric mean by 6. Find the numbers.
204. Harmonic series. An harmonic series, or harmonic
progression (H. P.), is a series of numbers whose reciprocals
form an A. P. A supposed H. P. may therefore be tested,
and problems in H. P. be solved, by an appeal to our
knowledge of A. P.
E.g., the numbers f, ^, -^j, ^'j, ••• form an H. P. because their reciprocals,
viz., I, 4, J^, -2/, ..., form an A. P.
Again, if we were asked to extend the H. P. f, \, y\, f^, ••• one or more
terms toward the right, we should need merely to form the corresponding
A. P., viz., I, 4, -^5% ■^, ..-, extend it as required (cf. Ex. 3, p. 308), and
then write the reciprocals of its terms.
EXERCISE CXXXVIII
1. If the 6th term of an H. P. is ^, and the 17th term is ^,
find the 37th term.
Hint. First find the 37th term in an A. P. whose Oth and 17th terms
are 3 and ^, respectively.
2. Insert 5 harmonic means between 2 and — 3.
320 HIGH SCHOOL ALGEBRA [Ch. XX
3. Assuming x to be the harmonic mean between a and 6, show
that = , and hence that x = — ^— •
X a b X a-^b
4. The arithmetical mean between two numbers is 5, and their
harmonic mean is 3.2. What are the numbers?
5. The difference between two numbers is 2, and their arith-
metical mean exceeds their harmonic mean by ^. Find the
numbers.
6. Given (b — a): (c — b) = a: x, prove that x equals a, b, or c,
according as a, b, and c form an A. P., a G. P., or an H. P.
7. If a and b are two unequal positive numbers, and ^ is their
arithmetical mean, G their geometric mean, and .H their harmonic
mean, show that : (1) A> G>Hj and (2) A:G=G: H.
CHAPTER XXI
MATHEMATICAL INDUCTION — BINOMIAL THEOREM
205. Proof by induction. An elegant and powerful form
of proof, and one that is very useful in many branches of
mathematics, is what is known as "proof by induction."
To illustrate : suppose it to have been found by trial that
a; — ^ is a factor of x^ — ^^, oi^ — «/^ and a;* — ^*, and that we
wish to know whether it is a factor of x" — y^^ x^ — y^^ •••
also. Actual trial with any one of these, say ^ — y^, would
show that it is exactly divisible hj x — y, but besides being
somewhat tedious, this division gives no information as to
whether a; — ^ is or is not a factor of x^ — y'^, ••• also ; each
successful trial increases the probability of the success of the
next, but it proves nothing beyond the single case tried.
That X — y is a factor of x^ — ^™, for every positive inte-
gral value of 7i, may be shown as follows :
Since a^^ - ^» = x(x''-^ - y''-^) + y""'^ (x - y),
therefore, if a; — ^ is a factor of x^~^ — y^~\ then it is a fac-
tor of the second member of this equation, and therefore of
x^—y^ also (why ?) ; i.e.^ if x — y is a factor of the differ-
ence of any two like integral powers of x and y, then it is a
factor of the difference of the next higher powers also.
But since, by actual trial, a; — ^ is already known to be a
factor of a;* — y^^ therefore, by what has just been proved, it
is a factor of a:^ — ^^ also ; again, since it is now known to be
a factor of a^^ — ^, therefore it is a factor oioc^ — y^ -, and so
on without end : ^.e., x — y is a factor of ot^—y"^ for every
positive integral value of n,
321
322 BIGH SCHOOL ALGEBRA [Ch. XXI
The proof just given is an example of what is known as a
proof by mathematical induction ; such a proof consists essen-
tially of two steps, viz. :
(1) Showing hy trial or otherwise the correctness of a given
law when applied to one or more particular cases^ and
(2) Proving that if this law is true for any given case, then
it is true for the next higher case also.
From (1) and (2) it then follows that the proposition
under consideration is true for all like cases.*
EXERCISE CXXXIX
1. Prove that the sum of the first n odd integers is n^.
Solution. (1) By trial it is found that 1 + 3 = 2^ and 1 + 3 + 5 = 32.
(2) Moreover, if i + 3 + 5 + . . . + (2 A: - 1) = A;^,
then, by adding the next odd integer to each member, we obtain
1+ 3 + 5 ... + (2 A: - 1) + (2 A; + 1) =: ^•2 + (2 A: + 1) = (A; + 1)2 ;
i.e., if the law in question is true for the first k odd integers, then it is true
for the first k + I odd integers also.
But, by actual trial, this law is known to be true for the first 3 odd inte-
gers, hence it is true for the first 4 ; and, since it is now known to be true
for the first 4, therefore it is true for the first 6 ; and so on without end :
hence the sum of any number of consecutive odd integers, beginning with 1,
equals the square of their number.
By mathematical induction prove that :
2. 14-2 + 3+.. -+71 = 1.71(71 + 1).
3. 2 + 4 + 6+.-. + 27i = 7i(7i+l).
4. 12 + 2^ + 32+.. . + 7l2 = ^n(71 + l)(271 + l).
* The student should carefully distinguish between mathematical induc-
tion, as here defined, and what is known as inductive reasoning in the
natural sciences. A proof by mathematical induction is, from its very nature,
absolutely conclusive. On the other hand, the inductive method in physics,
chemistry, etc., consists in formulating a statement of a law which will fit
the particular cases that are known, and regarding it as a laio only so long
as it is not contradicted by other facts not previously taken into account.
From the nature of the case step (2) above cannot be applied in physics, etc.
205-206] MATHEMATICAL INDUCTION 323
5. l« + 2« + 3«+"-+^' = i-n2(n + l)2 = (l+2 + 3+--+ri)'.
6. A + A4-A+..-+ '
1.2 2.3 3-4 n(n-\-l) n + 1
7. 1.2 4-2.3 + 3.4+...+w(n + l) = in(n + l)(w.-f-2).
8. a + ar + ar^-] f- ar^'-i = ^^i^— 1^ .
1 — r
9. a;" — ?/'' is divisible by ic + 2/ when 71 is even.
10. Having established (1) and (2) in the inductive proof of
any law, show the generality of the law by showing that there
can be no Jirst exception, and therefore no exception whatever.
206. The binomial theorem. The method of induction
furnishes a convenient proof of what is known as the bino-
mial theorem; this theorem, which was presented without
formal proof in § 112, may be symbolically stated thus :
wherein x-\-y represents any binomial whatever, and n is
any positive integer.
To prove this theorem by mathematical induction, observe
first that it is correct when n = % for it then becomes
2 ^ • 1
(x + ?/)2 = a;2 _|_ ^^ _^ rL_^ a;V ; i'e.,{x -^ yy^ = x^ -\-1xy -^ ?/2,
which agrees with the result of actual multiplication.
Again, if (1) is true for any particular value of n^ say for
n = k^ i.e., if
{X -f yy = x^-\-\ x'-^y + ^^f=^ x^'Y
+ ^^^-y-^> .-¥+-, (2)
324 HIGH SCHOOL ALGEBRA [Ch. XXI
then, on multiplying each member of (2) by x-\-y^ it be-
comes
{X + yy^^ = x'+^ + J x'y + ^^^"^^ x'-y
1.2-3 "^
which is of precisely the same form as (2), merely having
k-\-l wherever (2) has k. Moreover, (3) is obtained from
(2) by actual multiplication, and is therefore true if (2) is
true ; hence, if the theorem is true when the exponent has any
particular value (sa}^ A;), then it is also true when the exponent
has the next higher value.
But, by actual multiplication, the theorem is known to be
true when 7i= 2, hence, by what has just been proved, it is
true when ?i = 3 ; again, since it is now known to be true
when n = 3, therefore it is true when n = 4 ; and so on with-
out end : hence the theorem is true for every positive inte-
gral exponent, which was to be proved.
EXERCISE CXL
1. In the expansion of (x -f- ?/)", what is the exponent of y in
the 2d term? in the 3d term? in the 4th term? in the i2th
term ? in the rth term ? What is the sum of the exponents of
X and y in each term?
20r,-207] BINOMIAL THEOREM 325
2. In the expansion of (x + ?/)** what is the largest factor in
the denominator of the 3d term ? of the 4th term ? of the 10th
term ? of the rth term ? In any given term, how does this factor
compare with the exponent of y?
3. In the expansion of (aj-f?/)'*, what is subtracted from n in
the last factor of the numerator in the 3d term ? in the 4th term ?
in the 5th term ? in the 9th term ? in the rth term ?
4. Based upon your answers to Exs. 1-3, write down the 6th
term of (x + yy. Also write the 10th term ; the 17th term ; and
the ?-th term.
207. Binomial theorem continued. Strictly speaking, all
that was really proved in § 200 is that, for every positive
integral value of the exponent, the first four terms of the
expansion follow the law expressed by (1) ; that all the
terms follow this law will now be shown.
In multiplying (2) of § 206 by a; + ?^, the 2d term of the
product (3) is x times the 2d term plus i/ times the 1st term
of (2) ; so, too, the 10th term of (3) would be found by
adding x times the 10th term to ?/ times the 9th term of (2),
and the rth term of (3) by adding x times the rth term to
1/ times the (r — l)th term of (2).
But the (r — l)th and the rth terms of (2) are, respec-
tively,
1.2.3....(r-2) ^
and K^-l)(^-2)...(^-r+3)(y^-r + 2) ,_,^, , .
1.2.3. •(r-2Xr-l) ^ '
therefore the rth term of (3) is
' k(k - l)(Jc - 2) ... (k - r -\- ^)
1.2.3. ...(r-2)
^(^-l)(^-2)...(^-r + 3)(^-r + 2) 1 ,_,^, ^
1.2.3. ...(r-2)(r-l) J ^ '
(k+l}k(k-l) ... (^-r+ 3) ,_,^, 1
1.2.3. ...(r-1) "^ ^ '
326 IIIGII SCHOOL ALGEBRA [Cn. XXI
wliicli conforms to the law for the rth term expressed by (1)
of § 206. Hence the rth term, i.e.^ every term, in (3) con-
forms to the law expressed by (1), which was to be proved.
EXERCISE CXLI
1. Write down the expansion of (a+by-, also of {p — qf.
Explain why the alternate terms in the expansion of (^ — qf are
negative.
2. Write down the 1st, 2d, 3d, and 8th terms of {x + yf^.
3. Write down the 4th and 7th terms of {a — xy\
4. How many terms are there in the expansion of (x+yy^?
Write down the first three, and also the last three terms of this
expansion, and compare their coefficients.
5. Write down the coefficient of the term containing ay, in
the expansion of (a — yy^.
6. Expand (3 a^ - 2 xff ; compare Ex. 2, § 57.
7. Write down the 4th and 9th terms of (f ic — | ?/)".
8. How many terms are there in [a; ) ? Write down the
\ xj _
10th term. Also write down the 5th term of (J^-^J'^J.
9. Write down the term of (3 a^-^- 2 x'y, i.e., of (xy(S a^-2)^
which contains a^^.
10. Write down the term of f a^ ) which contains a".
3 a,
11. Expand (a^ + 3 a^x'^y, and write the result with positive
exponents.
12. Expand (l — x-{- of)* by means of the binomial theorem
(cf. Exs. 40-41, p. 176).
13. By applying the law expressed in (1) of § 206 show that
the coefficient of the (n + l)th term of (x -f yy is 1 ; also show
that the coefficient of every term thereafter contains a zero factor,
and hence that {x+yy contains only n-\-l terms.
207-208] MATHEMATICAL INDUCTION^ ETC, 327
14. Show that the sum of the binomial coefficients, ^.e., of 1,
71 n(n-l) n(n-l)(n-2) ■ o,,
r 2~~' 1.2.3 '••'^'^-
Hint. After expanding (a; -f y)**, let cc = y = 1.
15. Show that the sum of the even coefficients (i.e., the 2d,
4th, . . •) in Ex. 14 equals the sum of the odd coefficients, and that
each sum is 2"~\
Hint. In (x + 2/)** let x = 1 and y =— 1.
16. Show that the coefficient of the rth term in (x 4-?/)" may be
obtained by multiplying that of the (r— l)th term by ^~^"'~ ,
?' — 1
and thus show that the binomial coefficients increase numerically
in going from term to term toward the center.
17. Show that the coefficient of the rth term is numerically
greater than that of the (r — l)th term so long as r < ^ (n 4- 3) ;
and thus write down the term whose coefficient is greatest in the
expansion of (x + 3/)" ; and also in (x + ijy^.
208. Binomial theorem extended. It may be remarked in
passing that the binomial theorem (§ 206), which has thus
far been restricted to the case where the exponent is a posi-
tive integer, is greatly extended in Higher Algebra, where
it is shown that under certain restrictions it admits negative
and fractional exponents also. Although the proof of this
fact is beyond the limits of this book, its correctness may be
assumed in the following exercises.
EXERCISE CXLII
Using the binomial theorem, write the first 5 terms of :
1. (a;4-2/)i 3. (a-c)i 5. {l-as^^.
2. (l + w)i 4. (a-^b)-\ 6. (2m-Jc)-K
7. Write the 6th term of (3 r — sy^ ; also the 5th term of
{\-3x)\
8. Show that in such cases as the above the binomial theorem
leads to infinite series (cf. Ex. 13, p. 326). ,
328 HIGH SCHOOL ALGEBRA [Ch. XXI
9. Expand (1 — x)~'^ to 8 terms by the binomial theorem and
compare the result with the first 8 terms of the quotient
1^(1-0;).
10. Show that, when expanded by the binomial theorem and
simplified, (25 + 1)^ = 5 + ^^ - wo^ + sirio^ > compare this
result with VW as found by the usual method.
11. By expanding (9 — 2)^, find an approximate value of V7;
similarly, find an approximate value of VM (i.e., V27 + 4), and
of ^40 (i.e., ^^2 + 8).
209. The square of a polynomial. In § bQ it was pointed
out that, by actual multiplication, the square of a polyno-
mial consisting of 3, 4, or 5 terms equals the sum of the
squares of all the terms of the polynomial, plus twice the
product of each term by all those that follow it. It will
now be shown that if this theorem is true for polynomials of
n terms, then it is also true for those of w + 1 terms ; and
from this it will follow, as in § 205, that it is true for poly-
nomials of any finite number of terms whatever, since it is
already known to be true for polynomials of five terms.
Let a+h-[- c-\- -•■ -\- p -[- qhQ 2i polynomial of n terms, and
let (a+h-\-c-\- ••• +Jt?4-g)2= «2 + h^-\ \- q^ -\- 2 ah -{■ 2 ac+'-'
-\-1aq-\-1hc-\ h 2^g + [-2jt?g.
In this identity replace a everywhere by ic + ^ ; then the
number of terms in the polynomial in the first member will
become n + 1, and the second member will still consist of
the sum of the squares of all the terms of the polynomial,
plus twice the product of each term by all those that follow
it (the student should work this out in detail) ; therefore,
if the theorem is true for polynomials of n terms, then it is
also true for those of n + 1 terms, which was to be proved.
CHAPTER XXII
LOGARITHMS
210. Introduction. Early in tlie seventeenth century, two
British mathematicians, Lord Napier and Henry Briggs, con-
ceived the idea of expressing all real positive numbers as
powers of 10,* arranging the exponents of these powers in a
table for convenient reference, and then employing this table
to simplify certain arithmetical computations, especially
multiplication.
E.g., to find the product of 3.578, 7.986, and 48.67, we find
from the table that
3.578 = 10°^^, 7.986 = 10«-^23^ and 48.67 = lO^^^^^
whence 3.578 x 7.986 x 48.67 = 10«^^ x lO"^^^ x lO^-^^
__ -j^QO.5536+0.9023+1.6873 Tfi 3Q
we now find from the table that
10«-i^ = 1390.6,
whence 3.578 x 7.986 x 48.67 = 1390.6.
Thus, by performing an addition (of the exponents), we have
found the product of the given numbers.
Other advantages of such a table of exponents (loga-
rithms) will be shown later (§ 218) ; some necessary defini-
tions and principles must now be given.
211. Definitions. The logarithm of a number (iV) to any
given base (5) is the exponent (x) of the power to which
this base must be raised to equal the given number.
* That it is possible to do this, either exactly or to any required degree of
approximation, will be assumed in this chapter.
329
330 HIGH SCHOOL ALGEBRA [Ch. XXII
The logarithm of iV to the base b is usually written
logjiV; and the two statements
iV=5^ and log(,N=x
are, therefore, only different ways of saying the same thing.
E.g., •.•2^ = 8, .•.log28 = 3; •.•3^ = 243, .•.log8243 = 5
and •.• 10i-^'3 = 48.67, .-. logjo 48.67 = 1.6873.
EXERCISE CXLIII
1. rrom the equation 3^* = 81, find logg 81 .
Translate into logarithmic equations (cf. Ex. 1) :
2. 4^=64. 5. 2^ = 32. 8. 10« = 1.
3. 9^ = 81. 6. (|)' = ^V 9- 10-' = .001.
4. 10-* = 1000. 7. 2-^ = ^2- 10- (i)"' = 125.
Express the following statements in the exponent notation,
and then verify the correctness of each :
11. log7 49 = 2. 14. Iogiol0 = l. 17. log3i = -2.
12. log2l6 = 4. 15. logiol = 0. 18. logio.0001 = -4.
13. log.5 .125 = 3. 16. Iogiol0000 = 4. 19. loga256 = -8.
20. Find the value of the following logarithms : logg 27 ;
log2 64; log_8 64; log_6(-216); log4l; logio.l; log.ilO;
21. Between what two consecutive integers does each of the
following logarithms lie: logio83; logio2224; logio4; logio.007;
logio .1256 ? Explain your answers.
22. May the base of a set of logarithms be fractional ? nega-
tive ? May a logarithm itself be fractional ? negative ? May
negative numbers have logarithms ? Illustrate your answers.
212. Principles of logarithms. Since logarithms are expo-
nents (§ 211), therefore the principles of logarithms are
easily obtained from those governing exponents (§§ 171-175).
211-212] LOGARITHMS 331
Principle 1. TJie logarithm of 1 to any base is 0, and the
logarithm of the base itself is 1 ; i.e.^
logj 1 = and log^ 5 = 1.
The correctness of tliis principle follows at once from the
definition of a logarithm (§ 211), and from the fact that
Z>o=l and 51 = 6. [§§ 173, 9
Principle 2. The logarithm of a product equals the sum
of the logarithms of the factors ; i.e.^
log, (M/if) =log, M -{-log, /if.
For, if M= b"^ and iV^= ^>^
then M]sr= b^'^\ [•/ b'' - by = b'^+y
whence log<. {MN) =x-\-y = log^ M-\- log^ N.
Similarly, log (MZVP • • •) = log^ M+ log^, iY+ log, P + • • ••
Let the pupil translate Principles 3-5 below into verbal
language, and prove each in detail (cf. Principle 2 above).
Principle 3. log^^=log;, Af-log^/l^.
Hint. 11 M=h'' and N = &", then M^ N= b''-^.
Principle 4. log^ N^ = p' log^ N.
Hint. liN= &^, then N^ = (b^y = b^».
Principle 5. log^ ^N = ~ • logj N.
r
1 _ 1
Hint. If iV^= ^)^ then Vi\^=(^,-)^ 1</N=N'
EXERCISE CXLIV
Using Principles 1-5, express the following logarithms in
terms of log a, log c, and log e, the base h being understood
throughout :
1. log(ac). 3. log (ace^). ^ ^^
2. log(a^). 4. log (cV). * c*
HIGH 8CH. ALG. — 22
332 HIGH SCHOOL ALGEBRA [Ch. XXII
6. log-. alogVc. 11. log(aV^).
a
6/ lo 1 3 6
9. log ^ace. 12. log
lo i. 12 ^"^
a 10. log (a^c^). 13. logVce~l
Express each of the following by means of a single logarithm,
and explain [e.g., log c + log e = log (ce)] :
14. logc + loge. 16. 2 log a + 3 log e. is. i(loge — log 5 a).
15. log c — log e. 17. 4(logc — loga). 19. f log a -f 4 log 2 c.
If logio 2 = 0.3010, logio 3 = 0.4771, and logi„ 7 = 0.8451, find the
logarithms, to the base 10, of :
20.
6(i.e.,
3.
2).
25.
63.
30.
400.
21.
14.
26.
f.
31.
tV
22.
42.
27.
2i.
32.
V3.
23.
49.
28.
5(/.e., i^y
33.
<m.
24.
12(i.e.
,2^
'.3).
29.
30(?.e., 3 .
10).
34.
5-2^.
• (i)^.
213. Common logarithms ; characteristic and mantissa.
Logarithms to the base 10 (called common or Briggs loga-
rithms) possess many advantages over those having any other
base, and are used in all practical computations. In the
following pages the base 10 will be understood when no base
is written ; thus log 25 will mean log^^ 25.
Now since 100 = 1, 10^ = 10, 102 = 100, 10^=1000, etc.,
therefore log 1 = 0, log 10 = 1, log 100 = 2, log 1000 = 3,
etc.; and therefore the logarithm of any number between 1
and 10 lies between and 1, i.e., it is plus a decimal ; the
logarithm of any number between 10 and 100 is 1 plus a
decimal ; the logarithm of any number between 100 and
1000 is 2 plus a decimal ; etc.
Again, since 10-i=.l, 10-2=. 01, 10-3 =.001, etc.,
therefore log.l = -l, log .01 = -2, log .001 = -3, etc.,
and therefore the logarithm of any number between 1 and .1
is — 1 plus a decimal ; the logarithm of any number between
.1 and .01 is — 2 plus a decimal ; etc.
212-214] LOGARirUMS 333
The integral part (whether positive or negative) of the
logarithm of a number is called the characteristic of the
logarithm, and the decimal part (always positive) is called
the mantissa.
E.g.^ log 685 is 2.8357 ; the characteristic is 2, and the mantissa is .8357.
214. Advantages of the base 10. (i) It follows from § 213
that the characteristic of the logarithm of any number
between 10 and 100 is 1 (why ?); between 100 and 1000, 2 ;
between 10,000 and 100,000, 4; between .01 and .001, -3
(why?); between .001 and .0001, —4; etc.; i.e. (let pupil
fully explain why),
( 1) The characteristic of the logarithm of any number greater
than unity is less by one than the number of digits in its integral
part ; and (2) the characteristic of the logarithm of any number
less than unity is negative, and (^numerically} greater by one
than the number of ciphers preceding the first significant figure
of the given number.
(ii) Another great advantage of logarithms to the base
10 is that moving the decimal point to the right or left in
any given number makes no change in the mantissa of the
logarithm of that number.
^.^., if log 57.32 =1.7583,
then log 573.2 = 2.7583 ;. r§ 212,
for log 573.2 = log (57.32 x 10)= log 57.32 + log 10 LPrin. 2
= 1.7583 + 1=2.7583.
EXERCISE CXLV
1. Which of the following logarithms have negative charac-
teristics : log 79; log .315; log 5228; log 4.07; log .00098 ;
log .0231 ; log 14865.01 ? Explain.
2. How many units in the characteristic of each of the above
logarithms ? Explain.
3. How many places to the left of the decimal point has a
number if the characteristic of its logarithm is 2? 5? 0? 7?
Explain your answers.
334 HIGH SCHOOL algebra [Ch. XXII
4. How many ciphers has a decimal to the left of its first
significant figure if the characteristic of its logarithm is — 5 ?
— 1 ? — 3 ? Explain your answers.
5. If log 469 = 2.6712, show that log 469000 = 5.6712 and that
log 4.69 = 0.6712 [cf. § 214 (ii)]. Also point out the characteristic
and the mantissa in each of these logarithms.
6. If log 8.93 = 0.9509, find log .0893; log 893000; log 89.3;
and log .000893.
7. Show that logarithms to the base 10 have at least two prac-
tical advantages over logarithms to other bases.
215. Table of common logarithms. The mantissas (with
decimal points omitted) of the logarithms of all integers be-
tween 1 and 1000 are given in tabular form on pp. 341, 342.
This table omits the characteristics of the logarithms be-
cause these can be supplied by inspection (§ 214) ; but it
includes the mantissas of the logarithms of decimal fractions
as well as of integers — the mantissa of log 26.5, or of log
.00265, for example, is the same as the mantissa of log 265.
Note. Except where a number is an integral power of 10, the mantissa
of its logarithm is an endless decimal. Hence the mantissas in a table are
only approximate values, correct to three or more decimal places. The table
on pp. 341, 342 is called a "four-place table " because the mantissas are com-
puted to four decimal places. Such a table gives results less accurate than
those obtained from a six-place table, for example ; the degree of accuracy
required in any given computation determines the choice of table.
216. Use of tables. Given a number, to find its logarithm.
Write down the characteristic by § 214, before consulting the
table. Then :
(a) The number consisting of not more than three significant
figures. Find the first two figures of the number in the
column headed N in the table ; opposite these, and in the
column headed by the third figure of the given number, find
the required mantissa.
214-210] LOGARITHMS 335
Ex. 1. Find log 374.
Solution. By § 214 the characteristic is 2. On p. 341 oppo-
site 37 (in the N column), and in the column under 4, we find the
mantissa .5729 ; hence log 374 is 2.5729.
Ex. 2. Find log .835.
Solution. The characteristic is — 1 (§ 214). Now the man-
tissa of log .835 is the same as the mantissa of log 835 (§ 214),
which, found as in Ex. 1, is .9217 ; hence log .835 is — 1 + .9217.
Note. The mantissa being always positive, the sign of a negative charac-
teristic is (to prevent confusion) written over the characteristic. Thus we
write log .835 not as - 1 +.9217, but as 1.9217.
Another common notation is to add 10 to the negative characteristic and
then to indicate the subtraction of 10 from the entire logarithm. Thus
log .836 may be written 9.9217 - 10.
Ex. 3. Find log 6.
Solution. The characteristic is (§ 214)- The mantissa of
log 6 is the same as that of log 600, which is .7782 ; hence log 6
is 0.7782.
(5) The number consisting of more than three significant
figures. In this case we assume that the logarithm of a
number varies directly as the number itself. While this as-
sumption is not entirely correct (doubling 50, for example,
multiplies its logarithm by 1.17 H- instead of by 2), still for
small changes in a number, it leads to results sufficiently
accurate for many purposes.
Ex.4. Find log 2547.
Solution. The characteristic is 3, and the mantissa is the
same as that of log 254.7 (§ 214).
Now, mantissa of log 254 is .4048,
and mantissa of log 255 is .4065,
i.e., adding 1 to 254 adds .0017 to the mantissa of its logarithm,
hence adding .7 to 254 should add, approximately, .7 of .0017, i.e.,
.0012, to its logarithm, and hence
log 2547 = 3.4048 + .0012 = 3.4060.
836 IIIGTI SCHOOL ALGEBIIA
Ex. 5. Find loff 74.326.
[Ch. XXII
SoLUTio:^. The characteristic is 1, and the mantissa equals
the mantissa of log 743.26, which equals (let pupil explain why)
mantissa of log 743 -f .26 X (log 744 -log 743).
= .8710 + . 26 X. 0006,
= .8712;
hence log 74.326 = 1.8712.
EXERCISE
CXLVI
By reference to the table
verify
that :
6. log 416
= 2.6191.
9.
log
.00972 = 3.9877.
7. log 5
= 0.6990.
10.
log
5268 =3.7216.
8. log 83000
= 4.9191.
11.
log
.7436 =1.8714.
Find the logarithm of :
12. 513.
19.
7.
26. .1008.
13. 692.
20.
.009.
27. 3.141.
14. 3.47.
21.
4000.
28. 22220.
15. .81.
22.
36.02.
29. .000694.
16. 27.8.
23.
6215.
30. .011111.
17. .055.
24.
.3972.
31. 437910.
18. 200.
25.
851.3.
32. .0018952.
33. Write the logarithms in Exs. 24, 2&, 29, 30, and 32 in
two different forms (cf. Ex. 2, Note).
217. Given a logarithm, to find the corresponding number.
The number to which a given logarithm corresponds is
called its antilogarithm. Thus,
•. • log 53 = 1.7243, . •. antilog 1.7248 = 53.
Antilogarithms are found by reversing the processes of
§ 216 ; a few examples will make the procedure plain.
216-217] LOGARITHMS 337
Ex. 1. Find antilog 2.5587.
Solution. On consulting the table we find that .5587 is the
mantissa of log 362, and the characteristic 2 tells us that there
must be one cipher between the decimal point and the first sig-
nificant figure [§ 214 (ii)] ; hence
antilog 2.5587 = .0362.
Ex. 2. Find antilog 1.7493.
Solution. On consulting the table we find that
.7490 = mantissa of log 561,
and .7497 = mantissa of log 562,
these being the mantissas next smaller and next larger, respec-
tively, than the given mantissa. Hence antilog 1.7493 lies be-
tween 56.1 and 56.2 (the characteristic being 1).
Again, since the given mantissa, viz., .7493, is f of the way
from .7490 to .7497, therefore the required antilogarithm is
approximately f of the way from 56.1 to 56.2,
i.e,, antilog 1.7493 = 56.1 + f of 0.1
= 56.1 -f .043
= 56.143.
Ex. 3. Find antilog 3.1188.
Solution. antilog 3.1206 = 1320,
antilog 3.1173 = 1310
whence, subtracting, we obtain .0033 and 10
also 3.1188 - 3.1173 = .0015 ;
therefore antilog 3.1188 = 1310 + ^f of 10
= 1310 4- 4.5 = 1314.5.
EXERCISE CXLVII
Verify from the table that :
4. antilog 0.1875 = 1.54. 6. antilog 1.8454 = 70.05.
5. antilog 1.6021 = .4. 7. antilog 2.5221 = .03328.
Find the antilogarithm of :
8. 2.9605. 10. 1.8451. 12. 6.4983.
9. 0.5963. 11. 1.8401. 13. 8.0755-10,
338 HIGH SCHOOL ALGEBRA [Ch. XXII
14. 3.3997. 18. 3.7361. 22. 1.3019.
15. 4.2226. 19. 0.9002. 23. 5.9754-10.
16. 2.6512. 20. 2.9068. 24. 9.5327-10.
17. 1.8846. 21. 5.8049. 25. 4.6831 - 10.
218. Computation by means of logarithms.
Ex. 1. Find p, if j> = 47.45 x 3.514 x .0064.
SOLUTION
log p = log 47.45 + log 3.514 + log .0064 ; [§ 212, Prin. 2
but log 47.45= 1.6763, [§216
log 3.514= 0.5458,
and log .0064 = 7.8062 - 10 ,^ [§ 216, Note
therefore log p = 10.0283 - 10
= 0.0283;
and therefore p = 1.067. [§ 217
This product found in the ordinary way is .10671+.
Ex. 2. Find 3.041^
Solution, log (3.041^) = 4 x log 3.041 [§ 212, Prin. 4
= 4 X 0.4830 = 1.9320 ; [§ 216
therefore 3.041^ = antilog 1.9320 = 85.5. [§ 217
Obtained by ordinary multiplication 3.041* = 85.5196+.
Ex. 3. Find ^:0572.
Solution, log V.0572 = ^ x log .0572 [§ 212, Prin. 5
= I X 2.7574
= ix (1.7574 -3) t
= 0.5858-1 = 1.5858;
therefore V.0572 = antilog 1.5858 = .3853.
Obtained by the method of § 120, -^;0572 = .38529+.
* The form 7.8602 - 10 (instead of 3.8062) is used for log .0064 because,
in computation, negative characteristics increase the danger of errors.
t In order to divide 2.7574 by 3 without mixing positive and negative
numbers it is well first to write 2.7574 in one of the following forms:
1.7574 - 3, 4.7574 - G, 7.7574 - 9, etc., i.e., to add (and then subtract) some
multiple of 3 which will make the characteristic positive.
217-218]
LOGARITHMS
339
„ , J. 37.22 X (-19.86) n ,
'^•*- ^'^= (12.33y -^"^"'
Solution. In such examples we first find the numerical value
of the result by regarding all the factors as positive, and then
prefix the proper sign as determined by §§ 18 and 19. Thus,
ignoring the minus sign, we have
log X = log 137.22 + log 9.86 - 2 x log 12.33 [§ 212, Prin. 2 and 3
= 1.5707 + 1.2980 - 2 x 1.0910
= 0.6847 ;
therefore x=- antilog 0.6847 = - 4.84.
Ex. 5. Given 47.5^ = 293.64 ; find x.
Solution. On taking the logarithm of each member of this
equation we obtain
X . log 47.5 = log 293.64
^ log 293.64 .
log 47.5 '
2.4678
whence
I.e..
x =
1.6767
1.472.
Note. Equations in which the unknown number appears as an exponent
are called exponential equations. Such equations cannot be solved by the
methods given in the preceding pages, but are easily solved by the method
illustrated in the above solution of Ex. 5.
EXERCISE CXLVIII
By logarithms find the value of :
6. 376x58. 12. 380.7 -^ 9.8.
7. 2.29x8.7. 13. 10 -^ 3.141.
8. 69.5 x. 00543. 14. 3 -r- 5.963.
9. -42.37 X. 236. 15. 30.07 -?- .002121.
10. .2912x3.141. 16. .005918 -f- .0009293.
11. .0695 x .002682. 17. 13 x 753 ^ .06238.
340 HIGH SCHOOL ALGEBRA [Ch. XXII
By logarithms simplify :
18. 23\ 23. (2)8. 28. V675.
19. .08^1 24. (If)^. 29. ^:0500l.
20. .395^-1^ 25. (62)i 30. ^(.3192)«.
21. (-3.813)'. 26. (991.7)^. 3^^ -;/:i277^l7.
32. '^18^V2574.
22. (1.228)10. 27. (.1183)?
33.
34.
19x(-700) 4635^« X 200.4*
970 X 1.4 X .0616
36.
10123
3-1^1 X .0711 ^^ 13^n^2^5
.8331x51 • 57o^7)j2l
33^ 1.78 X. 0052x16. 2^x(#X^f
^ .339x4.315 38. ^2J^ ^.
39. If a, b, and c are the sides of a triangle, and s is one half
their sum, the area of the triangle is -\/s(s — a)(s — b)(s — c).
Find, by logarithms, the area of the triangle whose sides are
13.6 ft., 15.1 ft., and 20.1 ft. ; also the area of the triangle whose
sides are 260 ft., 319 ft., and 464 ft.
Solve for x (cf. Ex. 5) :
40. 16-"= 354. 43. 6"^ = 5'=+\
41. 7^ = 9.59. 44. 2'^ = 113-+!.
42. 28.8^ = 12750. 45. 152»<^«^ = 3275.
46. From (1), § 201, show that in a G. P. log r = ]2K1zl}^.^
n — 1
also find r when a = 10, w = 10, and I = 196830.
47. If A is the amount of P dollars at r % compound interest
for n years, show that A = P(l + r)" ; also solve this equation for
each letter it contains. (Cf. Ex. 25, p. 316, also Ex. 46 above.)
48. Find the amount of $700 for 5 years at 4% compound
interest; also the amount of $450 for 10 years at 3% compound
interest.
49. In what time will $ 800 amount to $ 1834.50 if put at com-
pound interest at 5 % ?
218]
LOGARITHMS
341
Table of Common Logarithms
N
1
2
3
4
5
6
7
8
9
lO
CXXX)
0043
0086
0128
0170
0212
0253
0294
0334
0374
II
0414
0453
0492
0531
0569
0607
0645
0682
0719
0755
12
0792
0828
0864
0899
0934
0969
1004
1038
1072
1 106
13
1 139
1173
1206
1239
1271
1303
1335
1367
1399
1430
14
I46I
1492
1523
1553
1584
1614
1644
1673
1703
1732
15
I76I
1790
1818
1847
1875
1903
1931
1959
1987
2014
i6
2041
2068
2095
2122
2148
2175
2201
2227
2253
2279
17
2304
2330
2355
2380
2405
2430
2455
2480
2504
2529
18
2553
2577
2601
2625
2648
2672
2695
2718
2742
2765
19
2788
2810
2833
2856
2878
2900
2923
2945
2967
2989
20
3010
3032
3054
3075
3096
3118
3139
3160
3181
3201
21
3222
3243
3263
3284
3304
3324
3345
3365
3385
3404
22
3424
3444
3464
3483
3502
3522
3541
3560
3579
3598
23
3617
3(>3^
3655
3674
3692
37"
3729
3747
3766
3784
24
3802
3820
3838
3856
3874
3892
3909
3927
3945
3962
25
3979
3997
4014
4031
4048
4065
4082
4099
4116
4133
26
4150
4166
4183
4200
4216
4232
4249
4265
4281
4298
27
43H
4330
4346
4362
4378
4393
4409
4425
4440
4456
28
4472
4487
4502
4518
4533
4548
4564
4579
4594
4609
29
4624
4639
4654
4669
4683
4698
4713
4728
4742
4757
30
477^
4786
4800
4814
4829
4843
4857
4871
4886
4900
31
4914
4928
4942
4955
4969
4983
4997
501 1
5024
5038
32
5051
5065
5079
5092
5105
5"9
5132
5145
5159
5289
5172
33
5185
5198
5211
5224
5237
5250
5263
5276
5302
34
5315
5328
5340
5353
5366
5378
5391
5403
5416
5428
35
5441
5453
5465
5478
5490
5502
5514
5527
5|3?
555'
36
5575
5587
5599
5611
5623
5635
5647
5658
5670
37
5682
5694
5705
5717
5729
5740
5752
5763
5775
5786
38
5798
5809
5821
5832
5843
5855
5866
^Hl
5888
5899
39
59"
5922
5933
5944
5955
5966
5977
5988
5999
6010
40
6021
6031
6042
6053
6064
6075
6085
6096
6107
6117
41
6128
6138
6149
6160
6170
6180
6191
6201
6212
6222
42
6232
6243
6253
6263
6274
6284
6294
6304
6314
6325
43
6335
6345
6355
6365
6375
6385
6395
6405
6415
6425
44
6435
6444
6454
6464
6474
6484
6493
6503
6513
6522
45
6532
6542
6551
6561
6571
6580
6590
6599
6609
6618
46
6628
6637
6646
6656
6665
6675
6684
6693
6702
^?'^
47
6721
6730
6739
6749
6758
6767
6776
6785
?Zt4
6803
48
6812
6821
6830
6839
6848
6857
6866
6875
6884
6893
49
6902
691 1
6920
6928
6937
6946
6955
6964
6972
6981
50
6990
6998
7007
7016
7024
7033
7042
7050
7059
7067
51
7076
7084
7093
7101
7110
7118
7126
7135
7H3
7152
52
7160
7168
7177
7185
7193
7202
7210
7218
7226
7235
53
7243
7251
7259
7267
7275
7284
7292
7300
73Sf
7316
54
7324
7332
7340
7348
7356
7364
7372
7380
7388
7396
N
1
2
3
4
5
6
7
8
9
342
HIGH SCHOOL ALGEBRA
Table of Common Logarithms
N
1
2
3
4
5
6
7
8
9
55
7404
7412
7419
7427
7435
7443
7451
7459
7466
7474
56
7482
7490
7497
7505
7513
7520
7528
7536
7543
7551
57
7559
7566
7574
7582
7589
7597
7604
7612
7619
7627
58
7634
7642
7649
7657
7664
7672
7679
7686
7694
7701
59
7709
7716
7723
7731
7738
7745
7752
7760
7767
7774
60
7782
7789
7796
7803
7810
7818
7825
7832
7839
7846
61
7853
7860
7868
7875
7882
7889
7896
7903
7910
7917
62
7924
7931
7938
7945
7952
7959
7966
7973
7980
7987
63
7993
8000
8007
8014
8021
8028
8035
8041
8048
8055
64
8062
8069
8075
8082
8089
8096
8102
8109
8116
8122
65
8129
8136
8142
8149
8156
8162
8169
8176
8182
8189
66
8195
8202
8209
8215
8222
8228
8235
8241
8248
8254
67
826i
8267
8274
8280
8287
8293
8299
8306
8312
8319
68
8325
8331
8338
8344
8351
8357
8363
8370
8376
8382
69
8388
8395
8401
8407
8414
8420
8426
8432
8439
8445
70
8451
8457
8463
8470
8476
8482
8488
8494
8500
8506
71
8513
8519
8525
8531
8537
8543
8549
8555
8561
8567
72
8573
8579
8585
8591
8597
8603
8609
8615
8621
8627
73
8633
8639
8645
8651
8657
8663
8669
8675
8681
8686
74
8692
8698
8704
8710
8716
8722
8727
8733
8739
8745
75
8751
8756
8762
8768
8774
8779
8785
8791
8797
8802
76
8808
8814
8820
8825
8831
8837
8842
8848
8854
8859
77
8865
8871
8876
8882
8887
8893
8899
8904
8910
8915
78
8921
8927
8932
8938
8943
8949
8954
8960
8965
8971
79
8976
8982
8987
8993
8998
9004
9009
9015
9020
9025
80
9031
9036
9042
9047
9053
9058
9063
9069
9074
9079
81
9085
9090
9096
9101
9106
9112
9117
9122
9128
9133
82
9138
9143
9149
9154
9159
9165
9170
9175'
9180
9186
83
9191
9196
9201
9206
9212
9217
9222
9227
9232
9238
84
9243
9248
9253
9258
9263
9269
9274
9279
9284
9289
85
9294
9299
9304
9309
9315
9320
9325
9330
9335
9340
86
9345
9350
9355
9360
9365
9370
9375
9380
9385
9390
87
9395
9400
9405
9410
9415
9420
9425
9430
9435
9440
88
9445
9450
9455
9460
9465
9469
9474
9479
9484
9489
89
9494
9499
9504
9509
9513
9518
9523
9528
9533
9538
90
9542
9547
9552
9557
9562
9566
9571
9576
9581
9586
91
9590
9595
9600
9605
9609
9614
9619
9624
9628
9633
92
9638
9643
9647
9652
9657
9661
9666
9671
9675
9680
93
9685
9689
9694
9699
9703
9708
9713
9717
9722
9727
94
9731
9736
9741
9745
9750
9754
9759
9763
9768
9773
95
9777
9782
9786
9791
9795
98CX)
9805
9809
9814
9818
96
9827
9832
9841
9845
9850
9854
9859
9863
97
9868
9872
9877
9881
9886
9890
9894
9899
9903
9908
98
9912
9917
9921
9926
9930
9934
9939
9943
9948
9952
99
9956
9961
9965
9969
9974
9978
9983
9987
9991
9996
N
1
2
3
4
5
6
7
8
9
INDEX
[Numbers refer to pages.]
Absolute, term, 190.
value, 19.
Addition, 8, 21, 29, 30, 115, 243.
Algebraic, expressions, 27.
fraction, 109.
numbers, 19.
sentence, 60.
sum, 21.
Alternation, 298.
Antecedent, 293.
Antilogarithm, ,336,
Approximate root, 181.
Arithmetical, mean, 311.
numbers, 19.
progression, 307.
series, 307.
Arranged polynomial, 43.
Associative law, 54.
Axioms, 56.
Axis, of coordinates, 219.
of imaginaries, 259.
of real numbers, 259.
Base, of logarithms, 329.
of power, 12.
Binomial, 27, 236.
, cube of, 76.
square of, 71.
theorem, 174, 323.
Brace, bracket, etc., 15.
Briggs logarithms, 332,
Character of roots, 278.
Characteristic of logarithm, 333.
Checking results, 31, 56.
Clearing of fractions, 126.
Coefficients, 27, 190, 236.
Commensurable numbers, 293.
Common, difference, 307.
factor, 98,
logarithms, 332,
multiple, 105.
ratio, 313.
Commutative law, 54.
Completing the square, 194.
Complex, fractions, 123.
numbers, 257.
Composite numbers, 77.
Conditional, equation, 55.
inequality, 289.
Conjugate, complex numbers, 257.
surds, 249.
Consequent, 293.
Consistent equations, 146.
Constant term, 190.
Constants, 301.
Continued, product, 25.
proportion, 296.
Coordinate axes, 219.
Cube, 13.
of a binomial, 76.
root, 176, 185, 187.
Cubic equation, 126.
Decreasing series, 316.
Degree, of term, 43.
of equation, 125.
Denominator, 109.
Determinate system, 165.
Difference, 8.
Discriminant, 278.
Dissimilar, radicals, 236.
terms, 28.
Distributive law, 54.
343
U4:
INDEX
Division, 11, 26, 45, 121, 250, 298.
Divisor, dividend, 12.
Elements of A. P. and G. P., 307
313.
Elimination, 147, 149.
Equations, conditional, 55.
consistent, 146.
determinate, 165.
equivalent, 127.
exponential, 339.
fractional, 130, 151, 198.
graph of, 221.
homogeneous, 208.
identical, 55.
inconsistent, 146.
independent, 146.
indeterminate, 144, 165.
in quadratic form, 205.
integral, 125.
irrational, 255.
linear, 125.
literal, 125, 128, 154.
locus of, 221.
numerical, 125.
of tlie problem, 62.
quadratic, 126, 190, 277.
radical, 255.
simple, 125.
simultaneous, 146, 206.
solution of, 57.
symmetric, 212.
Equivalent equations, 127.
Even, power, 170.
root, 176.
Evolution, 176.
Exponent, 12.
fractional, negative, zero, 265
laws, 38, 45, 171, 265.
Exponential equation, 339.
Extraneous roots, 131.
Extremes, 296, 311, 318.
Factoring, 78, 90, 282.
solving equations by, 94.
Factors, 10, 77, 78.
of quadratic expressions, 282.
Factor theorem, 92.
Finite, numbers, 52.
series, 307.
Formulas, for A. P., G. P., 308, 314.
for solving equations, 140, 277.
Fourth proportional, 296.
Fractional, equations, 130, 151.
exponent, 265.
Fractions, 12, 109.
clearing of, 126.
lowest terms of. 111.
General problem, 139.
Geometric, infinite G. P., 316.
means, 318.
series, 313.
Graph of an equation, 221.
Graphic solutions, 227.
Graphical representation of complex
numbers, 259.
Greater than, 287.
Harmonic series, 319.
Highest common factor, 98.
Homogeneous equations, 208.
Identical equations, 55.
Imaginary, numbers, 177, 235, 257.
unit, 257.
Improper fraction, 109.
Incommensurable numbers, 293.
Inconsistent equations, 146.
Independent equations, 146.
Indeterminate, equations, 144, 165.
systems, 165.
Index of a root, 176.
Induction, mathematical, 322.
^ Jnequalities, 287.
-^Infinite series, 307, 316.
Infinitely, large, 52.
small, 52.
Insertion of parentheses, 35.
Integral, equation, 125.
expressions, 42.
Interpretation, of results, 189.
of the forms
«-, .5, 301,
00
Inverse, operations, 9.
ratio, 293.
INDEX
345
Inversion of proportion, 297.
Involution, 170.
Irrational, equation, 255.
numbers, 235.
Known and unknown numbers, 125.
Laws, of exponents, 38, 45, 171, 265.
of operations, 54.
of signs, 25, 26, 177.
Less than, 287.
Letter of arrangement, 43.
Like, and unlike, radicals, 236.
terms, 28.
Limit, 301.
Linear equations, 125.
Literal, coefficients, 27.
equations, 125, 128, 154.
numbers, 1, 3,
Locus of an equation, 221.
Logarithms, 329.
table of, 341.
Lowest common multiple, 105.
Mantissa of logarithm, 333.
Mathematical induction, 322.
Mean proportional, 296.
Members of an equation, 55.
Minuend, 9.
Mixed expression, 109.
Monomials, 27.
Multiples, 105.
Multiplicand, multiplier, 10.
Multiplication, 10, 38.
Negative, exponent, 266.
numbers, 18.
term, 28.
Numbers, absolute value of, 19.
commensurable, etc., 293.
complex, 257.
constants and variables, 301.
finite and infinite, 52.
imaginary, 177, 235, 257.
known and unknown, 125.
literal, 1, 3.
negative and positive, 18.
opposite, 19.
Numbers, prime and composite, 77.
rational and irrational, 234, 235.
real, 177, 235.
Numerical, coefficient, 27.
equation, 125.
Odd, power, 170.
root, 176.
Operations, with literal numbers, 1.
with imaginary and complex num-
bers, 261.
Opposite, numbers, 19.
species, 287.
Order, of operations, 14.
of radicals, 236.
Parentheses, 15, 35.
Polynomials, 27.
square of, 75, 328.
Positive numbers, 18.
terras, 28.
Power, 12.
Powers of imaginary unit, 258.
Prime, numbers, 77.
to each other, 98.
Principal roots, 236.
Principles, of clearing of fractions,
126.
of elimination, 147, 149.
of inequalities, 287.
of logarithms, 303.
of proportion, 296.
Problems, 62.
directions for solving, 62.
general, 139.
Products, 10, 24, 38, 40, etc.
of fractions, 119.
of sum and difference, 72.
Progression, arithmetical, 307.
geometric, 313.
harmonic, 319.
Proof by induction, 321.
Proper fraction, 109.
Property, of complex numbers, 262.
of quadratic surds, 253.
Proportion, 295.
Pure, quadratic, 190.
imaginary numbers, 257.
346
INDEX
Quadratic equation has two roots, and
only two, 283.
Quadratic equations, 126, 190, 277.
form of, 205.
graphs of, 229.
roots of, 278, 279,
simultaneous, 206.
solution by formula, 277.
special devices for, 211.
surds, 253.
Quotient, 12.
Radicals, radical equations, 235, 236,
241, 255.
Radicand, 235.
Ratio, 293, 294, 313.
Rational numbers, 234.
Rationalizing factor, 249, 276.
Real numbers, 177, 235.
Reciprocal of a number, 110.
Relation between roots and coeffi-
cients, 279.
Remainder, 8.
theorem, 92.
Removal of parentheses, 35.
Review exercises, 67, 166, 284.
Root, principal, 236.
to n terms, 181.
Roots, of an equation, 56.
character of, 278.
extraneous, 131.
relation between coefficients and,
279.
Rule of signs, 25, 26.
Series, 307.
Signs, of aggregation, 14.
of deduction, 3.
of inequality, 287.
of operation, 1 , 10, 19.
of quality, 19.
Similar, radicals, 236.
terms, 28.
Simple equations, 125.
Simultaneous equations, 146, 206.
Solution of equations, 57, 146, 206, etc.
by factoring, 94.
Species of inequalities, 287.
Specitic gravity, 139.
Square, 13.
of a binomial, 71.
of a polynomial, 75, 328.
Square root, 176, 180, 188.
of binomial surd, 253.
of complex number, 262.
Standard form, of complex number,
257.
of quadratic, 190.
Subtraction, 8, 22, 32.
Subtrahend, 8.
Sum, 8, 21.
Summands, 8.
Surds, 235.
conjugate, 249.
Symbols, •.• and .-.,3,
> and <, 287.
Symmetric equations, 212.
System of equations, 146.
indeterminate, 165.
Table of logarithms, 341.
Terms, 27, 109, 203, 307.
Theorem, binomial, 174, 323.
Thermograph, 232.
Third proportional, 296.
Translation of common language into
algebraic language, and vice
versa ^ 60.
Transposing, 58.
Trinomial, 27.
Type forms, 71.
Unconditional inequality, 289.
Unknown numbers, 125.
Unlike terms, 28.
Variable, variation, 301.
Vary, 304.
directly, 304.
inversely, 304.
jointly, 304.
Vinculum, 15.
Zero, 52.
exponents, 266.
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