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III
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IN MEMORIAM
FLORIAN CAJORI
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SCHOOL ALGEBRA.
BY
C. A. VAN VELZER,
AND
CHAS. S. SLIGHTER,
PROFESSORS IN THE UNIVERSITY OF WISCONSIN.
MADISON, WIS.:
TRACY, GIBBS & CO.
COPYRIGHT,
C. A. VAN VELZER, CHAS. S. SLIGHTER,
1890.
Tracy, Gibbs & Co., Printers and Stereotypers.
PREFACE.
The present volume is icsued in the hope that it will assist the
teacher in the effort to get pupils to think and that it will induce pupils
to place the subject of Algebra on a rational instead of an arbitrary
basis, to work from principles rather than from rules.
In the first part of the work we have pursued what is termed the
inductive method, but we do not wish this understood as that method
which infers general principles from an accumulation of particular
cases. This is the inductive reasoning of the natural sciences, but
we believe it is never legitimate in mathematics. Induction, as we
use the term, means that method which proceeds from the particular
to the general. By particular cases, which gradually increase in
generality, the mind of the learner is prepared to appreciate the gen-
eral case, but this general case must so present itself to the learner's
mind that he sees that the truth stated must be so and cannot possibly
be otherwise.
It will be noticed that we have not thought it necessary to complete
one subject before taking up another, but subjects have sometimes
been treated in an elementary way at first and more completely at
some subsequent part of the book. We believe that by this plan
students can follow the work more easily and with more profit, and at
the same time we are enabled to treat some subjects, especially
Factors, Multiples, and Fractions, more fully than is ordinarily done.
We have placed the principles governing the use of parentheses
before the four fundamental operations of addition, subtraction, mul-
tiplication, and division, and have made the latter depend upon the
former. This we think enables us to treat the four fundamental
operations in a way which is more rational to beginners than is given
when the usual order is pursued.
'm^r^^-^r^*^ r\.
IV PREFACE.
The subject of equations is early introduced, and is distributed
through the book instead of being given together in one place. This
keeps up the interest in the subject and prevents the student from
getting the idea that he is learning a mass of theory which has no
practical application.
Indices and Surds appear after Quadratics for the reason that these
subjects are more difficult than Quadratics.
Additive and subtractive terms are distinguished from positive and
negative quantities, and the latter are postponed until after the four
fundamental operations.
This work contains about 3000 examples besides several hundred
problems and inductive exercises. Many of these are original and
many are taken from the German and French collections and from
the English examination papers. The answers are not printed in the
book for a reason that every teacher of Algebra can readily assign, but
the answers are issued in pamphlet form for the use of teachers only.
C. A. Van Velzer.
University of Wisconsin, Chas. S. Slighter.
December, 1890.
TABLE OF CONTENTS.
Bold face figures refer to the exercises and
light face figures to the pages.
CHAPTER I.
First Principles, .-----.
1, Illustrating how a letter may be used to represent a
number, i. 2, Leading to the idea of an algebraic ex-
pression, 3. 3, Leading to the idea of an algebraic equa-
tion and to the distinction between known and unknown
numbers, 4. 4, Symbolic expressions, 6. 5, Problems, 7.
6, Expressions in which several letters arc used to stand
for numbers, 10.
CHAPTER II.
Union of Terms and Removal of Parentheses,
15
7, Union of similar terms, 15. 8, Examples, 19. 9, Paren-
theses and their removal, 21. lO, Removal of parentheses;
general form a-\-[b-\-c'), 22 ; general form a-\-{b—c)^ 22 ; gen-
eral form a—{b-\-c\ 23 ; general form a — [b—c), 24 ; paren-
theses preceded by plus sign, 24 , parentheses preceded by
minus sign, 26. 11, Miscellaneous examples on the removal
of parentheses, 29.
CHAPTER III.
Addition, ----30
12, Addition of expressions, 30; examples, 31. 13, Arrange-
ment of work in addition, 31. 14, Examples, 33.
CHAPTER IV.
Subtraction, - -jg
15, Subtraction of expressions, 36; examples, 37. 16, Ar-
rangement of work in subtraction, 38. 17, Examples, 38.
1'8, Addition and subtraction of equals, 40. 19, Examples,
42. 20, Transposition in equations, 43. 21, Examples, 44.
22, Problems, 46.
vi CONTENTS.
CHAPTER V.
Multiplication, _._------5o
23, General definition of multiplication, 50 ; examples, 51.
24, Multiplication of monomials, 52. 25, Examples, 53.
26, Law of exponents in multiplication, 54. 27, Examples,
55. 28, Multiplication of polynomials by monomials, 56.
29, Examples, 59. 30, Arrangement of work in the mul-
tiplication of a polynomial by a monomial, 59; examples, 60.
31, Multiplication of polynomials by polynomials, 62. 32,
Examples, 63. 33, Arrangement of work in the multipli-
cation of two polynomials, 64. 34-, Examples, 69. 35,
Equations involving multiplication, 71.
CHAPTER VI.
Division, -- - • - Ti
36, Division of monomials by monomials, 73 ; the law of
exponents, 74. 37, Examples, 74. 38, Division of poly-
nomials by monomials, 76. 39, Examples, 77. 40, Di-
vision of polynomials by polynomials, 77. 41, Examples,
79. 42, Arrangement of work in division of polynomials
by polynomials, 81. 43, Examples, 85.
CHAPTER Vn.
Negative Quantities, ---88
44, Number and quantity, 88. 45, Opposite directions,
89. 46, How directions are distinguished, 90. 47, Posi-
tive and negative numbers, 92. 48, Illustrative examples,
94. 49, Addition, 97. 50, Subtraction, 99. 51, Multi-
plication, 100. 52, Division, loi.
CHAPTER VIII.
Parentheses, - - 103
53, Removal of parentheses, 103. 54, Insertion of paren-
theses, 104.
CHAPTER IX.
Elementary Factors, Multiples, and Fractions, - - 106
55, Factors, 106. 56, Highest common factor, 107. 57,
Lowest common multiple, 108. 58, Fractions, iii. 59,
Addition of fractions, 113; subtraction of fractions, 115.
60, Multiplication of fractions, 116. 61, Division of frac-
tions, 118.
CONTENTS. vii
CHAPTER X.
Simple Equations, -- 121
62, Definitions and general principles, 121. 63, Exam-
ples, 126. 64, Literal equations, 127. 64-a, Symbolic
expressions, 129. 64&, Problems, 132.
CHAPTER XI.
Simultaneous Equations, - 139
65, Definitions and general principles, 139. 66, Elimina-
tion by substitution, 141. 67, Examples, 143. 68, Elim-
ination by comparison, 144. 69, Examples, 145. 70,
Elimination by addition and subtraction, 146. 71, Special
expedients, 148. 72, Examples, 149. 73, Simultaneous
equations containing three unknown numbers, 152. 74,
Examples, 155. 75, Literal simultaneous equations, 157.
76, Problems producing simultaneous equations, 158.
CHAPTER Xn.
Powers and Roots, -- 163
77, Powers of monomials, 163. 78, Square of a binomial,
168. 79, Cube of a binomial, 169. 80, Square of a poly-
nomial, 171. 81, Roots of monomials, 173. 82, Examples,
177. 83, Square root of polynomials, 179. 84, Cube
root of polynomials, 184.
CHAPTER Xin.
Harder Factors, Multiples, and Fractions, - - - 187
85, Factors common to all the terms of an expression, 187.
86, Formation of certain products, 189. 87, Expressions
of the form X'—a^, 191. 88, Expressions of the form
x^-\-ax-\-l>, 193. 89, Expressins of the form a^—b^^ 195.
90, Expressions of the form a^-\-b^, 197. 91, Expressions
of the form x^-\-a-x^-\-a^, 198. 92, Expressions of the
form a^—b^y 200. 93, Expressions of the form a"-\-b'\ 208.
94, Miscellaneous factors, 212. 95, H.C.F. of expressions
which can be factored, 216. 96, H.C.F. of expressions not
easily factored, 217, examples, 225. 97, L.C.M. of expres-
sions that can be factored, 227. 98, L.C.M. of expressions
not easily factored, 229. 99, Fractions reduced to lowest
viii CONTENTS.
terms, 231. 100, Addition of fractions, 233. lOl, Sub-
traction of fractions, 234. 102, Multiplication of fractions,
236. 103, Division of fractions, 238. 104, Miscellaneous
fractions, 240.
CHAPTER XIV.
Quadratic Equations, 249
105, Preliminary topics, 249. 106, Pure quadratic equa-
tions, 252; examples, 253. 107, Affected quadratics, 255;
examples, 257. 108, Problems leading to quadratic equa-
tions, 261. 109, Equations solved like quadratics, 269.
110, Theory of quadratic equations, 270.
CHAPTER XV.
Simultaneous Equations Above the First Degree, - - 277
111, One equation of the first degree and one of the second
degree, 277; examples, 279. 112, Tv^^o quadratic equations,
280; examples, 282. 113, Miscellaneous equations, 283;
examples, 286. 1 14, Problems, 287.
CHAPTER XVI.
Theory of Indices, 291
115, Meaning of fractional exponents, 291, 116, Examples,
295. 117, Properties of fractional exponents, 296. 118,
Examples, 299. 119, Meaning of zero and negative ex-
ponents, 303. 120, Examples, 305. 121, Properties of
negative exponents, 306. 122, Examples, 310.
CHAPTER XVII.
Surds, - -314
123*, Definitions and general principles, 314. 124, To re-
move a factor from beneath the radical sign, 316. 125, To
introduce the coefficient of a surd under the radical sign,
317. 126, To integralize the expression under the radical
sign, 318. 127, To lower or raise the index of a surd, 320.
128, To reduce a surd to its simplest form, 320. 129,
Addition and subtraction of surds, 321. 130, Multiplica-
tion and division of surds, 323. 131, Powers and roots of
sures, 326. 132, Rationalization of expressions containing
quadratic surds, 327. 133, Rationalization of equations, 330.
CONTENTS. IX
CHAPTER XVIII.
Ratio, Proportion and Variation, 332
134, Ratio, 332; problems, 337. 135, Proportion, 339;
problems, 344. 136, Variation, 345; examples and prob-
lems, 348.
CHAPTER XIX.
Progressions, 350
137, Arithmetical progressions, 350. 138, Examples and
problems, 354. 139, Geometrical progressions, 357. 140,
Examples and problems, 360.
CHAPTER XX.
Binomial Theorem, 363
141, Laws of exponents and coefficients, 363. 142, Ex-
amples, 370.
SCHOOL ALGEBRA.
CHAPTER L
FIRST PRINCIPLES.
EXERCISE 1.
Illustrating how a Letter may be used to Represent a Number.
1. In Algebra, as in Arithmetic, the symbol for Plus
is -f- and the symbol for Minus is — .
1. How many dozen are 6 dozen + 4 dozen 4- 2
dozen ?
2. How many score are 6 score + 4 score + 2 score ?
3. How many hundred are 6 hundred + 4 hundred
+ 2 hundred ?
4. How many times 100 are 6 times 100 + 4 times
100 + 2 times 100 ?
5. How many times 10 are 6 times 10 + 4 times 10
+ 2 times 10 ?
6. How many times 7 are 6 times 7 + 4 times 7 +
2 times 7 ?
7. Six times a7ty number plus four times the same
n^imbcr plus two times the same number are how many
times that number f
2, In Algebra letters are often used to represent or stand
for numbers.
.2 FIRST PRINCIPLES.
3. In Algebra, as in Arithmetic, the symbol for
times is x .
In any statement like 6X100+3X100—5X100, /. c, in any state-
ment where addition, or subtraction, and multiplication occur to-
gether, the multiplications must always be performed before any
addition or subtraction takes place. Thus, 6X100-(-3 does not mean
6X103, but means 600+3.
8. If / stands for 10, how many times 10 are 4x/+
7x/— cSx/?
g. If /stands for 2, how many times 2 are 4x/+7x/
— 8x/?
ID. If t stands for 3, how many times 3 are 4x/+
Tx/— 8x/?
11. If 5 stands for 6, how many times 6 are 7x^+
5X5— 3x^?
12. If 5 stands for 7, how many times 7 are 7X5+
5X5—3X5?
13. Seven times a certam number plus five times the
safne number minus three times the same number are how
many times that mimber f
14. If n stands for a certain number, how ma^y times
that number are 7 X ?^ + 5 X ;z— 3 X ;^ ?
15. In question 14 can n stand for 17? for 100? for
25? fori? for J? for li?
In qjiestion z^, 71 ca7i stand for Ki^Y number WHATEVER,
provided it stands for the same niunber throughoitt question
and answer.
16. If a stands for a certain number, 8x« + 4x<2—
5 X <2 are how many times that number ?
17. If b stands for a certain number, 8x<^+4x/^—
hy.b are how many times that number ?
18. Could some other letter than a, b, n, 5, or / be
used to represent a number ?
ALGEBRAIC EXPRESSIONS. 3
4. The preceding questions suggest the following
principle :
A7iy letter may be icsed to represent or stand for any
number, provided that the same letter represents the sa?ne
number throughout the saine question and answer.
6. In Algebra it is usual to omit the sign X in a
product like 7 X w and write merely 7«. It is then read
*' seven n,'' instead of "seven times ^^," but of course it
always mea^is seven times ?^ ; and this meaning the learner
must keep in mind. Thus, if b stands for 31^, then 7^
stands for 7 times Z\\.
Evidently when the sign X occurs between figtires it
cannot be omitted. Thus we can write ^ib for 7x<^,
but we cannot write 731| for 7x31|-, for 731| has a dif-
ferent meaning already given to it.
6. The number written before a letter to show how
many times the number represented by the letter is taken,
is called the Coefficient * of the letter. Thus, in "b, 7
is called the coefficient of b.
ig. What does hb mean ? How much is this if b equals
10 ? What is the 5 called ?
20. What does 10?/ mean ? How much is this if n
equals 8 ? What is the 10 called ?
EXERCISE 2.
Leading to the Idea of an Algebraic Expression.
1. What does 7a— 3^ + 5<2 equal, \i a stands for 7?
2. What does 10i^4-4a— 9a equal, if a stands for 6?
* A more general definition of coefficient is given in Art. 14.
4 FIRST PRINCIPLES.
3. What does S5a—7a—Sa equal, if a stands for y\?
4. What does 9<2 + 6a— 7a— 4a + 2a equal, if a stands
for A?
5. What does 25;^ + | equal, if 7z stands for |-?
6. What does 2o?i—6n—^ equal, if n stands for ^?
7. Anything, whether short and simple or long and
complicated, which is or may be considered to be equal
to some number, is called an Expression.
The number to which the expression is equal is called
the Value of the Expression.
The number which a letter stands for is called the
Value of the Letter.
7. What is the value of the expression 10c-\-2c—5c—
6c-hSc, if the value of ^ is 4 ?
8. What is the value of the expression 12m—bm~Q>7}t
-\-Sm, if the value of m is 12 ?
9. What is the value of the expression 7a— 4a— 3a + a,
if the value of a is 2 ?
10. What is the value of the expression 16^+24^—
10^+25, if the value of d is yV?
EXERCISE 3.
Leading to the Notion of an Algebraic Equation and the Dis-
tinction BETWEEN Known and Unknown Numbers.
8. In Algebra, as in Arithmetic, the symbol for
Equals is =.
1. What is the value of 12a, if a= 2? if a=5? ifa=7?
ifa=|? if a=6i?
2. What must a equal if the value of 12a is 36 ? if the
value of 12a is 48 ? if the value of 12a is 72 ? if the value
of 12a is 8 ? if the value of 12a is 20 ?
ALGEBRAIC EXPRESSIONS. 5
3. If ^=8, what does Ix equal? If 7jtr=56, what
does X equal ? If lx=4S), what does x equal ?
4. If ;«r=9, what does \\x equal? If ll;r=99, what
does X equal ? If lljtr=121, what does x equal ?
5. If a=12, what does 3^-h5« equal? If 3^-f-5«=96,
what does a equal? If 3«-f5a=4, what does a equal ?
6. If ;z=6, what does 6?i-^5n-\-7i equal? If G7i-\-d?i
+ « = 72, what does n equal? If G?^ + 5;^^- 7^=108, what
does ?i equal ?
7. If 7<^— 6<^ + 3^=20, what is the value of d?
9. The statement of equality which exists between
two expressions is called an Equation, and the parts on
either side of the sign = are called the Members of the
equation. The expression on the left-hand side of the sign
= is called the Left or First Member, and the expres-
sion on the right-hand side of the sign = is called the
Right or Second Member.
8. If 3>&4-2/l' + /t=120, what is the value of k?
9. If 4;r-f-5;f— 3;f=3, what is the value of -r?
10. If 7jt:— 2jf-f-;f=6, what is the value of;*;?
11. If2j'=2^, what is the value of jk?
12. If 3^+2^=12, what is the valve of -^?
13. If r=f, what is the value of 12c-{-^—0c}
14. lfd=l^, what is the value of 8^^+5^—10 ?
15. If/=1.2, what is the value of 20/— 7/— 3/'?
16. If lOOze^— 78ze/+15ze'=74, what is the value of rr?
17. If 17«—5?<— 4/^=66, what is the value of 7^?
18. If ^=5, what is the value ot 25^—5^+5 ?
19. If /i=100, what is the value qf 20/i-6/i-3/i + 100?
20. If 18z'-|-llz;4-2z/4-9z/=80, what is the value of -j?
6 FIRST PRINCIPLES.
10. A careful inspection of the questions of this ex-
ercise shows that when there is only one letter in an
expression, two cases may arise : first, the value of the
letter may be given and the value of the expression re-
quired ; second, the value of the expression may be given
and the value of the letter required.
In the first case the value of the letter is known or
give?i, and in the second case the value of the letter is
unktiown or required.
Thus w^e see that in Algebra there are two kinds of
numbers, called respectively Known and Unknown,
either of which may be represented by a letter; and as it
is possible that both kinds of numbers will appear in the
same question, it is customary to distinguish between
them by representing the known numbers by the first and
intermediate letters of the alphabet, and the unknown
numbers by the last letters of the alphabet.
EXERCISE 4.
Symbolic Expressions.
1. A coat and hat cost |24 ; the hat cost $4. What
does $24- $4 stand for?
2. A coat and hat cost $24 ; the hat cost %x. What
does 124 -Ix stand for ?
3. A coat and hat cost $24 ; the hat cost %x and the
coat 5 times as much as the hat. What does %hx stand
for?
What does $24-|5x stand for ?
What does %x-\-%bx, or %^x, stand for?
4. If n stands for a certain number, what exprcssicn
will stand for a number which is 10 larger ?
PROBLEMS. 7
5. 1{ a stands for a certain number, what v/ill stand
for a number which is 25 smaller ?
6. There are two numbers ; the second number is five
more than twice the first number. If 71 represents the first
number, what expression will represent the second num-
ber?
7. Of two numbers the second is 12 less than 5 times
the first. If n stands for the first number, what will
stand for the second number ?
8. How many feet in n yards ?
9. How manj^ feet in ?i yards plus 5 A-ards ?
10. How many feet in 71 yards plus 5 feet ?
11. A room is a yards wide and twice as long as it is
wide. How many yards long is the room ? How many
feet long is the room ?
12. In jr years a man will be 36 years old. What is
his present age ?
13. A man is now 40 years old. How old will he be
a years from now ? How old will he be '2a years from now ?
How old was he Sd years ago ?
EXERCISE 5.
Problems.
I. A coat and hat cost $24. The coat cost 5 times as
much as the hat. What was the cost of each ?
SOLUTION BY ARITHMETIC.
Five times ^/le cost of the hat = the cost of the coat.
Once the cost of the hat = the cost of the hat.
Therefore, six times the cost of the hat — the cost of coat and hat.
But the cost of the coat and hat is $24.
Hence, six times the cost of the hat = $24.
Therefore, the cost of the hat = ^ of $24, or $1
Consequently the cost of the coat was $20.
8 FIRST PRINCIPLES.
SOLUTION BY ALGEBRA.
The cost of the hat is a certain number of dollars, and that number,
whatever it is, we may represent by a letter. For example, we may
say,
Let X = number of dollars the hat cost.
Then 5x = number of dollars the coat cost.
Hence, 5x-^x, or Qx, = number of dollars that both hat and coat cost.
But 24 = number of dollars that both hat and coat cost.
Therefore 6^ = 24.
Then x = 4,
and 5x =z 20.
Therefore the hat cost $4, and the coat cost $20.
The student should compare very carefully the two solutions of this
problem above given. It will be noticed that the arithmetic and
algebraic solutions are not so different from each other as would
appear at first sight. In fact, to change the first solution to the sec-
ond nothing need be done except to replace the words " i/ie cost of the
hat " by $x. In the arithmetic solution the phrase " the cost of the hat''
stands for a certain unknown number of dollars, which number of
dollars is represented by a single letter, x, in the algebraic solution.
2. The sum of two numbers is 72 and one number is
twice as large as the other. What are the numbers ?
Let jr=:the smaller number.
Then 2x=the larger number.
Therefore 'lx-^rx—Tl,
i. e. 3x=72.
Hence .^•— 24, the smaller number,
and 2.*"=48, the larger number.
3. Divide $65 between A and B so that B shall receive
4 times as much as A.
4. A rectangle is 3 times as long as it is broad, and
the distance around it is 64 feet. Find the length and
breadth of the rectangle.
5. A piece of timber 18 feet long must be cut so as to
give one piece 2 feet long and two other pieces, one of
which must be 3 times the length of the other. Find
how long each one of these pieces will be.
PROBLEMS. 9
6. A father said to his son, " Neiit year the sum of our
ages will be 70 years, and I will be 4 times as old as you
will be." What is the present age of each?
7. A man has $48, consisting of an equal number of
bank notes of the denominations of $1, $2, and 85. What
number has he of each ?
Let j:=the number he has of each. Then in $1 bills he has $x,
in $2 bills he has $2x, and in $5 bills he has $5^-.
8. John, Henry, and Mary paid 13.60 for their books.
Henry's cost twice as much as John's, and Mary's cost
three times as much as John's. Find the cost of each
scholar's books.
9. If you add together 3 times and 5 times and 7 times
a certain number you will obtain 315. What is the
number ?
10. Three persons subscribed 150000 to build a hospital.
The first two subscribed equal amounts, but the third party
subscribed 2 times as much as either of the others. How
much did each person subscribe ?
11. A man bought 10 turkeys, 10 chickens, and 10
ducks, paying §10 for all. A chicken cost the same as a
duck, and a turkey cost 3 times as much as a dnck. What
was the cost of each ?
12. A, B, and C form a partnership to do business. A
furnishes 4 times as much capital as C, and B furnishes 3
times as much as C. Altogether the three men put in
$24000. Wliat amount is furnished by each ?
13. A man and three boys did a piece of work for $16.
How should this money be divided among them, if we
suppose that the man did twice as much work as each
boy?
lO FIRST PRINCIPLES.
14. A man has a farm of 240 acres, of which there is
twice as much marsh land as wood land, but the rest of
the farm is 3 times as large as the marsh land and wood
land taken together. Find the number of acres each of
marsh land and wcod land in the farm.
15. Divide $1100 among A, B, and C so that A may
have twice as much as B, and B three times as much
as C.
16. A man raised 1730 bushels of grain, of which there
was 3 times as much oats as wheat, and 2 times as much
corn as oats. Find the amount of each kind that he
raised.
17. On a certain day a storekeeper took in $3G0, of
which there was 4 times as much in bank notes as in
coin, and 5 times as much in silver as in gold. Find the
amount of each kind of money that he received.
18. Of three brothers the middle one is 4 times and
the oldest one 5 times as old as the youngest. The sum
of the ages of the two oldest is 36 year?. Find the age
of each of the brothers.
ig. If each year I should double the money that I had
at the beginning of that year, in five years from now I
would have $63000. How much money have I now ?
20. There are three numbers, the sum of the last two
of which is 63. The second is five times the first, and
the third equals the difference between the Jfirst and
second. What are the numbers ?
EXERCISE C.
Expressions in which Several Letters ars used to stand
FOR Numbers.
11. We have already learned that any letter may be
used to represent any number, but it often happens that
EXPRESSIONS WITH SEVERAL LETTERS. II
different numbers occur in the same question. In this
case different letters may be used to represent these num-
bers, but here, as before, any 07ie letter must represent
the sa7ne number throughout question .an^ answer.
P^ind the value of the following expressions, if «=5,
^=4, ^=8, ^=2, ^=1:
1. 'da-\-2b-\-Ze—2e, 4. ha—b—c—Zd—Q,e.
2. Ae+?^a—U+bc. 5. I0e—2e+^b—a^d.
3. 2a-\-'ib-e-c+l. 6. 20a^-6^-2«.
12. Just as Q>xb is v/ritten 6^, so if we wish to ex-
press the product of two numbers represented by a and b
respectively, we would write it merely ab, instead oi axb.
Also abe means just the same as axbxc. This custom
of omitting the sign X between a figure and a letter, or
between two letters, is universal in Algebra.
Find the value of each of the following six expressions,
if fl;=2, ^=4, r=6, d=^^, e=d :
7. 7ae-\-Sbe+de. 10. 4abcd-^Sbcd—cde-\-5d—S-^.
8. Sabd—cd—ab+2. 11. bce+idab— bade +10— be.
9. 2bed-\-See—a—bbde. 12. abede+10 + bed+9—dee-\-8.
Find the value of each of the following eight expres-
rions, if a==6, b=^, e=4, n=\, s=l, t=^ :
13. 10at—2ens-\-6^—at. 17, aa-\-ab+ast-\-ee.
14. ab-\-be?it—4:S-\-20?i. 18. eee-\-aa-^ss—4ae.
15. 3et—7eii-\-2e7i—abt. 19. aa—2a-\-eee—Se-\-ssss—is.
16. Ses— 2 be— et-\- 12^. 20. aaee— 500— 7teee—6ab.
21. What is the value of <^<^^, if <^= 3? if ^=5? if/^=i?
if^=i?if^=|?
22. If <2=2, what is the value of aa ? of aaa ? of aaaa ?
of aaaaa ? of aaaaaa ?
12 FIRST PRINCIPLES.
13. When a product consists of the same number re-
peated any number of times, the product is called a
Power of that number, and is usually written in a
simplified form. Thus :
aa is written a^ , read a square or second power of a\
aaa is written a"^ , read a cube or third power of a;
aaaa is written a*, read a fourth ox fourth power of a\
and so on.
The small figure written above and to the right of a
number is called the Exponent or Index of the power ;
it shows how many times the number occurs in the
power.
According to this notation, a"^ would mean that a is used once as a
factor, but when a number is used only once as a factor it is cus-
tomary to omit the exponent altogether and write simply a instead
of «!.
23. Write 3 times x square ; 5 times y cube ; 5 times
the fourth power oi b \ 7 times the fourth power of (^ ; 9
times the fourth power oi b\ a times the fourth power of ^.
24. How would you abbreviate bbxxx ?
Just as ay^h is written ab, so b*Y.x^ is written b'^x*.
25. Write in the abbreviated form expressions 17 to
20 inclusive.
26. Find the value of ^a'^b'^-x^, if «=4, ^=3, and
x=^\.
Find the value of each of the following five expressions,
where a=2, <^=3, ^=1, ^=6 :
27. «2 4.32^^2_|.^2^ 29. a^J^b^—c^.
28. 2^2+3^^-4^^ 30. '^abc—b'^c—iSc^.
31. a3^3«2^ + 3«^2_|_^3^
14. When several numbers are multiplied together
the result is called the Product, and each of the numbers
EXPRESSIONS WITH SEVERAL LETTERS. I ^.
or the product of any number of them is called a Factor
of the product. Thus, if 5, a, b, b are multiplied to-
gether the product is oab- , and the factors of the product
are 5, a, b, b"^, 5a, 5b, ab, 5ab, 5b'^, ab'^.
Any factor of an expression is called the Coefficient:
or Co-factor of the product of the remaining factors..
Thus, in the expression a'^bc, a"^ is the coefficient or co-
factor of be, a'^b is the coefficient of r, ab is the coefficient
oi ac, a'^c is the coefficient of b, and a is the coefficient of
abc. Similarly, be, e, ae, b, and abe are the coefficients of
a'^, a'^b, ab, a'^e, and a, respectively.
When a product is made up partly of numbers repre-
sented by figures and partly of numbers represented by let-
ters, as 2 X ^ab'^, it is sometimes convenient to distinguish,
between these two kinds of factors. In this case we call
the numbers represented by figures nitmerieal factors and
those represented by letters literal factors.
Of course the product of all the numerical factors is the
coefficient of the product of all the literal factors, and the
former is often referred to as the Numerical Coefficient.
32. In abx"^ what is the coefficient of x'^l of bx'^1 of
ab"^. of abx? of a ?
33. In 12abn what is the numerical coefficient? of
what is it the coefficient ? What is the coefficient of bn ?
of2b?i} o{Aab?i? of 3?
34. Write five different factors of 15edx^, and tell
of what each factor named is the coefficient.
35. Write five different factors of ^a^?tj'^, and tell of
what each factor named is the coefficient.
15. In Algebra, as in Arithmetic, the symbol for
Divided by is -r-.
14 FIRST PRINCIPLES.
More often, however, division is represented by means
of a fraction where- the dividend is written above the line
and the divisor below.
Thus, a-^d is written 7-, and v/hcn written in this form
it is often read "a over 3."
Write the following in the usual algebraic notation, as
explained in Arts. 10, 13, and 14.
36. A known number divided by G.
37. A known number divided by anotjier knov/n
number.
38. An unknown number divided by three times a
known number.
39. Four times the square of a knoAvn number divided
by 5 times the cube of an unknown number.
40. Twice the square of a known number times the
cube of an unknown number over the product of two
unknown numbers.
Find the value of each of the following expressions,
if ^=2, 3=5, m=S, s=-l ^=|:
8ab . 403 ^ ah?i , 3 ^
ow
3m b has
^2 in 9^-/ , b'^s^in , 3
42. V-f. 47. -^ — V — r f-T
171 b 2 5 4
VI b _ 1 ^ 1 , , tm
43. —„- + —. 48. ^a■'m — ^ab^
a^ in L 6 a
a'^ b ^2 3 3^2^
a'' b a t T)
^ b a^ s" in 12
CHAPTER II.
UNION OF TERMS AND REMOVAL OF
PARENTHESES.
EXERCISE 7.
Union of Similar Terms.
16. When an expression is broken in parts just before
each of the signs + and — , each part thus formed is
called a Term of the expression. Thus, in the expres-
sion Sad-^Ac"^ ~2e—lo, the terms are
Sad, -i-ic\ -2e, -15.
We speak of the terms of an expression in a manner somewhat
analagous to the way in which we speak of the syllables of a word.
A syllable may not convey an intelligible idea when taken by itself,
but when joined with other syllables to make up a word, the whole
word does convey a definite idea. So a /tv;// taken by itself may not,
at this stage, express any idea, but the whole expression does convey
an idea. See Art. 7. Indeed, each term would, even now, convey a
definite idea were it not for the sign -|- or — which always goes with
each term after the first.
17. The terms of an expression which have the sign
-|- or no sign at all are called Additive Terms, and those
terms which have the sign — are called Subtractive
Terms. Thus, in the expression n-i-2d—4a + 5a—7d,
the terms ;^, -\-2d, and -{-oa are additive terms, and —4a
and —7d are subtractive terms.
In comparing several additive terms they are spoken
of as terms of the same s(^n, and several subtractive terms
are also spoken of as terms of the same sign ; but when
1 6 REMOVAL OF PARENTHESES.
additive and subtractive terms are spoken of together,
they are said to be terms of imlike or opposite signs.
Notice that when we speak of the signs, without any further quali-
fication, that it is only thejirst tivo of the four fundamental signs of
algebra, -\-, —, X, and -f-, that we have reference to.
18. Terms whose literal parts are identical and whose
signs and numerical coefficients may or may not differ are
called Similar Terms. Thus, in the expression 'i)a-d—
?>ad^—a~d-\-4.ad'^, the terms da'^d and —a'^b are similar;
also the terms —Zab'^ and ■\-\ab''-' are similar; but ^a'^b and
+ 4a^^ are 7iot similar.
1. Is the expression 12 + 2—8 + 5—4
the same as 12 + 2 + 5—8 — 4 ?
Is the last equal to 19 — 8—4 ?
Is this the same as 19 — 12 ?
2. Is the expression 2 + 4—5 + 20—14 + 11
the same as 2 + 4 + 20 + 11-5-14?
Is the last equal to 37 — 5 — 14?
Is this the same as 37 — 19 ?
3. Is the expression 8^— 2a— 4« + 5«— 2«
the same as 8^ + 5^-2^— 4a— 2a?
Is the last equal to 13a— 2a— 4a— 2a ?
Is this the same as 13a— 8a ?
4. Is the expression ;^ + 5a^^ — 2a(^^ + ISaZ)^ — 7 — 12a<^2
the same as n^hab'' ^Ihab"- -lab'' -\1ab''^ -1 1
Is the last equal to 7i-\-%)ab''- -lab'' — Xlab'' -1 1
Is this the same as ;^ + 20a^2 — 14a<^- — 7 ?
5. In a7iy expression whatever^ will the value of the ex-
pression be changed, if all the additive terms are written
first and the subtractive terms following ?
6. In any expression, can any number of similar addi- .
tive terms be replaced by a single additive term ?
UNION OF SIMILAR TERMS. 1 7
7. In any expression, can any number of similar sub-
tractive terms be replaced by a single subtractive term ?
8. A man had n dollars. He gained 23 dollars and
then lost 9 dollars. How much did he then have ?
How much more is /2-f 23— 9 than 71 ?
Write an expression of two terms which shall be equal
to« + 23-9.
The two terms +23—9 can be replaced by what single
term?
9. A man had n dollars. He gained 9 dollars and then
lost 23 dollars. How much did he then have ?
How much less is w-j-9— 23 than n ?
Write an expression of two terms which shall be equal
to w + 9-23.
The two terms -|-9— 23 can be replaced by w^hat single
term ?
10. How much more is « + 52-21 than a?
Write an expression of two terms which shall be equal
to« + 52-21.
The two terms +52—21 can be replaced by what single
term ?
11. How much less is fl; + 21— 52 than «?
Write an expression of two terms which shall be equal
to« + 21-52.
The two terms +21—52 can be replaced by what single
term ?
12. How much more is 100+17^-5^ than 100?
Write an expression of two terms which shall be equal
to 100 + 17«-5«.
The two terms ■\-VJa—ha can be replaced by what sin-
gle term ?
1 8 REMOVAL OF PARENTUESES.
13. How much less is 100 + 5a— 17^ than 100?
Write an expression of two terms which shall be equal
to 100 + 5^— 17a.
The two terms + 5a— 17a can be replaced by what sin-
gle term ?
14. How much more is n-\-2da—lla than ;? ?
Write an expression of two terms which shall be equal
to ;z + 29a — 11a.
The two terms + 20a — 11a can be replaced by what sin-
gle term ?
15. How much less is ;z + lla— 29a than n?
Write an expresGion of two terms v/hich shall be equal
to ;^ + lla— 29a.
The two terms + 11a— 29a can be replaced by what sin-
gle term ?
16. In any expression whatever, can two similar terms,
one additive and one subtractive, be replaced by a single
term?
What kind of a term is this, if the numerical coefficient
of the additive term is the greater ?
What kind of a term, if the numerical coefficient of the
subtractive term is the greater ?
19. The above questions suggest the following prin-
ciples :
/. T/ie order of terms in an expression is not fixed, the
value of an expression being U7ichanged if all the additive
terms be written first, followed by the subtractive teims.
II. Any number of similar additive terms in an expres-
sion may be replaced by a single additive teini, and any
number of similar subtractive terms may be replaced by a
single subtractive term.
EXAMPLES. 19
///. Two similar terms, one additive and one snbtractive,
may be replaced by a single term in which the munerical
coefficient is the difference of the numcidcal coefficients of the
two ter77is, and the sign is -\- or — the same as that 07ie of
the displaced terms havi^ig the gr'eater numerical coefficient.
EXERCISE 8.
Examples.
In each of the following expressions combine the similar
additive terms into a single additive term, and the similar
snbtractive terms into a single snbtractive term. After-
wards combine these tw^o terms into a single term.
1. 7^— 3a— 4« + 6«. 4. \^^-\-Zab—1ab-\-hab—Zab.
2. 1U-— 5;»:-f8;t:— 6;i;. 5. n-\-12y—lSy-{-2y—6y.
3. 50-6^+2^-20^+8^^. 6. 8^-3^+7^-9^-4^+8^.
7. a + 20xy—n0xy-o0xy-\-10xy.
8. a + 9a2-2«2_^5a2 + 8a2_7«2_9^.2^
9. 37 — 10m-\-nm—15m—Sm.
10. r>m—4:7n-^S?n—12m—77n-\-10m.
11. 10x^-7x^-j-Sx^ + Sx^-12x^-x\
12. 125 + 8j2_8_>'2^-6y2_20>/2_4r2^
13. 10 + 5a2^2_^2^2_7^2^2^5^2^2_^2^2^
14. n-\-20x'^y—6x''^y+x'^y—nx'^y—10xy.
15. 100+13j;i;2-23jjr2+33j/jt:2-43j/Ji'2.
16. Ga— 18^2^+4<^V-^V+10^V.
1 7 . 50a bed— 11a bed— Ala bcd-\- Sa bed.
18. ^x—^^x-\-x-\-fX.
^X - { -r+X+l X = 1 ^+|-;c+X - J T,
=^-^+«^+-6-^-ii^,
=-V--^-|-^. etc.
19. G + -Ja+|a-ia. 21. 5b+lb-ib-{-2b-U+Vj.
20. 1— y^a + |a + ia— a. 22. 10a—Sb+ib+^b—b—\b.
20 REMOVAL OF PARENTHESES.
Shorten the following expressions as much as possible
by a careful grouping and uniting of similar terms :
23. 99997 + 83752 + 3.
24. 9999 + 9998 + 9996 + 9095+4+2+1+5.
25. 17 + 18 + 19 + 20+21 + 22 + 23.
26. Sa + 7d-h5a. 23. 24/ + 30^+17^+35/.
27. x-i-oy+Sx. 29. Ufi-\-27ad-\-8n-^lSad-^l(j?i.
30. Sa-h^d'' +Sdc-i-Gdc-\-17a + lU'- -i-dd^ +2a+dc.
31. 5^ + 3^—3^. 36. 7a-j-6d—Aa.
32. 9^ + 3«— 9^.* 37. 9a--bd-\-a.
33. 4a— .r+A-. 38. 20— 7.r+9.
34. 84589 + 8783-4589. 39- 57-7a-9.
35. 28654 + 9999-18054. 40. 30+6r+5.
41. 18«-16^^+10a+14^r-16^-2<^.-
42. lQa''—Sad-Sad-6a--hl2ad.
43. 2xy—^2-\-\Qxy—82-\-1^2—12xy,
44. 8w— 10A' + 6;z— 4/5— 2;z + 2w— 8/5— 6w.
45. 5« + 8/iV-7^-2«-9<^V+2^-2a + 2^V+6^.
46. lOw + n— 5jr— 12— 4w— 3;t-+l + 9x— 5;;/.
47. 9a-7^+3^-8a + 7^-3r-5/^-8r.
48. ^a^b^c-7a^b''c''-^10a^b''c''-ha^b^c,
49. llxy+2ab—Axy—2bab-\-ab-]-10,
50. 25-25jc+2rj^'+13-30j^/+20ji;-8.
51. ;z2_2 + ^2^2-16r2+9;i24.5_|.^2^
52. lSx—5y+8z—5x+9y—llz—'6x—6y+2,
53. 54w— 62/Z + 18X— 62w— 6;r+42« + 10w+18w— 14j»;
54. 10;;z + ll-5«2_l2 + 4w-6a2 + l + 18a2_i0;;i.
55. ^-2/^^+18^^— 14^^-21=4, ^=3, and r=2 :
22 REMOVAL OF PARENTHESES.
5. (w-Ox(/ + ^)-r. 7. w + (/-/)-(^+r).
6. ^{rii^t^p)^{q-r). 8. {^m-t)-{p^-q-\-r),
9. w + 2(/-/')-(^-r).
10. (;;22 — /'2)_(^ + 2^) + r2.
11. 2(;;^2_/2) + 1(^4.2^) -f;.2.
Find the value of the following three expressions,
where ^=1, ^=2, <:=3 :
12. (2^ + ^)2+4(2^-2^)3. 13. 3(a + 3)2-(^ + r)2.
14. Qi{a-\-U)c—{b-\-c)abc.
EXERCISE 10.
Removal of Parentheseg.
GENERAL FORM a-\-{b-\-c).
1. How much greater is 10 + (7 + 3) than 10 + 7 ?
2. How much greater is 12 + (7 + 3) than 12 + 7?
3. How much greater is 25 + (7 + 3) than 25 + 7 ?
4. How much greater is « + (7 + 3) than a-j-7 ?
5. How much greater is « + (8 + 3) than a-j-8?
6. How much greater is « + (<5+3) than a-\-b}
7. How much greater is ^ + (^ + 5) than a-j-d?
8. How much greater is a-\-(d-^c) than a-^-d?
9. How much must be added to « + ^ to equal « + (^- /)?
10. Write an expression equal to a + (d-\-c) without
using a parenthesis.
GENERAL FORM a-^[d—c).
11. How much less is 10 + (7-3) than 10 + 7 ?
12. How much less is 12+ (7— 3) than 12 + 7 ?
13. How much less is 25 + (7—3) than 25 + 7 ?
14. How much less is « + (7 — 3) than a + 7?
15. How much less is « + (8— 3) than « + 8?
REMOVAL OV PARENTHESES. 23
16. How much less is a + (/^— 3) than a-{-d? Why?
17. How much less is « -I- (<^— 5) than <2 4-^? Why?
18. How much less is a-^(d—c) than a-\-d?
19. How much must be subtracted from a-\-l? to equal
20. Write an expression equal to a-{-(d—c) without
using a parenthesis.
GENERAL FORM a — [/>-\-c).
21. How much less is 10-(7 + 3) than 10—7?
22. How much less is 12 — (7 + 3) than 12—7?
23. How much less is 25-(7 + 3) than 25-7 ?
24. How much less is «— (7-f 3) than a— 7 ? Why?
25. How much less is a— (8-;-3) than a—S? Why?
25. How much less is «—(^ '1-3) than «—^? Why?
27. Plow much must be subtracted from a— I? to equal
a-(^ + r)?
23. How much less is a—(d-\-iJ) than a—d"?
29. How much must be subtracted from a — d to equal
30. Write an expression equal to «— ((^+5) without
using a parenthesis.
31. How much lees is a—(id-\-c) than a—d? Why?
32. How much must be subtracted from a — d to equal
a-(d+c)?
33. Write an expression equal to ^— (/> + <:) without
using a parenthesis.
34. The additive terms d and -i-c within the parenthesis
occur as what kind of terms when the expression is written
without a parenthesis ?
24 ' REMOVAL OF PARENTHESES.
GENERAL FORM a — [b — c).
35. How much greater is 10— (7— 3) than 10—7?
36. How much greater is 12 — (7— 3) than 12—7?
37. How much greater is 25— (7— 3) than 25—7?
38. How much greater is a—(J — Z) than a— 7 ?
39. How much greater is «— (8— 3) than «— 8 ?
40. How much greater is a — (^b—Z') than a—b}
41. Hov/ much must be added to a—b to equal
«-(/^-3)?
42. Write an expression equal to a—(b—o) without
using a parenthesis.
43. How much greater is a — (ib—b) than a—b} Why?
44. How much must be added to a—b to equal
fl-(^-5)?
45. Write an expression equal to a—(^b—b) without
using a parenthesis.
46. How much greater is a—{b—c) than a—b} Why?
47. How much must be added to a—b to equal
a-{b-c)}
48. Write an expression equal to a— (<^—<:) without
using a parenthesis.
49. The additive term b and the subtractive term —c
withi7i the parenthesis occur as what kind of terms when
the expression is written without a parenthesis ?
REMOVAL OF TERMS FROM A PARENTHESIS PRECEDED DY THE
PLUS SIGN.
50. When the sum of several numbers is to be found,
increasing one of the numbers has what effect on the
sum ? Decreasing one of the numbers has what effect on
the sum ?
PARENTHESIS PRECEDED BY PLUS SIGN. 2$
51. How much less is a-\-{b-\-c—d—f) than a-\-
{b+c-d+e-f) ?
52. How much must be added to a-\-(^b-\-c—d—/) to
equal ^ + (^-f<:~^+^—/) ?
53. Write an expression equal to a-\-{b-\-c—d-^e—f),
where e appears outside the parenthesis.
54. Can c be thus written outside the parenthesis in-
stead of ^ ?
55. Can a7iy additive term be thus taken outside of a
parenthesis ?
56. How much greater is a-\-(^b-\-c—d-\-e) than a-\-
{b + c-d^e-f)}
57. How much must be subtracted from a 4- (,b-{-c—d-^e)
to equal a-\-{b-\-c—d-\-e—f') ?
58. Write an expression equal to a-\-{b-\-c—d-\-e—/)^
where —/appears outside the parenthesis.
59. Can —d ho. thus written outside the parenthesis
instead of —fi
60. Can a7iv subtractive term be thus written outside
of a parenthesis ?
61. Can ^wjrterm, additive or subtractive, be taken out
of a parenthesis ?
62. Can any tzvo terms be taken out of a parenthesis
one after another ?
63. Can any number of terms be taken out of a paren-
thesis ?
64. Can all the terms be taken out of a parenthesis ?
21. These questions, if rightly answered and under-
stood, lead to the fact, that when a parenthesis is preceded
by a -\- sign the parenthesis viay be erased and the value
will not be changed.
26 REMOVAL OF PARENTHESES.
REMOVAL OF TERMS FROM A PARENTHESIS PRECEDED BY
THE MINUS SIGN.
22. When the difference of two numbers or expressions
is to be found, the number or expression from which some-
thing is subtracted is called the Minuend and the num-
ber or expression subtracted is called the Subtrahend
and the result is called the difference, or Remainder.
65. When the difference of two numbers is to be found,
increasing the subtrahend has what effect on the dif-
ference ? Why ?
Decreasing the subtrahend has what effect on the dif-
ference ? Why ?
66. How much greater is a—{b-\-c—d—f^ than a —
{b^c-d^e-f) ?
67. How much must be subtracted from a—{b-\-c—d—f)
to equal a—{b-\-c—d-\-e—f) ?
68. Write an expression equal to a—{b + c—d-\-e—f)y
where -{-e does not appear in the parenthesis.
69. When the additive term -f ^ is taken out of the
parenthesis, it becomes what kind of a term ?
70. Can -f<^ be removed from the parenthesis instead
of -f^? If it is so removed, what kind of a term does it
become ?
71. Can ariy additive term be thus removed from a
parenthesis preceded by a minus sign ?
What kind of a term does it become when it is thus
removed ?
72. How much less is a—{b-{-c—d-\-e) than a—
{b+c-d-^e-f) ?
73. How much must be added to a—i^b-^c—d-Ve) to
equal a—{b-\-c—d-{-e—f)'^
PARENTHESIS PRECEDED BY MINUS SIGN. 2/
74. Write an expression equal to a—{b-\-c—d-\-e—f^^
where the term —/does not appear in the parenthesis.
75. When the subtractive term — / is removed from
the parenthesis, it becomes what kind of a term ?
76. Can — ^ be thus removed from the parenthesis in-
stead of — y? When thus removed, it becomes what kind
of a term ?
77. Can any subtractive term be thus removed from a
parenthesis preceded by a — sign ?
78. Can any term, additive or subtractive, be removed
from a parenthesis preceded by a — sign ?
79. Can any iivo terms be removed, one after another,
from a parenthesis preceded by a — sign ?
80. Can any number of terms be removed from a paren-
thesis preceded by a — sign ?
81. Can all the terms be removed from a parenthesi.s
preceded by a — sign ?
82. When all the terms are removed from a parenthesis
preceded by a — sign, the additive terms become what
kind of terms ? The subtractive terms become what kind
of terms ?
23. These questions, if rightly answ^ered and under-
stood, lead to the fact that whenever any parenthesis is pre-
ceded by a — sign, all the terms within the parenthesis may
be taken out, provided that in doing so each additive term
be changed to a subtractive term and each subtractive
term be changed to an additive term ; or, what comes to
the same thing, whenever a parenthesis is preceded by a —
sigyi, the parenthesis may be erased, provided all the terms
within the parejithesis be changed fro7n additive to subtrac-
tive or from subtractive to additive as the case may be.
28 REMOVAL OF PARENTHESES.
Changing all the additive terms of an expression into
subtractive, and all the subtractiveinto additive, is spoken
of in algebra as changing all the signs of the expression,
although it will be observed that the first term of an ex-
pression has no sign at all. While this latter way of
speaking is not quite so definite as the former, still it is
much shorter and is very convenient.
24. We have already considered the case of paren-
theses, preceded by the sign -f and also by the sign — ,
but when a parenthesis stands at the beginning of an ex-
pression, it is usually not preceded by any sign at all.
This case, however, is so simple that it is very easily
settled.
In an expression like
a-{-b—c—d
we are first to add together
the numbers a and b, and
from this sum subtract c,
and then from this differ-
ence subtract d.
In an expression like
{a + b—c)—d
we are first to add together
the numbers a and b, and
from this sum subtract c,
and then from this differ-
ence subtract d.
From these two statements, which are exactly alike,
it is evident that when a parenthesis stands at the begin-
ning of an expression, the expression means exactly the
same thing that it would if the parenthesis were erased,
and from this it is evident that when a parenthesis is pre-
ceded by 7io sign at all, the parenthesis may be erased, just
the same as though the parenthesis were preceded by the
sign +, or in other words, when a parenthesis is pre-
ceded by no sign at all, it is treated just as though it
were preceded by the sign +.
MISCELLANEOUS EXAMPLES.
29
EXERCISE 1 1.
Miscellaneous Examples on the Removal of Parentheses.
Write each of the following expressions without using
any parentheses :
1. .;,7 + /^^-(/-f^^-r). 4. (^a^b-^c)-{q-r).
2. m + n+p-{q + r). 5. a^{a''-b)-{b'' +c).
3. {a-b) + {p-q)-\0. 6. a-b-^a'' ■^b--c'').
8. (^ab-a)-\-{;dab'' -2cd)~(cd'^ -a"-).
9. {Am-2n) — (?,p-2q + r).
10. m — {a-\-b) — {c—d).
12. (w--«2)-(/>+2^) + (6r2-fa/^-^3).
13. (a2_^^2^«^)-(^z;+5/^r-^/^r).
14. (lax— ^by) — {^ax— by') — 10.
15. C)a'^x-(lZby'^—babxy'-''dy^)^(Jx^-^y'^).
16. 21-(5-8y+13>/)-(6 + 16j'-15,r).
17. 75-(15-8)/+2l7)-(30+80>/-lo.v2).
18. (3«a--5y)-C2«y-|.r) + r75'-3)0.
CHAPTER III.
ADDITION.
EXERCISE 12.
Addition of Expressions.
25. When two or more expressions are to be added
together,. each expression may be enclosed in a parenthe-
sis, these parentheses written one after another, separated
by plus signs. Thus, if
a -^2, Oil — 1 and a-\-3
are to be added, w^e would indicate the sum by
(a + 2) + (3«-l) + (^4-3).
These parentheses ma}^ now be removed, and the sum in-
dicated thus : « + 3a + a + 2 — 1 + 3
which, by uniting the terms, may be written
5^ + 4.
I^et us find the sum of the three expressions
x-\-y, x—z and "Ix-^-Zy—^,
First, we enclose these in parentheses, write them one
after another separated by plus signs, and get
C-^+j) + {x-z) + (2.r + 3j/-2).
Second, we remove parentheses as in exercise 10, and get
x^-y-\-x—z^2x-\-?>y—2.
Third, we arrange these terms so that similar terms shall
come together, and get
x-\-x^1x-\-y-V'^y—z—1.
Fourth, we unite each group of similar terms into a single
term and get 4x-{-4y—z'—2,
and this is the simplest form possible for the sum of the
three given expressions.
ADDITION OF EXPRESSIONS. 3 1
26. It is easy to see that we could find the sum of
any number of expressions, whatever those expressions
may be, in a manner similar to that just pursued, viz. : en-
close each expression in a parenthesis, write these paren-
theses one after another separated by plus signs, and then
remove parentheses from the expression. Next arrange
the terms of the expression thus found, so that similar
terms shall come together, and then by uniting each
group of similar terms into a single term, we obtain the
simplest form possible for the required sum of the given
expressions.
EXAMPLES.
Find the sum of the following expressions :
1. x—?yy—4, 2x—ijj'--(j and ;«r-f 3y+12.
2. 3/2--2«<^-f5/^2^ a^—Aad—3d^ and a^-\-d'\
3. 4jry-_>^2_}.23, 6jrv+2y_50 and Sy--xy-2.
4. Jt:2-f2.rj'4-r-, x^--2xy-hj''' and x--j\
5. Gw— 4^/— r2, 2^/+2w-fG?'2 and li)s^—7r^-—m.
6. 3jr2+^j-f3>/2, x'^-Sxj+y- and ?>x''-\-^j\
7. a + b, 12c-j-Sd, Qd-4d and lOa-^d.
8. a-i-U, d+4c, c+4d and d-h4e.
g. oxy-\-2y, hxy—x, Sx—iry and 7xy—x—2y.
10. Sa—4d—ecd+2e, 10/^+3^— lOr^ and 9a —20^4- ^W.
11. (yc-{-7 — 4a—5d, ijd-\-7c—4—5a, and Ga + 7d—4c-o.
12. a^-^-P, a^-d^ and ad^-+aH-\-d\
EXERCISE 13.
Arrangement of Work in Addition.
27. In finding the sum of two simple expressions by
the method already learned, we place these expressions
in parentheses, writing them one after another, separating
32 ADDITION.
them by plus signs. But evidently it comes to the same
thing if, instead of writing the expressions one after an-
other within parentheses, we write them one below another
without parentheses, arranging the similar terms in the
same vertical column. We may then draw a line under
the last expression, and the example is arranged in ex-
actly the same form as in Arithmetic. Now the similar
terms in each column may be combined into a single term,
as we alread}^ know, and this term placed under the line
as one term of the sum. When all the columns are thus
treated all the terms of the sum are found. Thus, sup-
pose we are required to find the sum of
^a'^-^W-hcd, 6«2_2^2 and W-W-A^cd.
By the former method we write
Removing parentheses we get
2a2+3^2_5^^^(3^2_2^2_|.§^2_4^2_4^^^
Arranging similar terms one after another we get
2a2^e)^2_4^2_^3^2_2^2^8^2_5^^^_4^^^
Uniting similar terms, we get
By the latter method we write
2a2 + 3^2_5^^
6a2-2^2
--4a2 + 8^2_4^^
4^2_^9^2_l9^^
Now it is very evident that we have here exactly the
same terms to combine that we had by the other method,
after the parentheses had been removed, and the similar
terms brought together. Moreover, it is apparent that
these terms are combined in exactly the same order, giving
EXAMPLES. 33
of conrre the same result as before; and the only dif-
ference between the two methods consists in the arrange-
ment of the work. The similar terms being rather easier
to combine in the second arrangement, it is the more con-
venient arrangement for the beginner.
EXERCISE 14.
Examples.
I. 2.
«2-f ab-^ b"- 5ab-\-Gbc—7ca
W-?yab-lb'' ^ab-Abc+Zca
ia^+nab-^db ^ 2ab— bc-^bca
3- 4.
2x'^ — '2xy-\-^y'^ hax—lby-{- cy
Sx^--\-hxy-\-4y^ Sax-\-dby-^cy
x'^—2xy—^)y'^ ax—^by— cy
5. Find the sum of 5/;-f4/+3?^, 2/^-}-2/^-^^, and
7/^ + 3/+ 4/^.
If // be supposed equal to 100, / equal to 10 and u equal to unity,
this example illustrates nicely the analogy between the work the stu-
dent is now doing and the ordinary addition of numbers in Arithmetic.
Thus,
Hundreds. Tens. Units.
5^4-4/-f3« 5 4 3
2//-J-2/-I- w 2 2 1
1h\-V-\.\u n_ 3 4
14//+9/+8« 14 9 8
6. Arrange and add x-hy+^, 2x-\-?yy—22, and 3a-
7. Add 14«-6^-h3^~5arand S)a + lb-Ac-S)d.
8. KM a-\-b—c-^d Tind a—b—c^?yd.
9. Add ba'^-Zab + lb'', la'^^iab-bb'' , and a''~S)ab
+ 'ib\
34 ADDITION.
10. Add a-\-2d-\-oc, 2a—d—2c, b—a^c, and c^a — h.
Arrange thus: «-f-2<^-|-3^
la— b—1c
_ «_|_ /;_ c
_ a- /,-!- c
11. Add 6/5+8r— 5«, 8«— 3/;+4r, and 7Z^— 15r— 2^.
12. Add 3;/-f 2r+3^— 4/, 3r— 4^— 5/— 2«, and 5^—0/
^-12;^-10r.
13. Add x-\-Za-^2b-c, 2j-Sl?-\-2c+a, and 3^+3^
-2a-d.
3a-\-2/'- c-\-x
a-'6b-\-1c +2y
-la- b^?yc 4-32
14. Add 3ji:3-4jt;2-.r+7, 2;r3+Jt:2-f 3;tr-10, 2;tr2
-7;t'3-2ji;-14, and Zx^-\2x''A-\2^-hx.
In arranging expressions which involve different powers of the
same number, it is usual to place all of the terms containing the
highest power of the letter in the first column, all the terms con-
taining the next highest power in the next column, and so on. Thus,
this example would generally be arranged in this way:
S-r'— 4jc8— x-\- 7
-7^3_j_ 2jc2-2-jr-14
15. Add .r3+4jt:2 + 5.r-3, 2x^—7x'^ — Ux-h5, and
^2_^3_2 + i0jr.
16. Add Sx^—ix^-^x*, x^-{-x'^+x, 4x^-i-5x\ and
2x''-Sx-4:x\
17. Add 5x^-Sx^-hZx-8, x^--Sx^-\-Sl, 2x^-8x
-5.^2+2, and ix+lx^-d-SxK
18. Add Sx^-4xj^-\-y^-\-2x-\-Sy, lOxj-i-Sy^+dj^, and
bx''-Qxy+Sy^-\-Jx-7j^.
19. Add 4ad^+icde-6/i\ 2ad'--\-3/i^, dcde-7ad\
and ab''-^2cde-'5/i^.
EXAMPLES. 35
20. Add ^xy-V^ode-lfg, S/g-2xy, Sxj-Sde, Sde
~-^xy—2fg, and ^/g—2xy.
21. Add U'^a^-1aH+^ab''-^l^b^ 2inA 1\a^-2\a''b
-4ab''-12b\
22. Add 9ir+3f^+7.75 and 7ic-Uj%d-S.
23. Add|« + ^, ib-ia, ^aSb, and 3^-9^^.
24. Add a^+SaH-{-oab'^-\-b\ a^-3aH-hSab^-b^,
b^-3ab''-i-SaH-a'\ and a^-h6aH + Gab''-\~bK
25. Add 12j/—5a—7ax, 5ax-\-a—Sy, da—y—ax,
4ax—Sa-i-5y, and y-\-a—ax.
26. Add «3^5«/^2_^/!i3^ a^-10ab''-hb\ and 5a^2
27. Add x^—2ax'^+a'^x-^a^, x^-^Sax^, and 2«3
—ax'^—x^.
28. Show that if x=a + 2b—Sc, y=b-\-2c—Sa, and
^=f+2rt— 3^, then will x+y+z=0.
29. Add 3«4-2^— <:, Sb-\-2c-a, and 3^+2a— ^.
30. Add 2^-3^24.^2^ p^2c^ + Sd, and 3^3-2^-^.
CHAPTER IV.
SUBTRACTION.
EXERCISE 15.
Subtraction of Expressions.
28. When one expression is to be subtracted from
another, we may enclose each expression in a parenthesis
and separate the minuend from the subtrahend by a minus
sign. Thus, if « + 2 is to be subtracted from Sa—1, we
would indicate the difference by
(3«-l)-(« + 2).
These parentheses may now be removed, care being taken
that all the signs in the second parenthesis be changed
when the parenthesis is removed, and the remainder
written thus, 3^—1—^—2.
Now by grouping the terms, the same result may be
written 3^—^—1—2,
which by uniting the terms may b*^ written
2^-3.
which is the required difference.
Let us find the difference between
Sx''-4:xy+5y^ and 2x'^-2x_y-4j;\
First, we enclose each of these expressions in a paren-
thesis, writing the subtrahend after the minuend, with a
minus sign between them, and get
(Sx^-4:Xj^-\-5j/'')-i2x''-2xy-4y).
Second, we remove each of these parentheses, taking
care to change all the signs within the second parenthe-
sis, and get
dx^-4xy-j-5y-'2x'^-{-2xy-\-4y.
SUBTRACTION OF EXPRESSIONS. 37
Third, we arrange these terms so that similar terms
shall come together, and get
Fourth, we unite each group of similar terms into a
single term, and get
and this is the simplest form possible for the difference of
the two given expressions.
It is easy to see that we could find the difference of
any two expressions, whatever these expressions may be,
in a manner similar to that just pursued, namely, enclose
each expression in a parenthesis, write the minuend first
and the subtrahend second, with a minus sign between
them, and then remove parentheses. Next arrange the
terms of the expression thus found so that similar terms
shall come together, and, finally, by uniting each group
of similar terms into a single term we obtain the differ-
ence of the two given expressions in the simplest form
possible.
EXAMPLES.
1. From 17a-\-id—?>ctake ?>a-\-od—c.
2. From ^x-dy + 20^ take jir-4>'+3^.
3. What is the difference between (jd-\-25c and Qd—25c?
4. What is the difference between 8.r ^ H- 4j and Sx"^ —iy}
5 . From X-+ 2xy -\-y - take x - — 2xy -\-y ^ .
6. From4jf2 + 2;t:j4-o>'2 tixk^ x"^ - xy -\-2y'^ .
7. From 6j»;2-13a'-7 take 3.r2_^5jr-2.
8. From 2;»;— 11« + 10^— 5^— 23 take ba-^2c—10—U.
9. From 2A-'^+ji;2-35;r+49 take x^-2^x+A2.
10. From 4.x''-^x^-2x''+lx+% take x"^ -2x'' -2x'^
■~7a--9.
11. From a^-\-?>a''b^?^ab"'-\-b^ i2ikQ a^-^a'^b+Zab'^-b^
12. From 3;r+10 take 10— 3j/.
38 SUBTRACTION.
t
EXERCISE 16.
Arrangement of Work in Subtraction.
29. In finding the difference of two expressions by
the method ah'eady learned, we place each expression in
a parenthesis, writing the subtrahend after the minuend
with a minus sign between them, and then remove the
parentheses. But, evidently, it comes to the same thing
if, instead of writing the subtrahend with all its signs
changed after the minuend, we write the subtrahend
with all its signs changed below the minuend, placing
similar terms of the minuend and subtrahend in the same
veritical column, and then uniting similar terms exactly
as in addition. For example, if we wish to subtract
we arrange the work thus :
Minuend 9^2H-332_7
Subtrahend with signs changed — 2^2+4^^ -f 6
Remainder 1 a'^ -\-l b'^ — \
The signs of the subtrahend need not actually be
changed if the student will iraagine them changed as he
proceeds in the work.
EXERCISE 17.
Examples.
I. 2.
From 6:r-14ji/4-10 9^-43+8^
take 1x— ^y-\- 6 a-^1b^2,c
3- 4.
From 9«— 8^4-7c-\-4d^4e x+ y-{-hz—2 -\-n
II.
From 7?ya — r^2b—7\c-\-2\d—:)2x+\7y-\-r)^dz+n
take 54^— G0<^+81r+::]7^-f-18A-— :%>- t-99^-f 7
12. From 4x'^-\-2xj'+?>v' take Jt-— ji:)/+2:)'2.
13. From rt» + 3a2^+3«/^2_|.^3take«'^-3rt2^-f 3«^--/^''
14. From 2A-f ll^ + 10Z-5r-23 take 2^-10+5^-3^.
15. From ?>x^-2x'--\-?>x—4 take .^'3-4J»:2-8A-f 1.
16. From 72;r-*-78A-'^-10.r' + 17 take 2^-*+ 30x^-1 7.r
+ 10.r2.
17. From 3;»:'*+5A'*^-6.r2-7.r+5 take 2x*-2x''-h5x^
— Gjt- 7.
18. From 7;t:2— 8j»:— 1 take 5^-2— 6.r+3+.r2.
19. From4Ar*-3.r»-2A-2 — 7jt:+9 take .r* -2.^^-2.^-2
+ 7.r-9.
20 . From 5.^ 2 _^ g^^, _ 1 2 jr^ — 4j' 2 ^ake 2;i- 2 _ 7^^, _j_ 4^^,
21. From x^-\-Sxjy—j'^ ■^yz—2y'^ take x"^ -{■ 2xy + 5;r.^
22. From 7ji:*— 2.^2 + 2^-4-2 take 4jr3 — 2ji:2— 2.^-— 14.
23. From4.r3— 2.;r2— 2;*:— 14^ take 2jt:-'^— 8j»:2+4A-+a^.
40 SUBTRACTION.
24. From 3i«-24/!i 4-31^- 5^ take na + U-'i\c+Sd.
25. From (.)ay—5xy+2a'^x^ take 4jr;'— 3^) — a'^x-.
26. From J^ + J-^-f^-ll^+i take4cz-|^— V-^+W-l
27. From ^ + 2 take d—o.
28. From « + <^ take a-i-c.
29. From ^ + ^ take c-{-d.
30. From a — <^ take d—c.
31. From «"— ^^4-<^2 take Jtr2 4-^_)/+_>/2_
32. From 4al;y- — 5axy-^2a'^x" take «2;t:2_^^^_3^^^2^
33 . From 6x'^-\- 7xj —oy'^ — l 2xy2 — Sjz take 8xy — lyz
-\-Sx^—4y'^-\-6xy2.
34. From 2x-^na + 10d-rDc-2^ take 2c-10 + 5a-Hd.
35. From 4r^+62;^'^-26;;^-23;^2 take d?i''-2rs + 21m
•j-2n''-.Srs.
EXERCISE 18.
Additon and Subtraction of Equals.
1. How much less is x+i than ^+5? Ifjr4-5=12,
what does x-\-4 equal ? Why ? What does x-^S equal ?
Why ? What does x-\-2 equal ? What does x-\~l equal ?
What does x equal ?
2. If jt:+5=28, what does -r+3 equal ?^ Why? What
does x+1 equal ? Why ? What does x equal ? Why ?
3. If ^+30=50, what does ;r+10 equal? Why?
What does x-\-7 equal ? Why ? What does x+o equal ?
Why ? What does x equal ? Why ?
4. If jtrH- 12=44, write what x equals. What must you
do to each member of the equation ^+12=44 to get the
equation ;«;=32 ?
5. If ;r+ 23=48, write what x equals. What must you
do to each member of the equation ji;+23=48 to get the
equation you have just written ?
ADDITION AND SUBTRACTION OF EQUALS. 4I
6. If x-^27i—o7i, write what x equals. What must you
do to each member of the equation x+2n=6?i to get the
equation you have just written ?
7. If x-\-n=4:5, write what x equals. What must you
do to each member of the equation x-\-7t=4o to get the
equation you have just written ?
8. If x-\-?i—a, write what -T equals. What must 3^ou
do to each member of the equation x-^7i = a to get the
equation you have just written ?
30. In finding the value of x in the above equations
the student has made use of a ver}- evident mathematical
truth, which may be stated as follows :
If we take from equals the same niunber, or equal num-
bers, the remainders will be equal.
This is one of several equallj^ evident truths which are
known in mathematics as Axioms.
9. How much more is .r— 4 than x—b} If x—b=?),
what does x—\ equal ? Why ? What does ;r— 3 equal ?
Why ? What does x—2 equal ? Why ? What does .r-1
equal ? Why ? What does x equal ? Why ?
10. If ;r— 7 = 8, what does x—b equal? Why? What
does ji'— 3 equal ? Why ? What does x equal? Why ?
11. If ;«;— 20=25, what does x—lb equal? Why?
What does x—\0 equal? Why? What does x—4:
equal ? Why ? What does x equal ? Why ?
12. If .r— 11 = 13, write what x equals. What must
you do to each member of the equation .r— 11 = 13 to get
the equation .r=24 ?
13. If JT— 16=50, write what x equals. What must
you do to each member of the equation ;r— 16=50 to get
the equation you have just written?
42 SUBTRACTION.
14. If xSn=5n, write what x equals. What must
you do to each member of the equation x—S?i=5ji to get
the equation you have just written?
15. If x—n=15, write what x equals. What must
you do to each member of the equation x—7i = 15 to get
the equation you have just written?
16. If x—7i=a, write what x equals. What mUvSt you
do to each member of the equation x—7i=a to get the
equation j^ou have just written ?
31. In finding the value of x in the above equations
the student has made use of another very evident mathe-
matical truth, or axiom, which is usually stated as follows:
If we add- to equals the same number or equal 7iumbers^
the SU7US will be equal.
EXERCISE 19.
EXAMPLES.
Find the value oi x in each of the following equations,
by the addition or subtraction of equals :
1. ^_i9=32.
We have x- 19 = 32
Adding equals to each member 19 19
"Whence x=51
2. 5^+12 = 87.
We have 5-r-|-12 = 87
Subtracting equals from each member 12 12 '
bx—1b
Whence we know
x=Vo.
3. x-^ = ri.
7.
5.r+3=2S.
4. jt-+17=19.
8.
7x+5=26.
5. A-+21 = 69.
9.
5 + 6jr=29.
6. A-19=43.
10.
9.r-5=31.
TRANSPOSITION IN EQUATIONS. 43
11. ;i;4-140=191. 17. 194=ll;»;-26.
12. 18+^=35. 18. bx-\-2-hx=20.
13. ^—48=56. 19. 4ji:+54-7^+9— 8:r=16.
14. 25+x=38. 20. dx-hl2-6x-l^-{-2x=ld,
15. 2=;r-8. 21. 9A-+l?)-;t-=29.
16. 230=;»;-103. 22. 13:r+9-8a-=39.
EXERCISE 20.
Transposition in Equations
1. In the following, explain how each equation in the
second column can be obtained from the corresponding
one in the first column :
(1) x-h4=d. (V) .r=9-4.
(2) x+2c==5c, (2') x=5c-2c.
(3) x-\-2c=12. (3') x=12-2c.
(4) x+a==d. (4') x=:d-a.
(5) jt:-6=3. (5') x^n-i-i).
(6) x-2d=?yd. (6') x==Sd+2d.
(7) x-2d^l. (7') .r=.7 + 2^.
(8) x-a^b. (8') ji-=^-f^.
2. The additive terms -|-4, -f 2r, and +« from the /r/?
members of the equations in the first column appear as
what kind of terms in the right members of the equations
in the second column ?
3. The subtractive terms —6, —2d, and —aoi the left
members of the equations in the first column appear as
what kind of terms in the right members of the equations
in the second column ?
32. From the above work we learn the following
principle :
44 SUBTRACTION.
By the addition or subtraction of equals, we may cause a
terin to disappear from any member of an eqtiation and to
appear ivith its sign changed in the other 7nember of the
equation.
If we remove a term from one member of an equation
and make it appear in the other member, we are said to
Transpose that term. If we nse this word we maj^ re-
state the above principle as follows :
A7ty term in 07ie me7nber of a7i equation may be trans-
posed to the other 7ne77iber provided its sign be changed.
In this way of speaking, we are apt to keep in mind merely the
change which results in the equation and to lose sight of the addition
or subtraction of equals which causes transposition. The axioms must
always be appealed to when we are called upon to explain why trans-
position is allowable.
EXERCISE 21.
Examples.
33. To Solve an equation is to find the value of the
unknown number in the equation, and the process of
finding this unknown number is called the Solution of
of the equation.
As mistakes may be made in the solution of an equa-
tion, it is well for the student to test the results found, by
putting in the original equation the value obtained for
the unknow^n number in place of the letter representing
it. If the equation thus found is not true, a mistake has
been made and the solution should be re-examined.
This process of testing a result is called the Verifica-
tion of that result.
Solve each of the following equations :
I. 5ji;— 2=3.r-f 18.
EXAMPLES. 45
SOLUTION.
Transposing the terms 'dx and —2,
5x-Sx=lS-\-2.
Uniting terms, 2^=20;
whence jr=10.
VERIFICATION.
Substituting 10 for x in the original equation,
5X10-2 = 3X10-1-18,
or 50-2 = 30-1-18;
and, since this equation is true, the correct value of x has been found-
2. 30 + 5jt:=70. 4. 7j»:-8=41.
3. 9;«:-5=31. 5. 1oa'-13=107.
6. 28-8r=7.
SOLUTION.
Transposing the terms — 3t' and 7,
28-7=3/.
Uniting terms, 21 = 3/;
whence /=7.
VERIFICATION.
Substituting 7 for/ in the original equation,
28-21 = 7.
7. i9-r)jt-=4. 20. 5A-+18=3jt-+3S.
8. 107 — 13a-=42. 21. r>Qx—27=47x.
g. 76-19.^=0. 22. 9A' + 17=102-8jt;.
10. 25G-32j/=0. 23. bx-5==2x-^?y.
11. 8.r-f-7— -r=14. 24. lx-\-2^Ax^l.
12. 9.v-f 13--r=29. 25. 3.r-l = ll-.r.
13. i7^_j-i9_2v=64. 26. .r4-4=10-2.r.
14. 23jr— 18-f3j»:=8G. 27. 31 — 7.-r=41 — 8.r.
15. 3ji:+18=5x. 28. 38-2j'=9j/-39.
16. 19.r— 14=12.r. 29. 29a— 57=l()-r-5.
17. 19;r=l() + lLr. 30. 147 — 19j»;=122-14a\
18. 7.r=10() + 3-r. 31- 14jt:-f 23=19,r-2.
19. 30.r=80-10ji;. 32. 50 + 17ji-=295-18.ar
46 SUBTRACTION.
33. 5j*r-f 13-2jr=100-20jr-18.
34. 16:^+10— 21;r=45 — 10jr— 15.
35. o=U+x-8x-Sx-\-4-\-x.
36. 7—ox—10-\-Sx—7-\-Sx=x, •
37. 2x-ix=U-i-ix-h2.
EXERCISE 22.
Problems.
1. What number increased by 57 is equal to 98 ?
Let X represent the number.
Then, because the number increased by 57 equals 93, therefore,
Transposing, jf=93— 57.
Uniting similar terms, x — 36.
2. What number diminished by 26 is equal to 29 ?
3. What number increased b}^ 17 is equal to 35 ?
4. What number diminished by 19 is equal to 15 ?
5. If twice a certain number be increased by 12, the
result will be 30. What is the number ?
Let X represent the number.
Then, because twice the number increased by 12 equals 30, therefore,
2.v-fl2 = 30, etc.
6. If five times a certain number be diminished by 15,
the result will be 45. What is the number ?
7. Five times a certain number exceeds three times
that number by 22. What is the number?
Let X equal the number.
Then, since 5 times the number exceeds 3 times the number by 22,
therefore, 5.^:— 3-^=22, / etc.
8. If three times a certain number be added to the
number itself, the sum will be 36. What is the number ?
PROBLEMS. 47
9. If five times a certain number be added to eight
times that number, the sum will be 78. What is the
number ?
10. If three times a certain number be increased by 5,
the result is equal to the number increased by 25. What
is the number ?
Let x=the number.
Then uX-\- 5=ii times the number increased by 5,
and x-|-25=the number increased by 25.
Then, because 3 times the number increased by 5 equals the number
increased by 25, therefore,
3x-f5=.r-f25, etc.
11. If three times a certain number be diminished by
4, the result is equal to the number increased by 6. What
is the number ?
12. What number must be added to 787, so as to ob-
tain the same result as when the number is taken from
875?
13. What number is as much greater than 78 as it is
less than 108 ?
14. What number gives, when doubled, 7 more than
three times itself diminished by 16 ?
15. Having $78, I spent an amount such that I had
left 5 times as much as I spent. How much did I spend ?
16. A, B, and C together put $8781 into a business.
B put in $1000 more than A, and C put in $2000 more
than A. Find how much money each man put in.
Let a-=number of dollars A furnished.
Then .r-}-1000=number of dollars B furnished,
and j:-f"2000 = number of dollars C furnished,
and since altogether they furnished $8781, therefore,
x-\-{x-\- 1 000)-h( jc-f 2000) =r 8781.
Removing parentheses and uniting similar terms
iU-l- 3000- 8781, etc.
48 SUBTRACTION.
17. A boat' went 368 miles in three days. The second
day it sailed 32 miles further than it did the first day,
but the third day it sailed 24 miles less than it did the
first day. How far did the boat sail each day ?
18. John, Fred, and Dave have $4.35. Fred has twice
as much as John, and Dave has 75 cents more than John.
How much has each ?
19. A man dying left his property worth $52800 to
his four children. He gave his oldest child $3000 more
than the youngest, the next $2000 more than the youngest,
and the next $1000 more than the youngest. How much
did each receive ?
20. A merchant made $4222 in three years. He made
$1592 more the second year than he did the first, but the
third he lost all he made the first year and $350 more.
What did he make each year ?
21. If G3 be added to a certain number, the number
becomes 10 times as large. What is the number ?
22. A and B had equal sums of money. A doubled
his mone\^ and then made $25, while B tripled his money
and then lost $100. They then had equal amounts.
What sum did each have at first ?
Let X— the number of dollars each had at first.
Then, because A doubled his money, and also made $25, therefore,
2-*"-f- 2o=:the number of dollars A had finally.
Also, because B tripled his money and also lost $100, therefore,
3^—100 -the number of dollars B had finally.
Now, since A and B finally had equal amounts
3.r-100:=2x+25, etc.
23. If 10 times a certain number be diminished by 22,
there results the same as when 7 times the number is in-
creased by 23. What is this number ?
PROBLEMS. 49
24. A man doubled the money he had, and then made
$500. He then made an amount equal to 3 times what
he had at first, but losing $5400 he had nothing left.
How much did he have at first?
25. A man walked 47 miles in three days. He walked
8 miles more the second day than he did the first, and 10
miles more the third day than he did the second. Find
how far he walked each day.
26. A farmer rode in a carriage from his home to the
railroad, and then he rode on the cars 5 times as far as in
the carriage, and then on a steamboat 8 times as far as
in the carriage and on the cars together, when he had
traveled in all 270 miles. How far did he live from the
railroad ?
CHAPTER V.
MULTIPLICATION.
EXERCISE 23.
General Definition of Multiplication.
34. The original meaning of multiplication in Arith-
metic is that of repeated addition, and, with this meaning
in mind, we would define multiplication to be the taking
of one number as many times as there are units in an-
other. Thus, 3 multiplied by 5 means 3 + 3 + 3 + 3 + 3,
and I multiplied by 5 means f + f+f+f + f . As soon,
however, as the multiplier is a fraction, it is found that
this meaning of multiplication does not apply; for while
3 can be repeated 5 times, yet 5 caiuiot be repeated \ a
time, nor can f be repeated i a time. Now, although
the operation of multiplying | by f cannot be looked
upon as repeated addition, yet this operation does occur
in Arithmetic, and is called multiplication. It is plain,
therefore, that the word is used with some other mean-
ing than that originally given it, which new meaning
may be stated as follows : *
" To multiply one nutnber by another, we do to the first
what is do?ie to unity to obtain the second. ' '
Thus, suppose we are required to multiply 3 by 5. To make 5 from
unity, we must take unity 5 times, and hence to multiply 3 by 5 we
must take three, 5 times
Again, suppose we wish to multiply f by 5. To make 5 from unity,
we must take unity 5 times, and hence to multiply | by 5 we must
take two-thirds, 5 times.
*Boset Algebra Elementaire, Charles Smith's Elementary Algebra.
GENERAL DEFINITION. 5 I
Suppose we are required to multiply 5 by f. To make | from
unity, we must divide unity into 4 equal parts and take the result
3 times. Hence to multiply 5 by | we must divide 5 into 4 equal
parts giving | and take this result 3 times ; that is
5 multiplied by | ig |X3 or ^£-.
Finally, suppose we wish to multiply | by i. To make i from
unity, we must divide unity into 5 equal parts giving I and take this
result 4 times. Hence to multiply | by | we must divide | into 5
2 2
equal parts giving , or -— and take this result four times ; that is
t) X o 10
- multiplied by - is — — X4 or — .
o .) 3 X O A O
1. What must you do to unity to produce 8 ? Explain,
then, how 5 is multiplied by 8. Explain how % is mul-
tiplied by 8.
2. What must you do to unity to produce ^? Explain,
then, how 7 is multiplied by \. Explain how f is mul-
tiplied by \.
3. What must 3'ou do to unity to produce f ? Explain,
then, how 9 is multiplied by f. Explain how f is mul-
tiplied by ^.
Explain, by the general definition of multiplication,
how the product is found in each of the following cases:
4. 11 multiplied by 12. 13. 6 multiplied by 6.
5. 11 multiplied by 1.2 14. f multiplied by |.
6. 3 multiplied by |-. I5- f multipHed by |.
7. i} multiplied by |-. 16. 15 multiplied by f.
8. I multiplied by 3. 17. 1.6 multiplied by .25
9. f multiplied by f. 18. 12i multiplied by f.
10. f multiplied by f . 19. f multiplied by 12^.
11. f multiplied by -f. 20. 100 multiplied by .01
12. 4 multiplied by ^. 21. .001 multiphed by .01
52 MULTIPLICATION.
EXERCISE 24.
Multiplication of Monomials.
35. If an expression consists of but one term it is
called a Monomial, and if it consists of more than one
term it is called a Polynominal. Thus, Qab is a mono-
mial and oa'^—4:b-\-2 is a polynomial.
If a polynomial consists of just two terms, it is called
a Binomial, and if it consists of just three terms, it is
called a Trinomial. Thus, 3x~Axy-\-^ is a trinomial.
While binomials and trinomials are each polynomials, yet it is
usual to apply the word polynomial only to expressions of more than
three terms.
30. It is one of the laws of multiplication, discovered
in Arithmetic, that the product will be the same, no mat-
ter in what order the factors are multiplied together, or
as is briefly stated:
Multiplication may be performed in a?jy order.
This is called the Commutative Law of multiplica-
tion, and may be illustrated as follov/s:
2 X 5 X 7 X 12=12 X 7 X 2 X 5=7 X 2 X 5 X 12, etc.
|x5xi=4x|x5=5xix|=4x5x|, etc.
And if «, b^ and c stand for any juimbers zv/iatever, we
may say,
ahc=hca =cab—ach=hac=cba,
1. How many times 6 is 5 times twice 6 ? How many
times a certain number is 5 times twice that number ?
How many times ?^ is 5 X 2n ?
2. How many times .^r is dx8x}
3. How much is 8 X 11^?
4. How much is 25 X 4<^ ?
EXAMPLES. 53
5. How mucli is 15 x 12a;y ?
6. How many times 2x3 is 2x5x3? How many-
times ab is, axbb} How many times xy is xxSy'^
7. How much is x^ X 31)'- ?
8. How many times 2 x 3 is 5 x 2 x 7 X 3 ? How many
times ab is 3^ x 8*^ ? How many times xy is Ixx^yl
9. How much is Ojt'X llj'- ?
10. How many times abc is 9rX XOab}
11. How many times abxy is 21ay x 15<^.r?
37. From the above work we learn that :
T/ie p7-oduct of two monomials is found by imdtipJying
the numerical coefficients of the monomials to obtain the nu-
7nerical coefficient of the product, and by writing in succes-
sion the literal factors of the monomials for the literal part
of the product.
EXERCISE 25.
Examples.
Multiply
Multiply
I.
^iax by 9.
II. 14 by \ay.
2.
8«-U-by 11.
12. f\t.y|«.r3.
3.
i^/2 by 234.
13. mn by dy.
4.
8^2 jr by 80.
14. xy by ab2.
5.
lO^V- by 13.
15. ab'^ by c^y.
6.
12^.3 by |.
16. d'^-m by hax.
7.
25 by 2.r.
17. mn'^' by 9j'2.
8.
11 by \ab.
x8. la'^x by ^.
9.
90by6^;tr2.
19. 10ayby^2^
10.
15 by ^77in.
20. ^-w by 9^z/.
54 MULTIPLICATION.
Multiply Multiply
21. 12a^c by 4:in. 28. ^cx^ by ^Pj/.
22. llcfy^ by So-x"^. 29. fa'^^w^ by ^cx.
23. Gf?i^p^ by 7d;?2^. 30. ^d'^n^ by -f-^c^^n,
24. 15^-^"* by 3<^*2^. 31. -f^VbyyV^^^-
25. f^2;rs by f^3<5^ 32. i|^^2 by l^yH"^.
26. -fw^^es by |^>. 33. 4.6^2^5 by .Za'^b.
27. %b''- y by \a^y^. 34. .5«2;r by .6/^^j/.
EXERCISE 26.
Law of Exponents in Multiplication.
1. What is the product of aaa and aa, written in the
abbreviated form? What, then, is the product of «^ and^^ ?
2. What is the product of ccc and cccc, written in the
abbreviated form f What, then, is the product oic^ and c"^ ?
3. In y^ , how many times is y used as a factor? In
jj/*, how many times is y used as a factor? In y^ times
J*, how many times is y used as a factor? How, then,
would you write the product jm^ Xj/* in the simplest form ?
4. In tf^, how many times is a used as a factor? In
a^ , how many times is a used as a factor ? In a^ times a^,
how many times is a used as a factor? Write a^ Xa^ in
the simplest form.
How is the exponent in this simplest form obtained
from the exponents 5 and 3 ?
5. In ^^, how many times is a used as a factor? If n
stands for a certain whole number, how many times is a
used as a factor in a"? In a^ times a'\ how many times
is a used as a factor ?
How, then, would the exponent of the product be
formed from the exponents 5 and n ?
EXAMPLES. 55
6. If n stands for a certain whole number, how many
times is a used as a factor in a" ? If r stands for some
other whole number, how many times is a used as a factor
in «''? In «" times a' , how many times is a used as a
factor ?
How, then, would the exponent of the product of any
two powers of a be found from the exponents of the
factors ?
38. From the above we learn the Law of Exponents
in Multiplication, which is usually stated as follows :
The product of two powers of the same number is equal to
that 7iumber with an exponent cq^tal to the sum of the ex-
poneiits of the two factors.
39. In mathematics statements like the above are often
expressed in the symbolic language of Algebra, and when
thus expressed are called Formulas. The above law
expressed as Oi formula would be,
a" w = a"+^
Since a is a72/y mimber and 7i and r are any whole 7ium-
bers, this algebraic equation is equivalent to saying :
The product of two powers of the same number is eqiial to
that number wi^h an exponent equal to the sum of the ex-
ponents of the tivo factors.
EXERCISE 27.
Examples.
Multiply Multiply
1. 3«2 by 11^3^ 5, 5^2^2 by ^ab.
2. bx^ by Ibx''. 6. lab^ by Sa^b'^
3. 14^2 by cb^, 7. llxy'' by Sjt^^^
4. 14r^2 by b^. 8. axy by xyz.
$6 MULTIPLICATION.
Multiply Multiply
g. adc by dcd. i8. 20<^2^ by .3^2^.
10. 8ax^ by 6dx^. ig. 5w.i: by llb'^x^.
11. 12afy'^ by 10acy\ 20. S^i-^^^ by 20?ix^.
12. |'3 by «j'^.
14. Sa-x^ by da^pi-x"^, 23. 2Jy^;r by o^^^-j'"^.
15. 6/>2jt:5 by Dd^pKr\ 24. 33«^ by IQi^V*.
16. Sd^byl-aH\ 25. (;t-4-jiO' by 4(;r+7)^
17. 60xy' by .05a-3j. 26. 3(;r-5)2 by a(ix-5y,
EXERCISE 28.*
Multiplication of Polynomials by Monomials.
1. How much more than 200 is 2 times (100 + 12)?
How much more than 300 is 3 times (100+12) ? How
much more than i300 is 5 times (100+12)? How much
more than ?i hundred is 71 times (100 + 12) ?
2. How much more than 200 is 2 times (100 + ^)?
Write, then, what 2 x (100 + (^) equals.
3. How much more than 600 is 6 times (100 + r) ?
Write, then, what 6x(100 + r) equals.
4. How much more than 2a is 2 times (^ + 12)?
Write, then, what 2x(« + 12) equals.
5. How much more than 4a is 4 times (a-}-9)? Write,
then, what 4x((2 + 9) equals.
6. How much more than /la is 21 times (« + 9) ? Write,
then, what 7tX («+9) equals.
7. How much more than 91a is 7i times (^ + 15) ? Write,
then, what ?i X (a + 15) equals.
MULTIPLICATION OF POLYNOMIALS. 5/
8. How much more than na is n times {a-\-b)'^. Write,
then, what ;^ X {a-\-b) equals.
9. How much less than 200 is 2 times (100—12) ?
How much less than 300 is 3 times (100—12)? How
much less than 500 is 5 times (100—12)? How much
less than n hundred is n times (100—12) ?
10. How much less than 200 is 2 times (100— <^) ?
Write, then, what 2x(100— /^) equals.
11. How much less than 600 is 6 times "(100— r) ?
Write, then, what G x (100— <:) equals.
12. How much less than 2^ is 2 times (^—12) ? Write,
then, what 2 x (« — 12) equals.
13. How much less than 4« is 4 times (a— 9) ? Write,
then, what 4 x («— 9) equals.
14. How much less than na is n times (^—9) ? Write,
then, what ;zx(a— 9) equals.
15. How much less than na is n times {a—\h) ? Write,
then, what ny. (a— 15) equals.
16. How much less than na is n times {a—b) ? Write,
then, what ;^ x{a—b) equals.
40. The principles we learn above may be stated as
follows :
The product of the sum of tivo numbers by a third nu7nber
equals the sum of the products of each of the tzco numbers
by the third number.
The product of the differeiice of two ^lumbers by a third
number equals the difference of the products oj each of the
two numbers by the tlm^d 7iumber.
These principles may be stated in algebraic language,
by means of the following formulas :
fi{a-\-b) = tia-\-nb,
n[a—h)=7ia—nb.
58 MULTIPLICATION.
We will now proceed to the case where the multipli-
cand has more than two terms.
17. How much more than 5(a-\-b) is 5(« + <^4-3) ? How
much more than 5(« + ^) is 5(a-\-b-\-c)? Write an expres-
sion equal to 5(a-{-d-\-c) without using a parenthesis.
18. How much less than 5(« + <^) is 5(aH-<^— 3) ? How
much less than 6(a-\-d) is b(a-\-d—c) ? Write an expres-
sion equal to 5(a-\-d—c) without using a parenthesis.
19. How much more than l(a—b) is 7(a— <^+4) ? How
much more than l{a—b) \sl{a—b+c)l
Write an expression equal to l(a—b-\-c) without using
a parenthesis.
20. How much less than 7(a—b') is 7(<2— ^— 4) ? How
much less than 7(a—b) is 7(a—b—c)?
Write an expression equal to 7(a—b—c) without using
a parenthesis.
21. How much more than n(a + b) is n(ia + b-\-c)? How
much more than n(a—b) is 7i{a—b-\-c)?
Write what 7i(a-\-b+c) and n{a—b-{-c) equal without
using parentheses.
22. How much less than n(a-\-b) is n(a-] b—c)? How
much less than n{a—b) is n{a—b—c)t
Write what 7i{a-\-b—c) and n{a—b—c) equal without
using parentheses.
41. What we have learned may be expressed by
formulas as follows :
n'ya-^-h-^-c) =na-\-nh-\-nCf
n{a—b-\-c)=n'(5?/— 6r-|-4/). 10. 2arm(J'—a-{-r—7n).
5. 2jf(;r-4ji;2+6). 11. IxiZ-x-^-^x'^-^x^).
6. Aax{Z + 4:X-S)x''). 12. day{ll—4y-\-2y''-{-5y^^,
EXERCISE 30.
Arrangement of Work in the Multiplication of a Poly-
nomial BY A Monomial.
42. The following illustrations will be sufficient to
explain the usual arrangement of work when the product
of a polynomial by a monomial is sought :
Suppose it is required to multiply a-{-b—chy 5.
Multiplicand, a-\- b— c
Multiplier, 5
Product, ba-\-K)b—bc *
6o
MULTIPLICATION.
Suppose it is required to multiply x—Sx'^-\-2x^ by bx.
Multiplicand, x — Sx^ -\- 2x^
Multiplier, 6x
Product, 5;t-2 — 15x-' + 10;r*
Suppose it is required to multiply 6aj'^ by 2a-j'^ —Say
+ 5. Since ^mdtiplication may be pei'foi'mcd in any order ^
we arrange this just as though Gxy—12x'^y'^.
13. x^-Zx'^y-\-Zxy'^-2hYZxy'^.
14. Aab by 'la-d^+oad^—ld"^.
15. 2«-^-3r^3^i«^3_5by (3^^2^2^
16. 3a/^+4«2^-5^(^+6by 4a*fl?^
17. Ga^-4d''+2ad^-Sc''hy ^aHc,
18. 6;>;-^-8:r2^+2.t72 byx2j/3.
19. ^— 5^+ic— ifl'by 24«^r^.
20. 3^-^ + 9^'^r-^2_27by 1^2^.
Simplify each of the following expressions :
21. 4(4a—7d) + 7('2a—od).
4(4^^-7/0 = 16'' -28/'; 7(2/r-5^) = l4rt-35/^
Therefore, 4{U-'id)-\-l{2n-5^) = {]6(i-2S/>)-\-{Ua-'65^).
Removing parentheses and combining similar terms,
4{4a-/0-f-7(2) = }Ga-2Sl>; 7(2^-5/0 = 14^-35/^
Therefore, 4(4rt-7/0-7(2rt-5<^) = (16«-28/0-(14^/-35/0.
Removing parentheses, =1Q<7 —2H^—l4:a-\-33i.
Combining similar terms, ■=2ii-\-7/k
23. 5(x+5)-2Gr-4) + 3(2,r-l).
24. S(7a-4d)-4{oa-\-2b)-2(d-?ya').
25. a(a-\-b—c) — d(a—d-\-c)-i-c(a4-0).
26. 7.r(3_r + 4_y— G)— 4x7jt'— 3j'+14).
27. 14(3^ + 4^)-6(2«-6^)-4A-0r2-l).
28. 6a-'{a''-2ax)-4yiy+4xy)-^^a''-ay-).
29. «(rt — ^ + <:) + K^ — <^ + — ^(^ — ^ + <^)-
30. Sad(a-c)-dc(2d-Sa)-\-Qad^-d''(3a-2c),
31. 3.r(a + ^-2j/)-2X«-3^-3A-)-3/5(jt-+2j/) + 2ay.
62 MULTIPLICATION.
32. Qab{ab'^—Acd)abc.
Therefore, Qab{ab^- ^cd)abc= %a^b^c{nb^ - ^cd)
-^a^b^c-lia^-b^c^d.
33. 3(16-8;tr-4:r-H2:r-^)6x2.
34. 3«(2a2+3«2^-3«^2_432-)3^^
35. ab'^iax-bx -{-Qax^-bbx'^^iaH.
36. \xy-(S-xy'^ +dx^y-7y''^Sx\
37. 2aH(iax--bx+b'-yb\
EXERCISE 31.
Multiplication of Polynomials by Polynomials.
43. Just as it is customarj^ to write 7(10—3 + 5) in-
stead of 7x(10— 3-f-5), so usually the product of two
polynomials is indicated by writing it like
(14 + 2-9)(10-3 + 5),
instead of using the sign X , as in (144-2 — 9) x (10—3 + 5).
So the student must remember that it is the prod?id ot
two polynomials which is called for when they are en-
closed in parentheses with 720 sz^?i between them.
1. If a-i-b-\-c multiplied by ?i is a?i -{- b?i -\- c?7 , what is
a-{-b-\-c multiplied by (« + r) ?
Write a(7t-\-r)-{-b(^?t-hr)-\-c(7i + r) without using paren-
theses.
2. If a—b-\-c multiplied by 71 is an — b7i-\-c7i, what is
a—b-\-c multiplied by (;z + r) ?
Write a{7i-{-r) — b{7i-\-r)-\-c{7t-\-7^) without using paren-
theses.
3. If a-^b—c multiplied by 7i is a7i-\-bii—C7t, what is
a + b—c multiplied by (;^— r+/) ?
Write a(7i — r-]- t')-{-b{7i—7'-\-t)—c{7i—r-\- t) without
using parentheses.
EXAMPLES. 63
4. Could results similar to those above, be obtained
710 matter what polynomial is used as a multiplier ?
44. As the conclusion from the above, we have the
following principle :
The product of two polynoviials is the aggregate obtained
by placing one polynomial as a factor i7i each term of the
other polynomial.
EXERCISE 32.
Examples.
Find the product in each of the following :
I. (^-4) Cr-h9).
Placing the second polynomial as a factor in each term of the first
polynomial, we have x(jf-|-9) — 4(x-|-9).
Multiplying by x and 4, we obtain
(xs-fy.r)- (4^-1-36).
Removing parentheses, we get
,rS-l-9.r-4x-36.
Uniting similar terms, we have
Ar8+5-r-36
which is the required product.
Placing the second polynomial as a factor in each term of the firs!
polynomial, we have
Multiplying by 3(^?*, la and 9, we obtain
(6^3/;_j8rtV;)4-(4a8^--12r?/^)-a8.-?^^-54/;).
Removing parentheses, we get
6«V;-18^?~/^-}-4a2<5— 12(/^-18^^-f-54^.
Uniting similar terms, we have
^a^b—\^a^b—Z^ab^h^b
which is the required product.
3. (-r+3)(;r+7). 6. (;t:-3)(^--7).
4. (^+3)(-r-7). 7. (^+5)0r-5).
5. (jt:-3)(x + 7). 8. (a^+6)(:ir+6).
64 MULTIPLICATION.
9. (x+SXx-4). 20. (4d-5c)(Sd+4:c). '
10. (jtr— 5)(jt-— 5). 21. (Sa—4d)(a—d).
11. (^+12)(r-l). 22. (5.r+l)(7y-2).
12. (jc— 12)(;»;— 1). 23. (^— 5w,)(«H-37;?).
13. (x+lo)Cr-lo). 24. (8a + 5jr)(7^-4;t:).
14. (J^;-^7)(-^--18). 25. (2^-3Z')(5;r-7:iO.
15. (3jt:— 5j/)(3x4-5>'). 26. (#?— ;z)(x+j').
16. (3x— 5jr)(3j»;— 5>'). 27. (a + <^)(r— ^).
17. (3ji;-57)(5a--3jjO. 28. (2.r2-4;c+9)Cr-7).
18. (2a-2dX2a-d). 29. (3.r-4-2jr-6)(2.r-3).
19. (4;t:+9;/)(x— oj}'). 30- {x'^~xy+y^')(x—y').
31. (3;i:2-2.ry + 6)/2)(2;r + 3j/).
32. (a^-2ab-hd'-')(a-d).
33. (3x2-4;t- + 7)(ox2-a'--4).
34. (^2_|.7_^._5^(^_;^2_3_^_l_7^_
35. (Sa'^-5ad+2d'')(a'--7ad).
36. (2jrF— jj/— x)(a-— 3+>')-
EXERCISE 33.
Arrangement of Work in the Multiplication of Two
Polynomials.
45. Let us go through the work of multiphdng the
two polynomials 7x'^—Qx—^andox'^—bx-{-2 together.
We first place the second polynomial as a factor in each
term of the first polynomial,, and obtain
7x\Sx''-dx-{-2)-6xSx^-j-17x''-\-SSx-lS (4)
which is the required product.
This same work can be arranged in a convenient form,
as follows :
Multiplicand, 3x^— 6x + 2 (5)
Multiplier, 7x^-— Qx — 9 (6)
1st partial product, (21j«;*--3o;t:3 + 14^2) (7)
2d partial product, -(IS;*;^— 30;»;2 + 12;ir) (8)
3d partial product, — (27.^2 ^45j»;-f 18) (9)
where the parentheses are the same as in (2) above, the
only difference being that they are arranged so as to bring
sbnilar terms in the same vertical column. Removing these
parentheses and uniting the similar terms in the same
column, we have
Multiplicand, Zx"-- hx + 2 (10)
Multiplier, Ix''- ^x - 9 (11)
1st partial product, 2U-*-35j»;3 + 14j»;2 (12)
2d partial product, —X'^x^ -^Z^x"- — Vlx (13)
3d partial product, • -27jt:2-f 45;t;~18 (14)
Product, 21 j«;* - 53;r3 + 17;t:2 ^ 33;r- 18 (15)
which is the usual arrangement of work in the multipli-
cation of two polynomials.
46. The expressions (12), (13), and (14), which we
have called partial products, were obtained, as we have
just stated, in the following manner:
First. The expression ?>x'^ — hx-{-2 ivas placed as a factor
in each term of the expression 7;»:2— 6.v— 9.
66 MULTIPLICATION.
Second. The 7miltiplications by Ix"^, 6x, a?id d were per-
formed.
Third. The parentheses were removed.
Hence it follows that when the multiplicand is placed
as a factor in the additive term Tjt^, the result is the
partial product (12), which has its terms
additive, subtractive, additive,
the sa7ne as the multiplicand. But when the multiplicand
is placed as a factor in the subtractive terms —%x and
— 9, the results are the partial products (13) and (14),
which have their terms
subtractive, additive, subtractive,
which are just the opposite of those of the multiplicand.
47. From this reasoning we can readily formulate the
following method for finding the product of any two
given polynomials :
Neglect the sig7is of the terms, and miiltiply, in succession,
the multiplicand by each term of the fnultiplier, to obtain the
successive partial products.
Give' the terms of each partial product, the same sights
as those of the multiplicand when an additive term of the
multiplier is used, but give the terms of each partial pro-
duct just the opposite signs to those of the multiplicand whefi
a subtractive term of the multiplier is used.
Add the partial products thus formed, and the result is
the required product.
Thus, suppose (x^-\-'^x—^{x~ — ^-)X-\-1) is required,
x2+3x - 4
x^—hx + 2
—hx^ — \hx^^l{)x
2x8+ (;^_8
;c4— 2x3 — 17x3-f-2G^— 8
ARRANGEMENT OF WORK. 6/
The above way of obtaining the signs of the terms in the partial
products keeps the ''reasons why prominently before the student, but
in practice the signs are more often determined in another manner,
which we proceed to explain.
48. In the above we have seen that the signs in the
partial products will be the same as those of the multi-
plicand if the term used in the multiplier is additive.
Hence we have the two following cases :
If a term in multiplicand is additive, i. e. , has sign -f
and term used in multiplier is additive, i. e., has sign +
the resulting term in product is additive, i. e. , has sign +
If a term in multiplicand is subtradive, i. e., has sign —
and term used in multiplier is a^^zVzV^, i. e., has sign -f
the resulting term in product is subtradive, i. e., has sign —
We have also learned that the signs in the partial
products will be the opposite to those of the multiplicand
if the term used in the multiplier is subtractive. Hence
we have the two following cases :
If a term in multiplicand is ^^^///z' by i;;^ + |-r-6A
57. i^-#+J^by|«4-f^-i^.
EQUATIONS INVOLVING MULTIPLICATION. 7 1
EXERCISE 35.
Equations Involving Multiplication.
Solve the following equations :
1. 9(23-5y)=8(5>/-6).
Performing the multiplications by 9 and 8, we have
207— 4oy=40>'— 48.
Transposing the terms —48 and — 45j', we get
207+48r=40>'+45y.
Uniting similar terms, 255=^85v;
whence, y=^Z.
2. 6(jt:-5) + 2A'=8A'-2(jt-+10).
Performing the multiplications by 6 and 2, we have
(Ca;-30)+2a:=:8x-(2aH-20).
Removing parentheses,
()X— 30-|-2a:= 8a;— 2x— 20.
Transposing the terms —30, 8a:, and —2x, we obtain
6a; -[-2x- 8-r-f2.r.- 30-20.
Uniting similar terms, 2x=10;
whence, x=b.
3. 2(;ir-l) = G. 17. loCr-3)-17=103.
4. 4(;»;+5)=36. 18. 17(17~;t:)4-17=51.
5. 7(j^/-3)=14. 19. 6(37-.;»r) + 23=113.
6. 19(j»r-7)=57. 20. 5(2,r+7)-8=57.
7. 13(12-^)=2G. 21. 9(8^-5) + 13= 112.
8. 17(13-.r)=136. 22. 5(.r-l)=9A'-25.
9. 5(35-jt:)=105. 23. 5;»;+(7-2x)=ll.
10. 7(135-^)=85. 24. 8jtr-(3+5;r)=9.
11. 8(2:r+5)=lo2. 25. 8(5-^)=3(jj/-5).
12. 13(7r-61)=26. 26. 10(;t+2)=ll^+17.
13. 4(15-2.r) = 20. 27. 4(5jt:-3)=104-9;r.
14. 15(15-4;r)=45. 28. 8(10-;r) = 5(.;r+3).
15. 3(;i:-5) + 8=17. 29. 7(15-3jr) = GCr+4).
16. 3(j«:-3) + 5=23. :o. 8(9-2^0 =5(3;»;+2).
72 MULTIPLICATION.
31. 9(13-;r)-4;i;==5(21~2;r)-f9;r.
32. 199 + 15;»;-(4jr-5)=17(x+17)-13-22.
33. 8(3jt:-2)-7^-5(12--3jt:) + 28=8(3x4-2)-32.
34. 118^-13(54-ll^) + 15(3;»;-3)=.18;»;--7(^--5)-119
35. 7(3jr-6)+5(x-3) + 4(17-x)=44.
36. 2CW-x) + S{5x-i)=:12(S+x)-2(12-x),
37. 3(3^-2)~5(18-5;^) = 13(8;r-12)4-5(9;»;~ll).
CHAPTER VI.
DIVISION.
EXERCISE 36.
Division of Monomials by Monomials.
61. We may define Division as the process of undoing
multiplication. In multiplication two factors are given
to determine their product, while in division the product
and one of the factors are given to determine the other
factor. Thus, to divide 12 by 3, we must determine the
factor which, when multiplied by the given factor 3, will
produce 12.
In divison, the factor given is called the Divisor; the
product given is called the Dividend; and the factor to
be determined is called the Quotient.
1. What must 3a ?
3. What must hb be multiplied by to produce Z^abc^ ?
What, then, is Z^abc"- ^hb'>
4. What must «^ be multiplied by to produce «^ ?
What, then, is a^^a^'>
5. What must €>a^x^ be multiplied by to produce
r^a^'x'^ ? What, then, is X'^a'^ x'^ -^Za'' x^ ?
52- Since the dividend equals the product of the
divisor and quotient, it follows that the quotient of one
monomial by another monomial is found by removing from
the dividefid all the factors which ocair in the divisor.
74 DIVISION.
Thus ^abc-^2ab—A:C, because 2abx4:C=^8abc, and the
quotient Acis found by removing the factors 2, a, and b
from the dividend. If we wish to divide 2ab by xy, the
factors X and_y do not occur in the dividend, and conse-
quently cannot be removed from it, so we can merely m-
, ,. . . , 2ab
dicate the division thus: .
xy
53. Since the product of two powers of the same
number is found by adding the exponents, it follows that
the quotient of any power of a number divided by a lesser
power of the same number, is equal to that nuitiber with an
exponent equal to the exponent of the dividend minus the
exponent of the divisor. This is called the Law of Ex-
ponents in Division.
Thus, a^-r-a^=«^ because «^Xa^=^^. Also 2\a^bx^
'^Sabx'' = ^a''x^ because ^abx"- X?ja''x'' = Ua^bx\
54. The above principle may be stated in algebraic
language by means of 2i formula, for if we let a stand for
any number whatever, and n and r stand for a7iy whole
numbers whatever such that 71 is greater than r, we may
say,
because a"~'' x a' — a".
EXERCISE 37.
Examples.
Divide Divide
1. 6^ by 3. 6. 14r^by d.
2. 8b by 4. 7. 17m?i by m.
3. 1577in by 5. 8. 2Sxy by x.
4. 49^^ by 7. 9. 2o77ix by ox.
5. 12ab by a. 10. dAa^y by 6y.
EXAMPLES.
75
Divide
11. IS^^jt:^ by 63.
12. 28cy^ by 4c.
is. lOSm^c^ by 9;«^
14. I(y5dm'' by 15/^.
15. 76dx^- by IM
16. 51w2jj/ by 17;«2.
17. 6Smn^ by 9;«.
18. 145;;^;z-j/ by 29;/?.
19. 5cy by r>'.
20. 4Sm^7t by ;;^2;^.
21. Ilx^y hy 12x'\
22. 17;t:8j' by 17;»;».
23. 53;»:* by 5;t*.
24. Ilx^y- by Tljj''-'.
25. 2;«^2' by 2m^z.
26. 475^2 by 2^2^
27. 17a2<^ by a^^.
28. 423^2 by be'', •
29. 2«'m2 by ;;2.
30. 5a ^^"^ by a^.
31. Sm^fi^ by «^.
32. 15^/^"^ by 3«(^.
33. 5/^^jr* by ;i:^.
34. 18^2^-5 by 3j'^
35. Sm^7i^ by /z*'^.
36. 9m*jy^ by Sm'^y^,
37. ^^/^^ by «=^3"^
38. Ibv^x by 5z;2;,;.
39. «2^^3 ]3y ^^2
40. ^V;t* by d^x'^.
Divide
41. Id^x'^ by ^3 1-2.
42. aH^ hy ab.
43. fn^7iy^ by ;r^2^
44. 21z;*J»;2_y5 |)y 72;3;^2^
45. 6j»;7_>/9 by Zxy^.
46. 8^^ji:y bj2r2j3.
47. hXb^x^z by l?^^;^;^
48. 5a;i:^ by hax^ .
49. 15^^* by 5fl^3.
50. 225w^j^'^ by lowy*.
51. VlXx-'y^ by ll;i:-^j2
52. lOOS^^VoT^ by 18/^^^.
53. 306^«*^iJ/> by 18w/.
54. 10^2^ by 5 ab.
55. 140;t:3j/5 by Zhx^-y.
56. 108/i2j^/7 by 9/;>'«.
57. 702^5^* by 18^/.
58. XOmx^y'^ by SS^x'^y^,
59. f^2^4 by 5 xy,
60. fm^y^ by ^m*y^.
61. fa7^2 by yV^^.
62. T-V/^2^ by i/>.
63. ^d-^c^e by yV^r.
64. i/'V^^ by f/)=^^r.
65. ^ kr- by f /&/.
66. Ic'^x^ by Icx"^.
67. ia2^3 by -^\ab^.
68. 11^4^3 by -i-l^/z^ji:.
69. \ixy^ by Ji;r>/^
70. .o^h^ by .4/i*.
'J^ DIVISION.
EXERCISE 38.
Division of Polynomials by Monomials.
55. Since the product of a polynomial by a monomial
is found by placing the monomial as a factor in each term
of the polynomial, it follows that a poly^iomial is divided
by a monomial by removing from each term of the polynomial
all the factors which occur in the divisor.
Thus. (G«-9^+15r)-f-3 = 2^-3^+5^,>r (2^-3^ + 5^)
X 3=6«— 9^+15r, and the quotient is found by removing
the factor 3 from each term of the dividend.
Also, (9^S;r-15^2^2_^12a:r3)-f-3«;t:=3a2_5^^ 4.4^2^
because (3«^— 5ajt:+4jf2) y.Zax=^^a^ x—\ha'^ x'^ -{-Vlax^ y
and the quotient is found by removing the factors 3, «,
and X from each term of the dividend.
If we wash to divide IQa—dbx-^-Scx"^ by 4x, the factors
4 and x do not occur in some of the terms and conse-
quently cannot be removed from them, so we can merely
indicate the division in such cases, thus :
(^iQa-ddx + Scx')-r-ix=:^~-~ + ^-.
56. The Arrangement of the W^ork when a poly-
nomial is divided by a monomial is as follows :
Let it be required to divide 21?n'^x^ — S5am-\-7h'^m^
by 7;«.
Divisor, 7m I 21m'^x^ — S5a7n + 7h-m'^ Dividend,
Sm x^ — ba -f h'^m Quotient.
Let it be required to divide l(jx^ ^2Ax^ -20x'' by Ax^.
Divisor, " Ax^ | 16;t:^ + 24:t-'^— 20^" Dividend,
4x'^-h 6x — 5 Quotient.
Let it be required to divide 5x'*'—7x^y-j-4x^y'^ by 5x^.
Divisor, 5x^ I 5x'^—7x^y-^4x'^y'^ Dividend,
r'-i — I
Ix y-h^y'- Quotient.
DIVISION OF POLYNOMIALS. TJ
EXERCISE 39.
Examples.
Divide
1. 15«2_9a-5 + 18^» by 3«2.
2. UH-l^a'^b^-^-X^aH^ by ^a^b,
4. 6aH'^-Soa^b^c^+20adc'^ by oad.
5. 12a''x*y- -24ax^y* -18x''j^+6xy by 6;rj.
6. Sx^y--12x--16x by 4ji:.
7. 3a3^2-6a2^*4-12«^6^by 3^/^^
8. Sx^--{-l(Dax-{-6a\rhy4x.
9. 2«2^"-3«<^3_|_4^3^_^4 by 3a<^^
10. a^x^y—Sa'^dx^j+Sab'^xy^ — d^xy^ hy abxy.
11. 3A'*^+5jr3;j/-6;«:V^— -^y + 4y5 by 2x''y''.
12. 35:i:^H-15;«:2^--;r>/2--j3 by 5^2.
14. 5.r^j/— 25;t:2y2 _}_ \0xy^—8 by 5;rj/.
15. 4;>Y^r2+J^/2*r^— ^w^r^ by f/;«^r.
16. ZQx^y^-JrlSx^y^-Ux^y^ hy ^x^y^,
17. .6m'^x^—f7n'^x'^-{-.Somx^ by fwx^.
18. ^^y^ — '^x^y—zz^yhy\xy.
EXERCISE 40.
Division of Polynomials by Polynomials.
67. Suppose we wish to divide x'^-\-bx-\-Q by .a:+3.
The dividend x'^-\-bx-\-(S is the product of the divisor
x-\-Z and another factor (the quotient), which we wish
to find. Now, we can tmdo this multiplication if we can
write x'^-\-bx+Q so that {x-\-Z) will be a factor in each of
7^ DIVISION.
its terms, for, by the previous exercise, an expression is
divided by (:r+3) if we remove (^+3) as a factor from
each term. But we can say,
and, by using parentheses,
=^(^+3) + 2(:r+3).
Then, by removing the factor (x-\-Z) from each term, we
obtain
(jr2+5;i:+6)-^(;r+3) = *r+2,
which is the required quotient.
As another example, let it be proposed to divide
j»;2 + 14;»;-f 45 by x-\-^. Now this can be done if we can
write x'^-]-14iX-\-Ab so that (;r+9) will be a factor in each
of its terms; for an expression is divided by (jr + 9), if we
remove (:r4-9) as a factor from each term. But we
know
x- + 14;r+45=x2-i-9j»;+5^+45
and, by using parentheses,
=.:r(j»;+9) + 5(x + 9).
Then, by removing the factor (;r+9) from each term, we
obtain
(jt:2 + 14;r+45)^(jf+9)=j»;+5,
which is the required quotient.
Again, suppose it required to divide x'^-\-5x-\-4 by
jt:+4. This can be done if we can write x''-\-5x-\-x, so
that (x-\-4) is a factor in each of its terms ; for an expres-
sion is divided by (x-j-4), if we remove (x-i-4) as a factor
from each term. But we know
x'^-\-5x-{-4=x^-\-4x-\-x-i-4:
and by using parentheses,
=x(x+4) + (x+i).
Then, by removing the factor (.^+4) from each term, we
obtain (x'^-j-ox-}-4)-^(x-^4')=x-\-l,
which is the required quotient.
EXAMPLES. 79
It is noticed in each of the above cases that our process
consists in breaking the given dividend into parts, so
that the divisor is a factor of each part, and then remov-
ing that factor from each term.
58. We may formally state the above process as
follows :
One polynomial is divided by a secona polynomial, if the
first polynoinial be written so that the second polynomial is a
factor in each term of the first, and this factor removed.
If it is impossible to write a polynomial in this man-
ner, then that polynomial is not exactly divisible by the
proposed divisor, and the division must be merely in-
dicated. Thus, (;»:-^4-6jir+2)H-(;r+3) would be worked
as follows :
Using parentheses =^(ji;+3) + (oji--|-2).
Therefore,
EXERCISE 41.
Examples.
Divide Divide
1. :i:2+3;r4-2by ^+2. g. jt^+Qjr-fM by .r-f7.
2. x'^^-^x-k-^hy x-\-Z. 10. .;i:2-}-8^-fl5by j*r-f3.
3. x'^-'^^x^hhy x-\-h. ii. x^-\-\\x^1'^hy x-k-^.
4. .r2 + 7.r+10by .r-f-2. I2. .^'M 'J-^-f 18 by .^+6.
5. x'^-\-hx-\-^\yy x^Z.^ 13. :r2 + 10.r-r24 by .;»;4-4.
6. x'^-^-'dx^l^hy x^^. 14. A'2_^10.r+21by.r-t-7.
7. ;i:2-f6;ir+8by ^4-2. 15. .r^ + 12;<;-f-35 by .r-fS.
S.';i:2 + 8;t--f 12 by ;t:-f6. 16. x'^ ^-A^x-Yoby x-\-\.
8o DIVISION.
Divide
17. x'^-\-4:X—45 by x—5:
Using parentheses =x(x—ri)-{-9{x—5).
Removing the factor (x— 5) from each term
{x^-\-U-4o)^{x-\-5)=x-\-9,
the required quotient.
18. j»;2-f3jtr— lOby :r— 2. 21. ji:^— 4j»;— 21 b}^ Jt:— 7.
19. x^-\-Sx—4hyx—l. 22. x'^ +2x—S6 by x—5.
20. Ji;2-f3;»;-18by ;«;-3. 23. jr2-4.r-12 by ;i:-6.
24. Divide ;i:^ — 4;»;— 45 by .r-f-5.
x^-4x-45=x^-\-^x-9x-45.
Using parentheses =x[x-{-5) — 9{x-\-5). Art. 23.
Removing the factor {x-\-5) from each term
(.r8— 4a-— 45)h-(a-4-5) = x— 9,
the required quotient.
25. ;i;2-3jt:~10by .:r + 2. 28. ;r2-4.r-21 by ;t:+3.
26. .;r2— 3;*;— 4by .r+1. 29. j»;2 + 3jtr— 10 by ;t4-5.
27. .;i;2-f3:r-18by .r+6. 30. ^•-+2j«;-35 by ;i:+7.
31. Divide .;>;2 — 14.a; -1-45 by jir— 5.
x^-Ux-\-i5=x»-5x-9x-\-45.
Using parentheses =x{x—^)-9{x—5). Art. 23.
Removing the factor {x—5) from each term
{x^-Ux-\-4:5)^{x-5)=x-9
the required quotient.
32. a^-7a + 10 by«-2. 39. Sa^- + 19a-\-20 by a-\-5,
33. a^ — 5a-{-4:by a—1. 40. 4:a'^-\-Ua + 6 by 4a -f 2.
34. a'^-da-j-18 bya-3. 41. 4a''-\-2Sa-\-15by 4a-{-S.
35. a^-lOa-i-21 bya-7. 42. 3^^ ^10^-^3 by a-h3.
36. a''—7a + 10 byd;-5. 43. 5tf2 + lla + 2 by «-f-2.
37. ^2-10^ + 24 by«-6. 44. 2^^ + 11^ + 5 by 2«4-l.
38. 2a2-hl0« + 12 by« + 3. 45. 3a2 + 34«-i-llby3a-I-l.
46. 4^2 + 23^4-15 by 4^-1-3. ♦
47. 24a-—66a + 21 by 8a— S.
ARRANGEMENT OF WORK. 8 1
EXERCISE 42.
Arrangement of Work in the Division of Polynomials by
Polynomials.
59. Since division is the process of undoing multipli-
cation, we will exhibit in connection with each other the
arrangement of work in the two operations.
multiplication, or the direct operation.
Multiplicand, x -\- 4
Multiplier, .^- 4- ^
1st partial product, x'^-h 4;r
2d partial product, 9 ^+36
Product, x'^-i-lSx+Se
division, or the inverse operation.
Divisor. Dividend. Quotient.
x+A^ x''-j-rSx-hS6 ix+d
1st partial dividend, .r^-f 4j«; _
1st remainder, 9jt--f36
2d partial dividend, 9jc+3'6
2d remainder,
In the work in division the process is as follows : We
first arrange dividend and divisor thus,
x+A) x''-\-nx+S6 (
We next divide x^, the first term of the dividend, by x,
the first term of the divisor, which gives x as the quotient.
We now multiply the w/io/e divisor by x and put the
product, x'^-\-Ax, under the dividend. We then have
x^^:) jr2-f 13.^+36 {x
x^-\- Ax
by subtraction, 9.;i;-f36
Then divide 9jr, the first term of this remainder, by x,
the first term of the divisor, which gives 9 as the quotient.
Now multiply the whole divisor by 9 and put the product,
9a;-f 36, under the last remainder. We then have
6
82 DIVISION.
x+4. ) x^' + lSx+SQ ( x-{-d
x^-\- Ax
9jt:-f36
by subtraction,
whence the quotient is x-\-^.
B}^ comparing the above work in division with that of
multipHcation, the student will observe that the partial
products which, when combined, constitute the final
product in multiplication, occur in the work in division
as "partial dividends," which, when combined, equal the
original dividend. So that the process of division here
used consists merely in breaking up the dividend into the
■component partial products, from each one of which ojie
term of the quotient is obtained. Thus the above work
is merely a convenient way of breaking jt:^-f-13ji:+36 into
the two expressions, ( " partial dividends," )
which when written x{^x-[-A)-\-^{x-[-A)
•readily gives jt:+9 as the quotient by the method given
in the last exercise.
60. We give a few more examples where multiplication
and division are exhibited together, so that the student
may more clearly understand this method of undoing
multiplication.
(2) a + 1 . •
a +12
^2+ la
12^+ 84
a+7 ) «2 + i9^_j:'84 ( a-\-12
a"-\- la
12^ + 84
12^ + 84
ARRANGEMENT OF WORK. 83
(3) 2a -11
3^ + 4
6^2— 33«
8^-44
2^-11 ) 6^2_25a-44 ( 3« + 4
6^2-33^
8^-44
8«-44
(4) a +9
a — 5
-5^-45
fl! + 9 ) ^24.4^ — 45 (dr_5
a-+9g
-5^-45
Some prefer to write both divisor and quotient to the right of the
dividend. Thus :
(5)
Dividend,
15x*-44.r+32
-24X+32
-24a;-h32
3.X"— 4 Divisor,
5x-8 Quotient.
This arrangement saves space, and the divisor is where it is readily
multiplied by each term of the quotient.
61. In examples (4) and (5) above the student will
observ^e when the first term of the remainder is subtractive
that the corresponding term found in the quotient has the
opposite sign to the first term of the divisor. Of course
the reason for this is, that the divisor must be multiplied
by the term of the quotient to give the partial dividend,
84 DIVISION.
and it requires U7ilike signs to give minus in this multipli-
cation. If due attention be given to this fact no difficulty
will be found in obtaining the correct sign for each term
of the quotient.
In this statement it will be noticed that we have spoken of a term
having no sign at all as if it had the sign -|-. See Art. 17-
62. It is very important in the division of a polynomial
by a polyyioniial that both dividend and divisor be arranged
according to the powers of a common letter. It makes no
difference whether the arrangement be according to the
descending or the ascending powers of a common letter,
but both dividend and divisor should be arranged in the
sa77ie order. Any letter may be selected for this purpose,
but the letter which occurs the greatest number of times
in the given dividend and divisor, is naturally preferred.
If some powers of the selected letter do not occur in
the dividend, then it is well to leave a blank space in the
work for every such term. Thus:
(1) Divide a'^—b- by a + b.
a-^b)a''- ' -b'^ia-b
a'^-\-ab
-ab-b'^
-ab-b"-
(2) Also, divide a^ — b^ by a—b.
a-b') a^ —b^ (^a'^+ab+b'^
a^-a H
a'^b
oH-ab^
ab'^-b^
ab^^-b^
EXAMPLES. 85
EXERCISE 43.
Examples.
Divide
Divide
I.
x-' + 3jf— 40by ;r— 5.
8. ^2 +4^—45 by ^—5.
2.
X'-\-5x—(^ by x—1.
9. a^—4a^S2 by a— 8.
3.
,r2+7;r— 30by ;f— 3.
10. «2_}_7^_78 by «— 6.
4.
;r2+4;tr-5by;r-l.
II. «2_i2i by « + ll.
5.
x''-2x-eShyx-9.
12. ;i:2— ^jf— 6^2 by ;i-— 3«,
6.
x^-^7x—Ai by :r— 4.
13. ;i-2— 9^2 by A-— 3^.
7.
;^2_4by^_2.
14. .^-2-49J^'2 by jr+7)'.
15. a^+(yad+dd-
by a + 3^.
16. a^-—17ad + 7'2d-hy a—9d.
17. 2jr2-9A-+10 1
by 2a--5.
18. 3.r2 4-2jt;-l by 3;*:— 1.
19. 6j»;2 4-5;»:+21 by 3;t:+7.
20. 9a'2-64 by 3;f+8. '
21. 9;t:2-3;trj-2j2 by 3^-2^.
22. 4x^+4xf—3o_y'^ by 2jf+7>'.
23. 2x'^-^6ax—2Da'^ by x-^5a.
24. 4.r-+4«A-f «- by 2;i-+«.
25. oa6 + 15«5+5^ + 15 by « + 3.
5^7+15
5^-i-ii>
26. 42«*4-41a^-9«2_9^_l by 7«2_^8a + l.
7a»-|-8«+l ) 42^^44-4^3— 9rt*-9a— 1 ( 6a«-a — 1
42(z4+48^34- 6«*
— 7^?3 — 15rt* — 9a
— 1a^— 8«*— «
— 7a«-8rt-l
86 DIVISION.
27. Divide 21a^-^U^ by Za + 2b.
27^34-] 8^ V^
— 18^2^
12.//^ 2 +8^ 3
28. TiWidiQ a^ + b^ + c^ — 2>abc hy a + b-^c.
Arrange according to the descending powers of one of the letters,
say a. It is important to keep this arrangement throughout the work,
a-\-b-\-c ] a^ — ^abc-\-b^-\-c^ ( a^—ab—ac-\-b^—bc-\-c^
a^-\-a^b-\-a^c
-a^b-a^c —Zabc
—a^b —ab^— abc
~ —a^c-\-ab^ — tabc
— a^c — abc—ac^
ab^— abc-{-ac^-\-b»
ab^ _|-^3_j_32^
_ abc-\-(ic^ —b^c
* — abr — b^c—bc^
ac^ -^bc^--yc^
Divide
29. '2a^-^a'^-\-Za'^—Za^-\hya'^ — Za-^\.
30. l7n^—^m^-\-Zm^ — Zm-\-\hYm'^—^7n^-\.
31. G«5^-17a-jr2 + 14«jt3-3jt4 by 2^-3.r.
32. 4>/i_18>/3+22)/2-7j/+5 by 2y-5.
33. 4^-^+4a2-29a + 21 by 2^-3.
34. 45ji:'*4- 18-^3 +35j»;2+4jtr-4 by ^x^-^'lx-'l,
35. iV>-*«'^+if^'^'+i^^' by l-^ + i^.
37.x^-\-y^-{-Sxj—lhyx+j/—l.
38. «2_2^^ + ^2_^2^2^^-^-^ by ^-^+^-^.
EXAMPLES. Zj
Divide
39. a^ + b^-c^-2a''b'' hy a'^-y^-c'^,
40. l+x^+x"^ hy x'^ + l—x,
41. «-5 — 243 by «— 3.
42. l—Qx'^+bx^hyl — 2x-\-x'i,
43. j»;6-2^3j»;3+a« byjt:2-2a^4-a2.
44. Zx^-Zhy^x-'+^x+i.
CHAPTER VII.
NEGATIVE QUANTITIES.
EXERCISE 44.
Number and Quantity.
63. Anything which can be measured by a unit oi
the same kind is called a Quantity. Thus, 10 bushels
is a quantit}^, the unit being a bushel, and this unit
taken 10 times gives the quantity^ 10 bushels. Also 10
cords is a quantity, the unit in this case being one cord,
and this unit taken 10 times gives the quantity, 10 cords.
Also the abstract number 10 is a quantity, the unit in
this case being the abstract number 1, and this unit taken
10 times gives the quantity 10. Of course, the unit
itself is a quantity.
The word quantity as above defined, plainly includes
number, but while a number is a quantity, a quantity is
not always a number.
Five miles would be called a quantity and never be
called a number, but the number 5 may be called either a
number or a quantity indifferently.
The word quantity is usually used as here explained,
but some writers on Algebra never use the word quantity
to include number.
64. The answer to a problem in Algebra is often
something like 5 miles or 4 tons or 3 dollars, or some
other concrete quantity, but the reasoning is always con-
ducted by numbers, and so the letters used in Algebra!
always represent fiumbers, and the result reached is the
OPPOSITE DIRECTIONS. 89
number of miles or tons or dollars, or whatever it may
be, and then the name of the thing we are considering
may be added at the end to the number we have obtained
by working the problem.
EXERCISE 45.
Opposite Directions.
1. If a man travel east 30 miles and then west 20
miles, how far will he be from the starting point ?
2. If he travel east 30 miles and then west 40 miles,
how far will he be from the starting point ?
3. If the temperature is ?ero, and it rises 10 degrees
and then falls 6 degrees, how far will it then be from zero ?
4. If it rises 10 degrees and then falls 14 degrees, how
far will it then be from zero ?
5. If a man receive 50 dollars and spend 35 dollars,
the amount of money he has, differs from what he had
before by how much ?
6. If he receive 50 dollars and spend 65 dollars, the
amount of money he has, differs from what he had before
by how much ?
7. If a man travel east 30 miles and then west 20
miles, is he east or west of the starting point, and how
far?
8. If he travel east 30 miles and then west 40 miles, is
he east or west of the starting point, and how far ?
9. If the temperature is zero and it rises 10 degrees
and then falls 6 degrees, will it then be above or below
zero, and how far ?
10. If the temperature is zero and it rises 10 degrees
and then falls 14 degrees, will it be above or below zero,
and how far ?
90 NEGATIVE QUANTITIES.
11. If a man receive 50 dollars and spend 35 dollars,
has he more or less than he had before, and how much ?
12. If he receive 50 dollars and spend 65 dollars, has
he more or less than he had before, and how much?
66. It is plain that the last six of these examples are
very much like the first six, and yet these two sets differ
in one important respect. In the first six we are con-
cerned only with amount, in the last six something be-
sides amount is required, and this something we may.
for want of a better word, call Direction.
Illustrations might have been given involving other
kinds of quantity; as for example, degrees north or so2tth
of the Equator, longitude east or ivcst, time before or after
a given event, gain or loss, etc.
Plainly in those cases where we consider direction,
there are two directions, either one of which is just the
reverse, or opposite, of the other one; but where we are
concerned only with amount, there is no such thing as
reverse.
Time cannot be reversed. But time befoi^e can be re-
versed, and the reverse of it is time after.
Distance cannot be reversed. But distance east can be
reversed, and the reverse of it is distance west, etc.
EXERCISE 46.
How Directions are Distinguished.
66. Sometimes in Algebra w^e are not called upon to
consider anything but amount, or magnitude, and some-
times we are obliged to consider, both magnitude and
direction, and so we must in some way distinguish be-
tween the two opposite directions. To find out how to
distinguish between these opposite directions, let us con-
sider the following examples.
HOW DIRECTIONS ARE DISTINGUISHED. 9 1
The distinction between positive and negative is made
by means of the signs -f and — . To explain this let us
consider tv/o or three questions :
1. If a man has $1000, and he gains $500, and then
looses 1700, how much will he then have ? How much
less is IOOO + 0OO-7OO than 1000 ^
2. A man had a dollars and gained 500 dollars, and
then lost 700 dollars, how much did he then have ? How
much less is «4-o00— 700 than «? Write an expression
of two terms which shall be equal to ^ + oOO— 700.
The two terms +500—700 can be replaced by what
single term ?
3. A man, 100 miles east of St. I,ouis, travels east 50
miles and then west 75 miles, how far east ot St. Louis
is he then ? A man, a miles east ot St. Louis, travels 50
miles east and 75 miles west, how far is he then east of
St. Louis ? Is his distance east expressed by a -1-50—75 ?
Is it also expressed by a— 25 ?
67. Notice in these examples that two terms, one
additive and one subtractive, may be replaced by a single
term which in each example given has been a subtractive
term. Notice also, that in the expressions with three
terms, the additive and subtractive terms express oppo-
site directions, as in the last example the + sign was
used in connection with the distance traveled east, and
the — sign used in connection with the distance traveled
west.
We are naturally led to associate the + sign %vith one
direction and the — sign with the other direction.
In example 8, we could replace +50—75 b}^ —25.
Now we know that a journey of 50 miles east, followed
by one of 75 miles west, is equivalent to a single journey
92 NEGATIVE QUANTITIES.
of 25 miles west, so the term —25 which could be substi-
tuted for +50—75 comes to have a meaning, namely 25
miles west in this example.
In example 2, we could replace the two terms -^-500
—700 by the single term —200, and here, also, a gain of
$500 followed by a loss oi $700, is the same as a single
loss of $200, so that the term —200, which could be sub-
stituted for + 500—700, comes to have a meaning, namely,
$200 loss in this example.
Many other illustrations might be given, but these are
enough to show that we distmgidsh between the two oppo-
site directions by means of the signs + arid — .
We further .see that when the distinction is so made,
each term of the expressio7i co7nes to have a meariing^ and
so it does not ?natter which term is written first.
68. We see from this that the Signs Plus and
Minus have a Double Use in Algebra; first, to indicate
the operations oi addition and subtraction respectively;
-second, to distinguish between opposite directions as just
described. There is no danger of confusion arising from
this double use; the context will always make it clear in
which sense the signs are used.
EXERCISE 47.
Positive and Negative Numbers.
69. In Arithmetic we are concerned only with the
numbers 0, 1, 2, 3, 4, etc.,
and intermediate numbers ; but in Algebra we consider
besides these the numbers
0, -1, -2, -3, -4, etc.,
and intermediate numbers.
POSITIVE AND NEGATIVE NUMBERS. 93
We may represent the two classes of numbers con-
sidered in Algebra on the following scale :
. . -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . .
which extends indefinitely in both directions from zero.
The sign -f- perhaps ought to precede each of the num-
bers at the right of zero on this scale, but we will agree
that when no sign is written before a number the sign -}-
is understood.
70. Numbers to the right of zero in the above scale
are called Positive, and those to the left of zero are called
Negative, or, we might say,* numbers represented by
figures preceded by a + sign or no sign at all are positive
and numbers represented by figures preceded by a —
sign are negative.
71. In Algebra numbers are often represented by
letters, and we have already seen that a letter may stand
for a whole number or a fraction. We will now further
extend the signification of a letter by allowing that it
may stand for one of the numbers to the left of zero in
the above scale as well as one to the right of zero ; so
that while in the case of a number represented by figures
we can tell whether the number is positive or negative by
the sign that precedes it, yet in the case of a number
represented by a letter, we cannot tell by the sign before
it whether it is positive or negative.
A minus sign before a number always represeyits a 7121771-
ber of the opposite kind from that 7'epresented by the sa7ne
number ivith a plus sig7i or 710 sign at all before it.
We know that the number 5 is positive, but we do not
know whether a is positive or negative until we know
its value.
94 NEGATIVE QUANTITIES.
We know that — 5 is negative, but we do not know whether
— « is positive or negative until we know the value of a.
If «=3, then — «=— 3, and a is positive and —a is
negative; but if «=— 3, then — «=3 (because 3 is the
opposite of —3), and a is negative and — « is positive.
72. \^ie have seen that we are required to distinguish
between quantities opposite to each other and that this
distinction is made by means of the signs plus and minus;
for example, if +10 degrees means a temperature of 10
degrees above zero, then —10 degreed would mean a
temperature of 10 degree below zero, and if +10 miles
means 10 miles north of the equator, then —10 miles
would mean 10 miles south of the equator, and if +10
rods means 10 rods east of a given point, then —10 rods
would mean 10 rods west of the same given point, and
if +10 be 10 units of miy kind in any sense, then —10
would be 10 units of the same kind in just the opposite s&nso:.
In each case one of these quantities is positive and the
opposite one is negative. Either direction may be selected
as positive, and then, of course, the opposite direction
will be negative. But it is almost alwa3'S easiest and
best to select the quantity about which we are inquiring
in any given problem as positive.
Anything that increases the quantity we have chosen
as positive will also be positive, and anything that de-
creases our selected positive quantity will be negative.
EXERCISE 48.
Illustrative Examples.
I. One day a man travels east 100 miles, the next day
he travels west 250 miles, and the third day he travels
east 175 miles; how far is he then east of his starting
point ?
ILLUSTRATIVE EXAMPLES. 95
SOLUTION.
because the problem asks how far east the man is of his starting
point, we take distance east to be positive, and therefore, distance
west to be negative. We associate distance east with the -)- sign, and
distance west with the — sign. The distance the man is east of his
starting point will, therefore, be represented by
100-250+175
which is equal to 25. Hence, the man is 25 miles east of his starting
point.
2. One day a man travels east 100 miles, next day he
travels west 250 miles, and the third day he travels east
50 miles; how far east is he then from his starting point?
SOLUTION.
Because the problem asks how far east the man is of his starting
point, we take as before distance east to be positive, and of course,
distance west to be negative, and asssociate distance east with the -)-
sign and distance west with the — sign.
The distance the man is east of his starting point will then be rep-
resented by 100 - 250-f-50.
Now, these three terms may be' replaced by —100, hence we say
100-250+50= — 100. Therefore, the distance from the starting point
is represented by —100, which by our interpretation of negative quan-
tities means 100 miles 7vest of the starting point.
SECOND SOLUTION.
Let us now take distance ivest to be positive, and therefore distance
east negative, and everywhere associate distance west with the +
sign, and distance east with the — sign. Upon this assumption the
distance the man is west of his starting point will be represented by
— 100+250-50
which is equal to 100. Hence, as before, the man is 100 miles west of
his starting point.
3. A boy rows his boat up stream 3 miles in an hour,
and then rests an hour, when he floats down stream 5
miles, he then rows up stream again 3 miles in the next
hour; how far is he above the starting point ?
Between what two directions are we required to distinguish in this
problem ? Which is it most natural to take for the positive direction,
up stream or down stream ?
g6 NEGATIVE QUANTITIES.
4. A boy rows his boat up stream 3 miles in an hour
and then rests an hour, when he floats down stream 6
miles, he then rows up stream again 2 miles in the next
hour; how far is he above his starting point?
5. A boy rows his boat up stream 3 miles in an hour
and then re.ots an hour, when he floats down stream 6
miles, he then rows up stream again 2 miles in the next
hour; how far is he then below his starting point ?
6. A merchant was in business 3 years; the first year
he lost $2000, the second year he gained $500, and the
third year he gained $2600. What was his profit in
business ?
7. A merchant was in business 3 years; the first year
he lost $3000, the second year he gained $500, and the
third year he gained $2000. What was his profit in
business ?
8. A merchant was in business 3 years; the first year
he lost $3000, the second year he lost $500, and the third
year he gained $2500. What was his profit in business ?
9. A merchant was in business 3 years; the first year
he lost $3000, the second year he lost $500, and the third
year he gained $2500. What was his loss in business ?
10. A merchant was in business 4 years; the first
year he lost $2550, the second year he gained $1025, the
third year he gained $575, and the fourth year he gained
$900. Did he gain or lose, and how much ?
11. On a cold morning the temperature was 10 degrees
below zero, and it rose 12 degrees during the day; what
was the temperature at evening ?
12. On a cold morning the temperature was 10 degrees
below zero, and it rose 12 degrees during the day, and
fell 6 degrees during the night; what was the temperature
the next morning ?
negative quantities in addition. 97
73. Fundamental Operations with Negative
Numbers. Until the beginning of the present chapter the
letters used always stood for positive numbers, but now
a letter may stand for either a positive or a negative
number. Moreover, a letter standing alone with a minus
sign before it has a meaning now, but to know the mean-
ing of a letter with a minus sign before it, as, for instance,
—a, we must first know what a means when preceded by
a -f sign or by no sign at all, and then give to —a the
opposite interpretation. As letters now have a much
larger significance than before, we must briefly re-examine
the four fundamental operations of Algebra, viz.: Ad-
dition, Subtraction, Multiplication, and Division, to see
if the results previously obtained, on the supposition that
the letters stood for positive numbers, hold when the
letters stand for negative as well as for positive numbers.
EXERCISE 49.
Addition.
74. Addition is the process of finding the result of
two or more numbers taken together. The result found
is the sum.
This definition agrees with all before found, when all
the numbers used are positive, and, as we shall see, is suf-
ficiently broad to include the case of negative numbers.
If we wish to add -f 25 and —15 together, we get for
our result -f 25-15 or +10.
This is called adding —15 to -|-25, and is easily seen
to be the same as subtracting -fl5 from +25.
With this definition of addition, it is easy to see that
to find the sum of two or more numbers, we supply +
signs to terms having no signs, and then, write them
down with signs unchanged, and combine terms if pos-
sible, the same as in Chapter III.
7
98 NEGATIVE QUANTITIES.
To add 7, 5, and —8 we supply + signs before the
terms 7 and 5, and write
+ 7 + 5-8.
When the sign of the first term is + we may omit the
sign if we like, but we must never omit the — sign, and
never the + sign in any term but the first.
Addition is nicely illustrated by the scale of algebraic
numbers. For example, if we wish to add 6 to —4 we
begin at —4 on the scale and count forward 6 spaces,
arriving thereby at 2. Again, to add —6 to —4, we
begin at —4 on the scale and count backward 6 spaces,
arriving thereby at —10.
Now, if we use letters, we can easily find the sum by
the method here given.
Suppose we wish to add ^, — ^,and —c. Supplying +
sign before a, and writing down with signs unchanged
we get, -\-a—b—c.
Suppose next we wish to add a-\-b, b—c, and^— /.
Supplying + signs before terms <2, b^ and ^, and writing
down with signs unchanged we get,
-\-a-\-b-\-b—c-\-e—f
or a-\-2b—c-\-e—f.
Exactly the result obtained by the method of Chapter III.
Any result reached in Chapter III could easily be tested
by the notion of addition here presented, and found cor-
rect. Therefore, the methods and results of Chapter III
hold, whether the numbers used are positive or negative.
One important difference may now be noticed between
addition in Arithmetic and Algebra. In Arithmetic ad-
dition implies augmentation, but in Algebra this is not
necessarily the case.
NEGATIVE QUANTITIES IN SUBTRACTION. 99
Examples.
1. Add a-\-d-\-c, —Za—b-\-c, and —a—b—4iC.
2. Add —r—2s—Zt, —r-^t, and — 2r— 2^.
3. Add a—b-\-c, —a-\-b-\-c, and a-\-b—c.
4. Add«4-^, — 1^, — 1^, and 4^.
5. If x=a-\-b—'2c, y=a—2b+c, and 2'=— 2a + ^+^
show that x+y-\-2=0.
6. If x=a + b—c, jy=za—b-\-Cy and ^=--a4-^+^ show
that x-}-jy-i-2=a-\-b-\-c.
EXERCISE 50.
Subtraction.
75. Subtraction is the reverse of addition /. e.j it is
the process of finding a number called the remainder,
which added to the subtrahend will give the minuend.
For example, 12 is a number, which added to —5 will
give 7; therefore, —5 subtracted from 7 gives 12.
It is easy to see from this meaning of subtraction that
to subtract one number from another, we chayige the sign
of the subiraheiid and then unite it to the 7ninuend.
This is exactly the method of Chapter IV ; therefore,
the methods and results of Chapter IV hold, when the
letters used stand for negative as well as positive numbers.
Examples.
1. From jr2+jj/2 take x'^—y'^.
2. From x-\-a—W^ take —x—Sa + b^,
3. From «--^4-<^— ^take ^z+^— ^4-^.
4. From 2?i^+Sa^ — r^—s^ take n^—a^-\-r^—2s^.
5. From ^24. 2^3+^2 take a^-2ab-hb\
6. 'Brova. uvw'^^2tiv^w-\-S2i^vw take Suvw'^-{-2uv'^w
lOO NEGATIVE QUANTITIES.
7. From the sum of a^-{-d^ and —2ad subtract the sum
ofa2— ^2and3^3
8. From x^ -^-ax^ -^a'^x-}-a^ subtract 2ax'^—a^x, and
from this difference subtract 2ax'^—a'^x.
9. What must be added to r^-\-s^ + t^ to produce 3 ?
10. What must be subtracted from adc^ to produce m-\-r?
11. What must 9ad be subtracted from to produce —ad?
EXERCISE 51.
Multiplication.
76. The definition of multiplication given in Chapter
V is sufficient to include the case of negative numbers.
What is the product of 6 multiplied by — 3 ? To pro-
duce —3 from unity we take unity three times and reverse
the result. Hence to multiply 6 by —3 we must take 6
three times and reverse the result, giving —18.
What is the product of —6 multiplied by 3 ? To pro-
duce 3 from unity we take unity 3 times ; hence to mul-
tiply —6 by 3 we take —6 three times, giving —18.
What is the product of —6 multiplied by — 3 ? To pro-
duce —3 from unity we take unity 3 times and then reverse
the result ; hence to multiply —6 by —3 we take —6 three
times, giving —18, and reverse the result, giving -f 18.
From these illustrations it is evident that numbers can
be multiplied as in Chapter V whether the letters used
stand for positive or negative numbers. It is also evident
that we have here another and very nice demonstration
of the ''Law 0/ Signs'' in multiplication.
Examples.
Find the product of
1. 10 multiplied by —4. 3. —| multiplied by — y\.
2. —-25 multiplied by 4. 4. adc multipYied by —a^d'h.
NEGATIVE QUANTITIES IN DIVISION. lOI
5. —fms^ multiplied by ^m^s.
6. a^-\-d^ multiplied by —a^ + d^.
7. —habc—Zrs multiplied by f)abc—Zrs.
8. —«—^—r multiplied by —a4-33— 9^.
9. 5^3 + 6^3 multiplied by — 7r+25— /.
10. —x^y^—xz^-\-yz^ multiplied by ^xy—\y^,
EXERCISE 52.
Division.
77. Division is the reverse of multiplication, that is,
it is the process of finding a number called the quotient,
which multiplied by the divisor, equals the dividend.
We may express this in the form of an equation thus,
Divisor X Quotieiit = Divideiid.
As here arranged this is a case of multiplication where
the divisor is the multiplicand, the quotient is the multi-
plier, and the dividend is the product. As we know in
multiplication, that when the multiplier has the sign +
the signs of the multiplicand and product are alike, it
follows here that when the quotient has the sign -f the
dividend and divisor must have like signs, or stated in
the reverse order, when the signs of the dividend ayid
divisor are alike the sign of the quotient is + .
It is also easy to see that when the quotient has the
sign — the signs of the dividend and divisor are unlike,
or stated in reverse order. Whe^i the signs of the dividend
and divisor are imlike^ the sign of the qnotieyit is — .
These two statements put together give the Law of
Signs in Division, viz.: In division like signs give + and
unlike sig7is give — .
102 NEGATIVE QUANTITIES.
Examples,
1. Divide 10 by —5. 5. Divide —63 by —7.
2. Divide 5 by —10. 6. Divide a} by —a.
3. Divide 27 by —3. 7. Divide aH^ by — ^*.
4. Divide —63 by 7. 8. Divide —aH^c^ by a^b'^c
9. Divide m^r^s'^ by ^ms^.
10. Divide 6^3__^2_i4^^3 ^^^ 3a2-f-4«— 1.
CHAPTER VIII.
PARENTHESES.
EXERCISE 53.
Removal of Parentheses.
78. The subject of parentheses has already been con-
sidered to some extent, and we have already learned that
an expression within a parenthesis is to be looked upon
as a single number just as though it were represented by
a single symbol.
Now, it may happen that an expression within a paren-
thesis is itself an expression which contain^ a parenthesis,
so we would have a parenthesis within a parenthesis. In-
deed, we may have several parentheses one within another.
These complicated expressions present no difficulty, for
we can take the parentheses one at a time, and if w^e know
how to remove one, we may do this and then remove an-
other, and so on until all are removed. For example, if
we wish to remove the parentheses from
we begin by removing the inner parenthesis first and
write the expression in the form
and now by removing the remaining parenthesis we write
the result in the final form
a-\-b—c-\-d-\-e,
79. In removing parentheses it is usually best to re-
move the innermost parenthesis first, and then the inner^
most parenthesis of all that remains, and so on until all,
or as many as may be desired, are removed.
I04 PARENTHESES.
80. When several parentheses are used one within
another, they are often made of different shapes and
sometimes of different sizes to prevent confusion. Some
of the forms used are, ()>{}»[]• Sometimes a hori-
zontal line, called a Vinculum or bar, is drawn above an
expression instead of using a parenthesis. Thus,
means the same as <2-f(;tr—jj/).
Remove the parentheses from the following expressions :
1. a-\-[d-ic-\-d)'].
2. a-^ld—(c-{-d—e)-{-2'].
3. Sa-^2d-li5d^-4)-(U-2)l
4. w2+«2_(^3_|-^3^5(2:r-fl)]-r).
5. 4;r-f3>/-5;tr-[2)/-(6;r-6>/)].
6. 4a-(Qa-l5a-(ia-2a)'\).
7. Sx-(4y-l6x-(Qy-7x)]).
8. Sx+i-4:yj-l5x-C6y-7x}]).
9. 5a^-(4d^-[S(_a^ + d'')-4.(x-2)])-{-2.
EXERCISE 54.
Insertion of Parentheses.
1. If the parenthesis be removed from a + {d—c-{-d),
what is the result ?
2. If the parenthesis be removed from a — (^—b-\-c—ci)
what is the result ?
Notice how the expressions within the parentheses in these two
examples are related, and notice, also, how the results are related.
3. If from any expression a parenthesis preceded by a
-f sign be removed, what is the effect on the terms
originally within the parenthesis ?
INSERTION OF PARENTHESES. I05
4. How, therefore, can any number oi terms be brought
within a parenthesis, preceded by a -f sign ?
5. If from any expression a parenthesis preceded by a
-— sign be removed, what is the effect on the terms orig-
inally within the parenthesis ?
6. How, therefore, can any number of terms be brought
within a parenthesis, preceded by a — sign ?
7. Enclose the last three terms of a^-\-d^^c'^-\-5 in a
parenthesis, preceded by a + sign.
8. Enclose the last three terms of a^ + b^ — ^* + 5 within
a parenthesis, preceded by a — sign.
9. Enclose the last two terms of a- — /5^— c*+5 within
a parenthesis, preceded by a + sign.
10. Enclose the last two terms of a'^-\-b^—c^-\-b within
a parenthesis, preceded by a — sign.
11. Enclose the last three terms of a-\-b—Ac—^e'^-\-Qr^
— w* — 16 within a parenthesis, preceded by a + sign.
12. Enclose all but the first term of a-f ^— 4<:-f5^^-f 6r^
— ;2* — 16 within a parenthesis, preceded by a — sign.
13. Enclose the third and fourtli terms of a-\-b—Ac-{-be'^
+ 6r^— w^ — 16 within a parenthesis, preceded by a —
sign, and the fifth, sixth, and seventh terms of the same
expression in another parenthesis, preceded by a — sign.
14. Fill out the blank parenthesis in the equation
«2 + (2^_l)=a2-( ).
15. Fill out the blank parenthesis in the equation
aH-\bcd-Zx''-Qt^-Si)'\=a''b+(^ ) + 7/*-9.
CHAPTER IX.-
ELEMENTARY FACTORS, MULTIPLES,
AND FRACTIONS.
EXERCISE 55.
Factors.
81. A definition of factor has been given, (see page
12), and we have already learned that an expression may
have several factors. For example, the different factors
of lO;^^ are
2, 5, 10, X, 2x, 5x, 10;r, x\ 2x^, 5x^.
Of these factors 2, 5 and x may be called Prime, because
they cannot be further factored.
The expression lO;^^ contains the prime factor x twice,
so all the prime factors of IOjt^ are 2, 5, x, x; and as
any expression equals the product of all its prime factors,
we have
10ji:2 = 2x5;r;i;.
When an expression is written as the product of all its
prime factors, it is said to be Resolved into its Prime
Factors.
Resolve the following eight expressions into their
prime factors :
1. SOx^y\ 3. SSad^c^. 5. ISSr^-s^. 7. 2431uv^w^,
2. 150^5^^ 4. 51m^r\ 6. 42dadn. 8. 25dx^j;2*,
9. Find three of the factors BOx'^y^ which are not prime.
10. What are the different prime factors of 15a ^ ?
11. What are all the prime factors of \ha^ ?
12. Resolve %a^b^c^ into its prime factors.
HIGHEST COMMON FACTOR. lO/
EXERCISE 56.
Highest Common Factor.
1. Does the factor a belong to each of the two expres-
sions ha'^-x and lax""" ? Does the factor x belong to each
of these two expressions ? Does the factor ax belong to
each expression?
When the same factor belongs to two or more expres-
sions it is called a Common Factor of those expressions.
When a -brime factor is common to two or more ex-
pressions it is called a Common Prime Factor of those
expressions.
2. If 2 and 3 are each factors of an expression, is 6 a
factor of that expression ?
3. If a and b are each factors of an expression, is ab a
factor of that expression ?
' 4. If a and b are each comnioyi factors of two or more
expressions, is ab a common factor of those expressions ?
5. Is the product of any number of common factors of
two or more expressions, a common factor of those ex-
pressions ?
82. The product of all the common prime factors of
two or more expressions is called the Highest Common
Factor of those expressions. The abbreviation H. C. F.
is frequently used to stand for the highest common factor.
83. From the definition of the H. C. F. it follows at
once that the way to find H. C. F. of two or more expres-
sions, is to resolve each expressioti into its prime factors,
afid take the product 0/ all those which are coifmioji to all the
expressions.
I08 FACTORS, MULTIPLES, AND FRACTIONS.
Find the H. C. F. of the following expressions ;
1. x^jy and x^j'^. 5. 7uvw and lOv^wx"^.
2. ^abc^ and lla^b. 6. x^y, ^xyz and lOx'^2-.
3. 6r^2 and 15r-^/^. 7. Zx'^yz, Ibxzw^ and \2xyw.
4. la'^bc^ and 14a^f^. 8. 4mn^, Smn^'x, and 12:r?;2;?®
9. 50^3^3^% 75aHcd, and 80 a^M.
10. 5^5*/^ Tr^^jt:*, 9r2;i;3, and Urs^x.
11. 35;»;3jK, 63aJr2^^ and 70abxy2.
12. 17«5^6rV8, 51;;z3a*/^3^5^S and Sira^d^c^dK
13. ^2jK^, AT^/^^, ;rj'2'2, and x'^y'^z'^.
14. 2rs^uv, Zr'^stu, A.rs'^tv, and Sr^^y^/ze^^.
15. lu^v^w'^ , S5m?iu'^v^y ISQr^u^v^w, and 77iiv^w.
16. 21«3^4^6^8, 63^9/^7c6^jj/-3', 84«2^3^52^^^^ and
49«6^V;r.
17. 99;t:2, 187^^jK, 2537<^, 143^2;, and 275;*;*.
18. 2a^-n^x^, 12bn^y, lOOc^fz^x, and Ae^i^xy^.
19. IGa^^jr^, SOb'^ny^, ZQabr^x\ and 28^^r^^
20. lZx^y^z\ dOx^yz, Uxy\ 360>/*^, and ZQxs.
EXERCISE 57.
Lowest Common Multiple.
84. When any number or expression is multiplied
by something, the product is called a Multiple of the
given number or expression. Thus, 10 is a multiple of 5,
50 is another multiple of 5, 100 still another multiple of
5, etc.; 10 is also a multiple of 2 as well as of 5, 50 is a
multiple of 25 as well as of 5, etc.
LOWEST COMMON MULTIPLE. IO9
1. Is 100 a multiple of 5 ? Is 100 a multiple of 10?
Is 100 a multiple of 2 ?
2. Is 2^2 a multiple of 2 ? Of « ? Oi a^ ?
3. Is «<^2^- a multiple of a? Of ^? Of ;tr ? Of «^ ?
Of«^? Of adx?
85. From these questions it is evident that the same
expression may be a multiple of several different expres-
sions, in which case it is called a Common Multiple ot
those expressions.
4. Is ad'^x a common multiple of ad, ax and abx ?
5. Is a'^bx also a common multiple of ab, ax and abx"?
«
6. Is a^b^x'^ Si common multiple of the same three ex-
pressions ?
86. From what is here given, it is plain that two or
more given expressions may have more than one com-
mon multiple. Indeed, if any common multiple of two
or more given expressions be found, then if this common
multiple be multiplied by any number whatever, the
result will also be a common multiple of the given
expression.
Any common multiple of two or more expressions
contains all the prime factors of each of the given ex-
pressions.
That common multiple which contains the leasf number
of prime factors is called the Lowest Common Multi-
ple of the given expressions. The abbreviation I^. C.
M. is frequently used to stand for the lowest common
multiple.
no FACTORS, MULTIPLES, AND FRACTIONS.
87. From what we have had it follows that to find
the L. C. M. of two or more expressions we proceed as
follows :
Resolve each expressioji mto its prime factors and form a
product in which each of these prime factors occurs as many
times as it occurs in that one of the given expressio7is in
which it occurs the greatest 7iumber of times.
Find the L. C. M. oiZa'^x'^y and ZOax'^,
Sa^x^y — 'daaxxy;
'60ax^ = ^X2X5axx.
The prime factors are 2, 3, 5, a, x, y. The prime factor 2 occurs
once in the second expression, hence it occurs once in the L. C. M.
Similarly, the prime factors 3 and 5 occur once in the L. C. M. The
prime factor a occurs once in the second expression, but twice in the
first expression ; hence it will occur twice in the L. C. M. Similarly,
the prime factor x occurs twice in the L. C. M, Finally, the prime
factory occurs once in the L. C. M. Collecting results, we see that
the L. C. M. is equal to 2X3X5 «« ^ ^/, or 30a^x~y.
Find the L. C. M. of the following expressions:
1. Ux'^y'^ and42xy^z. 4. 17a'^t?'^c- and 17 a^d^c^.
2. 7xjyz and Sax'^y^z'^. 5. 27x'^y^2^ and 2m'^x^y.
3. dadc and 15a'^d^x^y. 6. 14x'^y'^, 9adc, and 7 xys.
7. 42xy^2:, ^ax'^y^z^, and bw^.
8. a'^b'^c'^x^y^z^ , abcu^v^w"^ , and uvwxyzabc.
9. hax, lOay, 25b'^z^, lOOa'^c^, and 50abcxy^z.
10. brst, 12, rt, 30, 20r/2, Ibst^ , and 4s^t.
11. 14rV6, 2Uc\ 70a''bc, 10^oad\ and ZUcd'^ .
12. 2bcd\ 15ad^, 14b^c^, lOb^d^ 21 a'' c^, and S5bd.
13. 5ac\ lOac', 7d\ ^a'^b'', lb b^c^, and 14ad'^ .
14. 30«2^V3, 70ac''d\ 42a''b^d\ l^hb^c'^d^ , and Zhc^dK
15. 20jr|/2^, 4hxz'^v'', ZQzu'^v^, Sxyzu'^v^, and 5 u^v^,
16. QOzv, 45xz^v, dOy'^zuv, 9xz7c'^v, and 30;t:z;^.
FRACTIONS. Ill
17. Multiply the L. C. M. of Sa-xy and 2ax^j'^ by the
H. C. F. of the same expressions.
i8. Divide the product o{ a-dx and ad by the H. C. F.
of a'^dx and ad, and compare the quotient with the ly. C.
M. of a-dx and ad.
19. Divide the L,. C. M. of 2a'^x^y and 3 ax'^jy by the
first of the two expression, and compare the result with
quotient obtained by dividing the second of the two ex-
pressions by the H. C. F. of the two expressions.
20. Divide the product of ba^x^j^ and 15adx*2, by the
L. C. M. of the same two expressions, and compare the
quotient with the H. C. F. of the same two expresssions.
EXERCISE 58.
Fractions.
88. We have already used the fractional form j as
another way of writing a-i-d, so that -r is an expression
of division. We have already learned that in any case
of division the divisor multiplied by the quotient equals
the dividend, or, in the language of fractions, precisely
the same thing may be written,
Denominator x Quotient = Numerator.
89. From this equation it is plain to see that if the
denominator remains unchanged, multiplying the numer-
ator by any number multiplies the quotient by the same
number, and dividing the numerator by any number
divides the quotient by the same number, or, as it is
more often stated, multiplying the numerator dy any num-
der multiplies the fraction dy that numder, and dividing the
numerator dy any 7iumder divides the fractioji dy that
numder.
112 FACTORS, MULTIPLES, AND FRACTIONS.
90. Again, from the same equation,
Denominator X Quotient = Numerator,
it is also plain that if the numerator remains unchanged,
multiplying the denominator by any number divides the
quotient by that number, and dividing the denominator
by any number multiplies the quotient by that number,
or, as it is more often stated, mtiltiplying the denominato?
by any number divides the fraction by that number, and
dividing the deiiominator by any number multiplies the
fraction by that number.
91. Once more, from the same equation.
Denominator X Quotient = Numerator,
it is plain that if the quotient remain unchanged, multi-
plying the denominator by any number multiplies the
numerator by the same number, or, stated in another
way, multiplying both numerator and denominator by the
same number does 7iot alter the value of the fraction. It is
also evident from the same equation that dividing both
numerator and denominator by the same number leaves
the quotient unchanged, or, stated in the usual form,
dividing both numerator and denominator by the same num-
ber does not alter the value of the fraction.
92. When the numerator and denominator of a frac-
tion are each divided by all the factors common to both
numerator and denominator so that the resulting numer-
ator and denominator contain no common factor, the
fraction is then said to be in its Lowest Terms.
Of course, then, to reduce a fraction to its lowest terms
we divide both numerator and denominator by every factor
common to both, i. e., by the H. C. F. of the numerator and
denominato? .
ADDITION OF FRACTIONS. " II3
Reduce the following fractions to their lowest terms :
7. T
I.
oax
\Ux'^'
2.
32^2 j«:2
^'^bcx '
Vlabc
^ ZSixz^ ' 187aiijir9j/5-
25wV*£^ 209r3^V*z;
Ibcde' ' liy^axy ^' SOlaH^z'
S00x*y^zu^
EXERCISE 58.
Addition of Fractions.
93. If two or more fractions have the same denom-
inator, the fractions may be added by adding the
numerators and placing the sum over this common de-
nominator; but if the denominators are not the same we
must multiply the numerator and denominator of each
fraction by such a number as will make all the denom-
inators the same, and ^/ie?i add the numerators, and place
the sum over this common denominator.
94. The process of changing the numerators and de-
nominators of fractions, so that each fraction shall pre-
serve the same value it had before, while the denominators
are all made alike is called Reducing to a Common De-
nominator.
For example, suppose we wish to add — - and — to-
m^ mn
gether. We must reduce to a common denominator,
which, of course, must be a common multiple of w-
and mn. Any common multiple will do, but the lowest
common multiple is preferable.
114 FACTORS, MULTIPLES, AND FRACTIONS.
The lowest common multiple is plainly vi'^n\ hence, we
must multiply the numerator and denominator of the
first fraction by ?2, and the numerator and denominator of
the second fraction by m. We then have
Plainly, then, the sum of the two fractions is
Aiabn-\-^mx
m'^71
Hence, we may write the equation,
ab 4:X _ 4ab?i + 4mx
m'^ 7nn m'^71.
If we had three fractions to add together, would the
common denominator be a common multiple of each of
the given denominators ? Would there be any preference
for the lowest common denominator ? Why ?
If any mmiber of fractions are to be added together,
what would you prefer to take for a common denominator ?
State the method to be pursued when two or more frac-
tions are to be added together.
Add the following fractions :
2« , 3« cd ^ cd m^ r'^ , r'^s'^
and 7y-. 3. --^. and -— x. 5. -^— - and
X 2x' ' x'^ y xy^' ' bxy Ibx
ab ^ X abc , ab^^c^ _ Aiab . 4cd
2. — and — T— . 4. and - — —-:^. o. ^^r— , and t^—,.
m ni^n xyz xyz"" Ilea liao
Ub'' ^ Iz lOa'-b^ ^ 7
_ \Qrst , IQrst 12uv , 13^/
8. — and -^ . 10. -tr^-- and -.-^ — .
a X y"^ Auv 12
x"^ m^ u 14xy , 7
12. — ^, — ^, - — ^-, and -^-
3*
SUBTRACTION OF FRACTIONS. II 5
EXERCISE 59.
Subtraction of Fractions.
If one fraction is to be subtracted from another and the
fractions have the same denominator, we may subtract
the second numerator from the first and place the re-
mainder over this common denominator ; but if the
denominators are different, we must first reduce the
fractions to a common denominator and then perform the
subtraction.
^ Sx , Ax ^ \babc ^ , xyz
1. From TT- take 5— I. 3. From take ., r , .
2a 3^2 ^ xyz \habc
Za'^x , 4cx ^ lOaHc. Vlab
2. From —^rr- take ^r^- 4- From take
2b 3^2- t- ;^5^^^ ""^^^20^2^
5. From— p— take ^2_^2-
(xbc^ cdc"^
6. From y-j^ take -7-7^.
bcd^ dej ^
7. From -^- take j^-.
12 i xyz
„ ^ '?^ab , V2xz
o. From jr r, take 771 — ^-.
\)xyz~ Ibxz-u
^ Ma . Qx
9. From fTz take 7^^ — ^— r.
12mr^ . 14;z^2
10- I^rom ^^^^:— y- take -,
20;«V 2l7i's
Il6 FACTORS, MULTIPLES, AND FRACTIONS.
EXERCISE 60.
Multiplication of Fractions.
95. We are to multiply -r by -7.
Now to multiply by -3 means to multiply by c, and di-
vide the result by d. Therefore to multiply t by - we
first multiply t by c, and then divide the result by d.
We may multiply a fraction by multiplying the nu-
, a ac
merator; hence, -7X^=-r.
o
We may divide a fraction by multiplying the denom-
. , - ac . ac
inator; hence, -r-i-«=T3-
oa
/TM r a c ac
Therefore, -7 X -7 = -r>
a oa
Hence to multiply two fractions together multiply the
numerators together for a new numerator^ and the denom-
inators together for a 7iew demoninator.
96. Suppose we wish the product of three fractions,
r ■ ^ ace
as for instance, -7 x -7 x >.
d f
We may multiply the first two together, as just ex-
plained, giving ^
and we can then multiply this result by the third fraction,
by the method just explained, giving
ace
'bdf'
ATvt, c ace ace
Therefore, x - X -r= 7-^ .
b d f bdf
MULTIPLICATION OF FRACTIONS. II7
As this can be extended to any number of fractions,
we have :
The product of any nuviber of fractio7is is found by mul-
tiplying all the numerators together for a new numerator,
and all the de7iominators together for a new denomhiator.
Multiply the following fractions :
^ab . 5fd a^x"^ bxy , Qbys
'• A^i ^"'^ 6i^- 3. -j^,, --.-. and ^.
4xy , Suvw 7ir u^ x , v^
2. 7;-^ and — —T— . 4. , 1, and — r.
b 7iv Vlxyz 21V wx* xz^
Arstu'^v lOx'^y labu'^v'^
buvxy ' \^u^v^-t' rsx
143 nrs Six^z 2w^ bab^ Icd^ %e£
' ISlxyz' '6\)wt' Ss ' ^' 6xy Siw' ^^^ 70w
a^ a^ a^ a^
x^' x^' x^' x^'
^2^2^2 2 ^ 7ia a b bx ^ y^
9. — :j — , , and -irr- ^°- ""' » > and -^.
1 7n'w a- be x y ay b^
a* x'^ y'^z , w'^xyz
II. i, , ■^, — ^, and ~-,
X* y z-w^ abc
abc a^x bz^ cy
12. , -ry-, --, and 5—3.
xyz b^z xy^ xy^z^
2rst^ 42jr^2 3^,^^,^ l^r'^x
^3. 1---1. 1 A .. > o^.- > and
4xyz^' Uy-^' 27yz' SSrxz^'
571VW 187ivz^ Vl7ivxy
14- ^r~i 1 — ^ -■ " o > and -r:: r.
Vlxyz lovw^y Sovwz*
Idabrst^ 51az>s'^x IQawx^z
\l7ivn^xy 64:b7(r'^wy' d7ivyz^
m'^n^r^s^ bux^fy"^ , m^r^
16. , — = — f^, and TT — .
xyz ' Icz^ ' 9r
Il8 FACTORS, MULTIPLES, AND FRACTIONS.
i8.
axys d'^ti'^x'^ w^x . tvw
7 -' ^^-^' — 1-> a^^ ■
Duvw r-y^z-' yz* xyz
abk mr r'^kv , l(Smii
19. ■ ^, -^, — - — ^, and -p .
uvx^ u^ wx"^ bxyz
ax b'^z A.WX . rxy
by c^y mz- syz
EXERCISE 61.
Division of Fractions.
97. We are to divide -.- by -3.
a
We may write the quotient in the form of a fraction,
where the numerator is itself the fraction -r and the de-
nominator is the fraction — .
a
a
Hence —= quotient.
Let us multiply both numerator and denominator of this
fraction by bd. We know this will not change the value
of the quotient.
Hence
Therefore
a
1''
bd=
abd
'' b~~
=ad,
c
bd=
bed
d
--be.
quotient=
ad
''be-
a
T
c
ad
~ be'
DIVISION OF FRACTIONS. II9
This result ma}^ be obtained by multiplying the fraction
-. by the fraction — , which last is the divisor inverted.
c
Hence, to divide one fraction by another, invert the
terms of the divisor and multiply.
2.1- \z ^. .^ Imr ^ ^r'^t
,,- by -^~. 3. Divide —7.7- by -^-^.
6v bv btx Ix^
_. ., rst . 2nv _. ., 34jt-i' 2y^z
2. Divide — — by — ^ — . 4. Divide ^^ by -^ — r,
uvw 6sw oliiv 6vw^
^. .^ \Omk'' ^ ZUs^
5. Divide -^r-j- by — ,,^ ^ .
b;'5^ blx-2
6. Divide ^,— bv
5to - lOk'-xz
—2a.
Dividing both sides by —2/;/
_a-|-3
~ m '
which is the value of x.
ax—b bx-\-c
2. =aoc.
c a
Multiply both sides by ac
a[ax — h') — c[l)x-\-c^^=^a.^bc*.
Remove the parentheses
a^x—ab—bcx — c^r=za'^bc*.
Transposing known numbers to right side
a^x — bcx^=^a'^ bc'^ -\-ab-\-c^ ,
Uniting with a parenthesis
{a^—bc)x—a^bc^^ab^c*.
Dividing both sides by {a^—bc)
■ _ a^bc^-\-ab-\-c^.
^~ a^-bc '
3. x-\-a=d, 8. 77ix=r.
4. Sx-\-6n==dr. 9. mx-\-p—r.
5. a-\-b—x—^. 10. V8-\-ax—b=2x.
6. x-\-a—b—c. II. x-\-hx—b==2a.
7. J)/— 3 + ^=8. 12. Za-^2y—U=-hy—b.
13. 6a-8^+6«j/=4«-6^-2^+8m'.
14. hmx—^a—\\b=Z7nx—^b-\-lc—hmx—2a.
SYMBOLIC EXPRESSIONS. 1 29
15. 12(x-c) = 7(ic+x). 20. 5(5c-2x') = 4x-Qc.
16. S(a—5x)=4:(Sx—2c). 21. a(bx—c)=ac—adx.
17. 7(a—x)=6(d—x). 22. ;^;r— r=;ir.
18. (d—l)j=d—j/. 23. ad—dy=-my—am.
19. (l — /^)ji:= «—;»;. 24. /)(;t:— l)-f •^=^— /.
25. r(l4-jr) + «(r+/>)=r;i:+w/+;ir.
26. «r— ;?(_>/+ 1)+j=?2(2—jk).
27. (^+l)j»;+a/^=^(«+^) + «.
„ U{^x-a) , (jt:-^2) (4a-\-cx)d
20. p 1 r^r-7 = ^ .
ba loo oa
nx r—x , 7i(r—x)
^9. V— 2^-+-3^ = "-
30.
"lax— 2b ax—a_ax 2
3^^ 2b~~~b ~ 3*
^4-<: , a ,^ a{a-V)-b{b-V)^-c
^ X^ X ^ X
EXERCISE 64a.
Symbolic Expressions.
1 14. In solving a problem in Algebra we must not only
select some letter to stand for the unknown number, but
we are required to find expressions which will sj'^mbolize
all other numbers which occur in the problem. Such
may be called Symbolic Expressions. Thus, if the
sum of two numbers is 100, and if x stands for one of
them, then the expression 100— ji: stands for the other
number. If x is the price of one horse, then V^x stands
for the cost of 10 horses; if 5 yards of cloth cost x dol-
lars, then the cost of one yard is represented by the ex-
X
pression -, etc. Some drill in the formation of symbolic
expressions will be of help in the solution of problems.
130 SIMPLE EQUATIONS.
1. The sum of two numbers is 85. The first number
is 8, what is the second ? The first number is n, what is
the second ?
2. The sum of two numbers is a. The first number is
5, what is the second ? The first number is b, what is
the second ?
3. A train travels at the rate of 20 miles per hour for
3 hours; how far does it go? A train travels at the rate
of r miles per hour for 3 hours; how far does it go ? A
train travels at the rate of r miles per hour for / hours;
how far does it go ?
4. What must be added to 100 to make a ? What must
be added to x to make a ?
5. One factor of 100 is 10; what is the other? One
factor of 100 is x\ what is the oth^?
6. What two numbers differ from 100 by 7 ? What two
numbers differ from 100 by ^ ? What two numbers differ
from nhy xt
7. How much will n apples cost at c cents apiece ?
8. If one apple costs 2 cents, how many can you get
for X cents ? If one apple costs c cents, how many can
you get for x cents? How many can you get for d
dollars ?
9. How many hours will it take to go x miles at 4
miles per hour ? How many hours will it take to go x
miles at r miles per hour ? How many minutes will it
take?
10. A train goes 150 miles in h hours; what is the rate
per hour of the train ? A train goes m miles in h hours;
what is the rate of the train ?
11. The rate of a train is r. How far will it go in
time / ?
SYMBOLIC EXPRESSIONS. 131
12. What is the interest on d dollars for t years, at 5
per cent.? What does the principal and interest amount
to for this time ? What is the interest on d dollars for /
years at r per cent.?
13. A man can do a piece of work in 10 days. How
much of it can he do in one day? A man can do a
piece of work in n days. How much of it can he do in
one day ?
14. A can do a piece of work in 9 days and B can do it
in 12 days. What part can each do in one day ? What
part can both, working together, accomplish in one day ?
A can do a piece of work in a days and B can do it in b
days. What part can both, working together, accom-
plish in one day ?
15. A pipe will fill a cistern in 7 hours; what part runs
in during one hour ? Another pipe will fill the cistern
in 5 hours; what part runs in during one hour? The
cistern holds x gallons. How many gallons does each
pipe carry in one hour ?
16. One pipe will fill a cistern in a hours, and another
pipe will fill it in b hours. What part does each pipe
carry in one hour ? What part do both pipes together
carry in one hour ?
17. The digit 5 stands in tens' place; what number is
expressed? The digit, represented by x, stands in tens'
place; what number is expressed ? A digit represented by
X stands in hundreds' place; what number is expressed?
18. If the first and second digits of a number are 5
and 7 respectively, what is the number? If the first and
second digits of a number are represented by a and a + 2
respectively, what is the number ?
132 SIMPLE EQUATIONS.
19. The three digits of a number beginning at the
right are represented by .r, x+2, and x—S; what is the
number expressed by them ?
20. Write three consecutive numbers of which 7 is the
first. Write three consecutive numbers of which n is the
first.
21. Write three consecutive even numbers of which 6
is the first. Write three consecutive even numbers of
which 2n is the first.
22. Write three consecutive odd numbers of which 5
is the first. Write three consecutive odd numbers of
which 2« + l is the first.
23. Write three consecutive numbers of which n is the
greatest. Write three consecutive even numbers of which
2n is the greatest. Write three consecutive odd numbers
of which 2n-{-l is the greatest.
115. It is well to note that if n stands for any whole
number, then 2n is the symbolic expression for any even
number, since it is exactly divisible by 2, and 2n + l is
the symbolic expression for any odd number, since when
divided by 2 the remainder is 1.
EXERCISE 64&.
Problems
116. The Student has already solved a sufficient num-
ber of problems to give him a general idea of the way
problems are solved in Algebra. He will find his expe-
rience embodied in the following:
Directions for Solving Problems in Algebra.
/. Represent one of the unknown numbers, preferably the
one whose value is asked for, by x.
PROBLEMS. 133
//. Make symbolic expressions to represent each of the
other unknown numbers mentioned in the problem.
III. Find, from the problem, two of these symbolic ex-
pressions that are equal to each other.
IV. Solve the equation thus formed.
Solve each of the following problems :
1. The sum of two numbers is 33 and their difference
is 7. Find the numbers.
Let «= one of the numbers,
then 33—^= the other number.
And because the difference of the two numbers is 7; therefore,
a;-(33-;»r) = 7.
Removing parenthesis x—ZZ-\-x='J.
Transposing and uniting terms, 2;'=10, how would you find the
value of _>/ ? of ^ ?
6. How much more is x+2y than x-\-y} How much
more is 28 than 25 ? If jt:+^=25 and if ;»:+2_>/=28, what
is the value of jk ? Knowing the value of y, how would
you find x from x-\-y=25 ?
7. If 2;i;4-:j^=18 and if Sx+y=2S, what is the value
of ;»: ? Knowing the value of x, how would you find y
from2;r4-jK=18?
8. If 5x—2y=22 and Sx—2y=10, what is the value
of ;tr ? Knowing the value of x, how would you find the
value of y from Sx—2y=10 ?
9. If 4:X—7y=l and 4x—5y=S, what is the value
of jK ? Knowing the value of y, how would you find x
from ix—5y=S ?
10. lix—2y-\-A and x=Sy-\-2, make an equation which
will not contain the unknown numbers. If 2j/-l-4=3y+2,
find the value ofy. Knowing the value ofy, how would
you find x from x=2y-\-4: ?
11. If 6x=5y-\-ll and 6x=4y-{-12, make an equation
which will not contain the unknown number x. What
value of y will satisfy the equation just found ? Know-
ing the value ofy, how can you find x from 6x=6y-{-ll ?
ELIMINATION BY SUBSTITUTION. 141
12. If 3;^=4jf— 11 and 3>'=j<;— 2, make an equation
which will not contain the unknown number jk. Solve
the equation thus found. Knowing the value of x, how
can you find j^ from Sy=x—2 ?
120- In each of the above cases we have found that
the values of two uyiknown numbers can be found from two
equations C07itaini7ig the two unknown 7imnbers.
121. When two or more equations containing several
unknown numbers are so related that they are all satis-
fied simultaneously by the same values of the un-
known numbers, the equations are called Simultaneous
Equations.
122. Of course, if we can obtain from two equations
containing two unknown numbers, a single conditional
equation containing but one of the unknown numbers,
the value of this unknown number can be found in the
way explained in the last chapter. When from two
equations containing two unknown numbers, we obtain
one equation containing but one of the unknown num-
bers, we are said to Eliminate the other unknown number.
We will explain three methods of elimination: I. By
Substitution. II. By Comparison. III. By Addition
and Subtraction.
EXERCISE 66.
Elimination by Substitution.
123. Examples 2, 3 and 5 in the last exercise are illus-
trations of elimination of an unknown number by sub-
stitution. We give a few more examples worked by this
method.
142 SIMULTANEOUS EQUATIONS.
(1) Given 2j»:=6y-38 and _>/ + 23=3jt: to find x andj^.
From the first of these equations the value of x in
terms of y is found to be
x=?>j-ld. (1)
Substituting Sy—19 for x in the second of the given
equations we get
j/-f23=9j/-57. ^ (2)
Transposing jj/— 9y=— 57 — 23. (3)
Uniting terms and dividing both sides by —1
8)/=80, ■ (4)
whence jy=10. (5)
Substituting this value for^ in (1) we get
;t:=30-19=ll,
whence ;i;=ll andj/=10.
To verify, we substitute these values in the original
equations and get
22=60-38 and 10+23=33.
(2) Given 7^ + 3^=100 and 3;t:— jf=20 to find x and jj/.
From the second equation the value of y in terms of x
is readily seen to be
j=3jc-20. (1)
Substituting 3;i:— 20 for jy in the first of the given equa-
tions, we find 7^+9;r-60=100, (2)
whence by transposing and uniting terms
16jt:=160. (3)
Therefore, ^=10. ' (4)
Substituting this value for ;r in (1) we have
jj/=30-20=10,
whence .:r=10 and_>'=10.
EXAMPLES.
143
124. It is easy to see that the method of elimination
used in these examples may be applied in the case of any
system of simultaneous equations. Hence we learn how-
to eliminate an unknown number by the Method of
Substitution :
From either equation express 07ie of the unknown num-
bers in terms of the other, and substitute this value in the
other equation.
EXERCISE 67.
Examples.
Solve the following by the method of substitution :
1. ;^-f-4>/=37.
2jir+5y=53.
2. ;ir+5jj/=573.
x-\- jv=181.
3. 1x-Zy=\m.
2x-\- jj/=156.
4. Bx+4y=25S.
y=5x.
5. 2x-{-9b=llj/.
;»;_3_>/=0.
6. 6x—4y=6.
^x=^ly.
7. ;r=3jj/-19.
;j/=3x-23.
8. x+y^UI,
x-y^lbZ.
9. 3.r+2j/=:7.
/Jx— y=5.
10. x-\-5y=47.
x-h j/=lo.
11. x-\-5y=S5.
3^+2jK=27.
12. ox-{-7y=101.
7x— y=^55.
13. x=lQ—4y.
j^=34-4;»:.
14. 2;t:=ll + 9>'.
3;t:-12j/=l5.
15. 8x— j/=22.
2;ir-3^=0.
16. Sx-j-4y=18.
Bx-2y=^70,
144 SIMULTANEOUS EQUATIONS.
EXERCISE 68.
Elimination by Comparison.
125. We give a few examples of elimination of an
unknown number by the method of comparison.
(1) Given Sx=7S—y and 2x=y-{-S2, to find x and jy.
73—0/
From the first equation, x= — ^. (1)
From the second equation, ;r= ^ "" . (2)
Therefore 73-^^yJ^_ ^^^
Clearing of fractions, 2(73-_>/) = 3(j+32). (4)
That is, 146-2_y=3_r+96. ,(5)
Hence, by transposing and uniting terms,
5r=50, (6)
whence j/=10. (7)
Then, from (1), ^=^ — =21- (8)
Hence ;r=21 andjK=10.
(2) Given9j/+8^=41 and lljtr— 7jj/=37, to find ;»; andjK.
From the first equation, j/= — ^ . (1)
From the second equation, y= ^ . (2)
Whence — g — = tj • (3)
Clearing of fractions,
7(41_8jt:) = 9(ll;t:-37). (4)
That is, 287-56;»;=99;i;-333. (5)
EXAMPLES.
145
Hence, by transposing and uniting terms,
155^=620.
Therefore x=i,
41—32
Then, from (1), y= — ^ — =1.
Hence
x—4: and y=l.
(6)
126. These examples are sufficient to teach us how to
eliminate an unknown number by the Method of Com-
parison :
Express the same unknown number in terms of the other
from each equation and equate the expressions thus found.
EXERCISE 69.
Examples.
Solve the following by the method of comparison :
1. 5:r=63-8>/.
7;»;=39-3^.
2. 2^=29— 3:r.
5>/=20+3;ir.
3. 3;«r4-^=31.
5x=15 + 2jy.
4. 4Ar=19-f7>/.
5. 16y-3;»:=5.
5;r4-28>/=19.
6. ;r+7=24.
x—y—lQ.
10
7. 5j«r-3j=13. .
19x-oy=75.
8. 7X+ 5y=60.
13j»;-11j/=10.
9. 3;ir-f2j/=32.
20;«r-3jj/=l.
10. nx-7y=S7.
8;»;H- 9^=41.
11. 10.r+ 9j=290.
12;«;-lljK=130.
12. x-hy=579.
;r_v=333.
146 SIMULTANEOUS EQUATIONS.
EXERCISE 70.
Elimination by Addition and Subtraction.
127. The elimination in the following examples is
done by the method of addition and subtraction.
(1) Given x-{-y=67d and x— j/=333 find x Sindy.
Adding the left members and the right members of
these two equations together, we obtain
^+jj/=579 (1)
x-y=SS^ (2)
2a'=912 (3)
whence ;r=456 (4)
Subtracting the members of (2) from the corresponding
members of (1), we get
2jj/=246
whence y=12o.
Therefore, ;r=456 and j/=123.
(2) Given 15:r— 8>'=30 and Sx+2j'=16 to find x and j.
Multiplying both members of the second equation by
5, we have for the two equations
15x- 8y=30 (1)
15;t:+10r=75 (2)
By subtraction —18;/=— 45 (3)
whence J='^i' (4)
Hence from (1) 15;t:— 20=30. (5)
Therfore ^=3 J. (6)
Or the value of x may be obtained in another way.
Multiplying both members ot the second of the given
equations by 4, we have for the two equations
lDx-8y=30 (7)
12.^4-8;/= 60 (8)
By addition 27ji:=90 * (9)
whence ^=|^ or 3|-.
ELIMINATION. I47
(3) Given ll;f+12>'=100 and 9;tr+8>'=80 to find
X andj^.
Multiplying both members of the first equation by 9,
and both members of the second equation by 11, we
obtain
99j^:+108>/=900 (1)
99.;ir+ 88>/=880 (2)
By subtraction 20>/= 20 (3)
whence jK=l. (4)
Substituting this value for jk in either of the original
equations, we find x=S
whence jK=land^=8
128. The above examples show us how to eliminate
an unknown number by the Method of Addition and
Subtraction :
Multiply both members of the equations by such numbers
as will make the 7iumerical coefficieyits of one of the u7iknow7i
munbers the same in the resultiiig equatio7is ; the7i by addi-
tion or stibtradion we ca7i form a7i equatio7i C07itai7ii7ig
only the other u7ik7iow7i mwtber.
Solve the following by the method of addition and
subtraction :
1. x+y=ZO. 5. 4;i:+3j/=97.
;r-j/=6. 7x-{-Sy=127.
2. 5x-{-7y=17Q. 6. 24x-i- 7y=27.
5x-Sy=46. 8:r-33j=115. ,
3. x-i-5y=^57S. 7. 2;t:+3j=41,
x+ '=181. 4x+2j/=54.
4. x-\-4y=S7. 8. 5x-\-7y=17.
2x-{-oy=oo. Tx— 5;'=9.
148 SIMULTANEOUS EQUATIONS.
9. 7:r-3>/=27. X2. 13j/- 7x=9d.
5x-6j;=0, 9^-28^=52.
10. 2Sx-\-15j=4:\, 13. lGx+ 17r=274.
48jt:+45>/=18. 24;r-105>'=150.
11. 6;i;-7j/=42. 14. 21x+ 8jj/=66.
7;t:_6>/=75. 28:i:-23>/=13.
EXERCISE 71.
Special Expedients.
129- The student will find that elimination by addi-
tion and subtraction is in most cases the shortest method.
Occasionally, however, an example will be found which
is more readily done in some one of the other ways.
Sometimes, too, special expedients will still further ab-
breviate the processes. We give a few examples of this.
(1) Solve 3;r+7^=29, 1x-\-Zy=^Al.
By adding the members of the two equations, we
obtain 10^^4-107=70. (1)
By subtracting the members of second equation from
those of the first, we have
4:x-iy=12. (2)
From (1) we get x-i-y=7, (3)
From (2) we get x—y=S. (4)
Whence, by addition and then by subtraction,
x=o and_>'=2.
(2) Solve -^—g^ = l, — -f-^=6.
To solve these we must first clear each of fractions,
giving 9j»r— 8jj/=12
and 14j»r-f oj/=36
which can now be solved in any of the usual ways.
EXAMPLES. 149
(3) Solve =1, 1 =16.
X y X y
If these be cleared of fractions, the resulting equations
will involve the product xy, and we would have equa-
tions of a kind we have not yet considered. But by con-
sidering: — and - as the unknown numbers, we may solve
X y
by the methods already used. For example, by multi-
plying both members oi the first equation by 2, we get
for the two equations
l«-^=2 (1)
X y ^ ^
LV^o^ie. (2)
X y ^ ^
28
By subtraction --=14, (3)
2
whence -=1 or r=2.
y
Therefore from (1) j»:=3.
We could have eliminated x by dividing both members
of the second given equation by 2.
EXERCISE 72.
Examples.
Solve the following by any method :
1. x^-y=m. 4- 3;r-f-8>/=19.
x—y=iS. ox— y=l.
2. 2x-{-y=ll, 5. 3;t:-fcS>/=59.
Sx-y=4:. 6jr4-5)/=107.
3. Sx-h4y=SS. 6. 8x-lDy=S0.
5x+4y:^107. 2x+ 3^=15.
ISO
SIMULTANEOUS EQUATIONS.
7. 5;i:4-8>'=101.
9;^;4-2j=95.
13
, 2;i: 4- 7^-34=0.
5;t:-|.9^_51=0.
8. Sdx-{-27y=105.
52;t:4-29j/=133.
14.
, 3j/-4;t:-l = 0.
18_3.r=4^.
9. 72^4-14>/=330.
63:r+ 7>/=273.
15-
2x+^y=^Al,
3;r4-2j/=39.
10. 2x—7y=S.
4y-9x=ld.
16.
19^-21jj/=100.
21;r-19j/=140.
II. 8jtr4-3j/=3.
12:r4-9jj/=3.
17.
17jr-i;^=2.
13jt: 4-17^=236.
12. 69j/-17-r=122.
14:ir-23j^/=31.
18.
f^4-iy=17.
19. 10;i:4-|=210.
10>/-|=290.
23.
i4-A=6.
;r y
2-1=0.
X J/
20. ~ + ly^2bl.
|4-7jt=299.
24.
^+^=3.
X y
25.
1+2=10.
X y
i+^=20.
26.
^+^=19.
^-^=13.
X y
EXAMPLES. 151
27. 4+^=—. 31. — +f:=7.
4 • -^ • 3.r ' 5j/
tv J 1^
5"* ^x 10/
28. o+^=o- 32. -^-+-^=5.
8 ■ 2 3* -^ 8 ■ 6
2"^3 6
•^-fr^=I :^i^_£:i^=io.
29. ?+|=31. 33. ?:^+3=^.
4 ' 6~ ^*
X y
3 ""8'^ 2* ^ 4 ~2'4'
X y_\ 3 .^-2^ /^^ ^ J
1. 1_11 4.r+5j
;»;"^j~30- ^'*- 40
111 1 2jf-j/
X y 30' . 2 3 ^-^'
35.-4 4- = ^-V-
2.r+j^ 9jr— 7 _ 3jK+9 4;>:4-5y
2 8 ~ 4 16 •
36. -y ^=3^-5.
152 SIMULTANEOUS EQUATIONS.
EXERCISE 73.
Simultaneous Equations Containing Three Unknown Numbers.
130. If we have two equations containing three un-
known numbers, such as
we can eliminate one of these unknown numbers by the
methods already explained, giving one equation contain-
ing two unknown numbers. Thus, in this particular
case, by multiplying the members of the second equation
by 2 and subtracting, we would find
7r-192'=-17.
Since an indefinite number of values will satisfy one
equation containing two unknown numbers; therefore,
an indefinite number of sets of values will satisfy two
equations containing three unknown numbers.
Suppose, however, that we have three equations con-
taining three unknown numbers, as
2,r+3y-5^=9 (1)
jr-2j^+7/-19^=~17. (4)
Multiplying both members oi (2) by 3, and subtracting
from (3), we obtain
5r-23j=-31. (5)
Now we can eliminate _y from (4) and (5) by multiplying
both members of (4) by 5, and both members of (5) by 7,
giving 35y— 95^=— 85 (6)
35^- 161^= -217, (7)
THREE UNKNOWN NUMBERS. 1 53
whence, by subtraction, 66^=132, (8)
whence z=2.
Substituting 2 for z in (4)
7y-38=-17, (9)
whence j^'=3,
and substituting j=3 and ^=2 in (1), we find x—b.
Therefore, ;r=5, jj/=3, and z=2.
Here we notice that we have been able to find the
values of three unknown numbers from three equations.
131. It is evident that we can proceed in a similar
way in any case of three equations containing three un-
known numbers. That is, to solve three simultaneous
equations containing three unknown numbers :
/. Obtain froyn two of the equations a7i equation which
contaifis only two of the tinknown numbers^ by any method
of elimination.
II. Fro?n the third given equation and either of the
former two, obtaiji anothe> equatiofi which contains the same
two U7iknown numbers.
III. Fro7n the two eqtiations coniaiyiing two unknown
numbers thus fou7id, fina the values of these unknown
numbers.
IV. By sicbstituting these values in one of the give7i
€quatio7is, the value of the re7nai7iing unk7iown number
may befo: nd.
132. We will further illustrate this subject by work-
ing a few examples. It should be observed that while it
makes no difference which one of the unknown numbers
we eliminate first, yet the work is often lessened by the
selection for this purpose of that one of the unknown
numbers whose numerical coefficients have the smallest
L. C. M.
154 SIMULTANEOUS EQUATIONS.
(1) Solve 4x- 5y-\- ^=6. (1)
7;i:-lljj/+2^=9. (2)
x+ j^/+32'=12. (3)
The unknown number z has the smallest numerical
coefficients, and it is easier to eliminate it than any of
the other unknown numbers. Multiplying both members
of (1) by 2, we have
8^-10)/+ 2^= 12. ^ (4)
Subtracting (2) from this, x-j-y^S. ' (5)
Now multiply both members of (1) by 3, giving
12x-15>/+3^=18. (6)
Subtracting (3) from this, we find
ll.r-16>/=6. (7)
We have now to find the values of x and j^ from (5)
and (7). Multiply both members of (5) by 11, giving
ll^+llj/=33. (8)
Subtracting (7) from this, we find
27y=27, (9)
whence j=l.
From (5) x=2
and from (3) 2 + 1 + 3^=12, whence ^=3.
Therefore x=2, j=l, and ^=3.
The student may verify these in the original equations.
(2) Solve A'-fj=5. (1)
y-hz=7. • (2)
x+z=6. (3)
This is quickly solved by special expedient. Thus
add the members of the three equations together, giving
■ 2x-\-2y+2z==lS
or x-\- y-\- ^•=9. (4)
From (1) x-\-y=b, therefore from (4) ^=4.
From (2) y-\-2=l^ therefore from (4) x=2.
From (3) x-^2=Q>, therefore from (4) y=Z.
EXAMPLES. 155
(3) Solve -+-+-=4. (1)
X y z
X y z
^+L2_10=4. ■ ■ (3)
X y z
Here we should consider -, — , and- as the unknown
X y z
numbers. Subtracting (1) from twice (2), we get
^^^=4. (4)
y z
Subtracting (3) from three times (2), we get
y z
We are now to find - and - from (4) and (5). Sub-
y z
tracting (4) from (5),
?-.
whence
^=5.
From (4)
^=4.
From (1)
x^Z.
EXERCISE 74.
Examples,
Solve the following simultaneous equations :
1. x-\-y=S7. 3. x-h y+ z=ZO,
x+z=2o. Sx-^4y+2z=50.
y-\- ^=22. 27;»;+9>/+3^=64.
2. 2x+y =5. 4. 5j*r4-7y— 2^=13.
;»:4-3^=16. 8x-{-Sy-\- ^=17.
6y- z=10. x-4y+10z=2S.
156 SIMULTANEOUS EQUATIONS.
5. 5;^-3y=l.
7. 3;i:+2>/+ ^=23.
9y-2z=12.
5;»:4-2jK+4^=46,
8;^+3^=17.
10;i:+5>/H-4-3'=75.
6. x+y-z=17.
8. 4x—2y+52=lS.
x-\-2-y=lZ.
2x+4:y-Bz'-==22.
yJ^2-X=1.
6x+7y- ^=63.
9. X
+ ^ + ^
1^=23.
Zx
_ y^2z=\\.
X
•+4y~
^=4.
M=^^-
X4. i+i=l.
X y
H='-
i+i=2.
12+7=^-
V z 2*
'•.-fe-
2,13
^5- ^+ri-
^r-
i-^=2.
^r^-
i+i=i
.. f +1+1=62.
16. — \ =1.
X y z
f+f+l=^^-
^+i+^=24.
X y z
f+M=^«-
X y z
3. 258-H+f-
3_4 1_38
X hy z 5 *
304-£=f+|.
Sx 2y^z 6*
296-f=£+|.
4 1 4 161
5x 2y ' ^"~ 10 •
LITERAL SIMULTANEOUS EQUATIONS.
157
EXERCISE 75.
Literal Simultaneous Equations.
Solve the following :
I. x-\-y=2a and x—y=2d.
By addition,
2x-
:2^+2/^
whence
X-
: ^Z+ /=x.
6. adx+cdy=2.
d-d
ax-cy^-^^^
8. f-^=^
1 , 1
X y
71 \
9. -4— =«.
1 1
— r.
X y
X y
10. :rH-jj/=2«.
13. «.r4-<^j^— '=r.
14. .r— «;r+j=0.
^-^=^.
^y— /J + -3'=0.
bx^-z=t.
x-^z=t.
12. ;«;+jj/H-2'=^.
15. jr+«^+«2^+«^ = 0.
;i;— ^+2'=^^.
jr+^^+/^-^+^^=0.
;c+J^— 2'=r.
^+0'+<:^'S' 4-^^=0.
X
n
y~
r
y^
J-
z
9
1
1
17. -=
-a-
—
' X
y
1
1
— =
^b-
y
z'
1
1
— =
-c—
z
X
158 SIMULTANEOUS EQUATIONS.
a b c
16. x-\-y-\-z=a. 18. — I =n.
•^ X y z
a b c _
X y z
X y z
19- -+-+-=3.
X y z
X y z
s2a b c ^
X y z
EXERCISE 76.
Problems Producing Simultaneous Equations.
1. The sum of two numbers is 70, and their difference
is 24. Find the numbers.
Problems exactly similar to this have been worked with the use of
one unknown number. We will now work it using two unknown
numbers.
Let ic=the first number,
and let jj/ = the second number.
Then, because the sum of the numbers is 70, and the difference of
the numbers is 24 ; therefore,
■^+7= 70,
and re— 7=24.
Solving these, we find ic=:47 and 7=23.
2. Find a fraction such that if we add 1 to the numer-
ator, it becomes equal to \, but if we add 2 to the de-
nominator, it becomes equal to \.
Let x=the numerator of the required fraction,
and.let jj'=the denominator of the required fraction.
Because the fraction with 1 added to the numerator equals \,
therefore, -—-=—. (1)
y 1 ^ '
PROBLEMS. 159
Because the fraction with 2 added to the denominator equals ^,
X 1
therefore, —r^=-7:- (2)
From (1) and (2) we get, by clearing of fractions,
2a:+2=j (3)
^x=y^2. (4)
Eliminating/ we get x=4, whence from (3)^ = 10. The fraction is,
3. Says A to B: " Give me $49 and I will have just as
much money as you." Says B to A: " Give me $49 and
I will have 3 times as much money as you. ' ' How much
has each ?
Let :c=the number of dollars A has.
Then j=the number of dollars B has.
If B gives A $49, A would have x-|-l9 dollars and B would have
;' — 49 dollars. Because they would then have equal amounts, there-
fore, x4-49=j-49. (1)
If A gives B $49, B would have y-\-^2 dollars and A would have
x-49 dollars. Because B would then have 3 times as much as A,
therefore, 3(jt -49) = r+49. (2)
Eliminating J from (1) and (2), we get
2j:-196 = 98, (3)
whence x=\Al. (4)
From (1) we then find ^=245.
Therefore, A has $147 and B has $245.
4. A man bought two kinds of cloth, 7 yards of the
first kind and 11 yards of the second kind, and paid $47
for both. If he had bought 11 yards of the first kind and
7 yards of the second kind he would have saved $4.
What was the price per yard of each kind of cloth ?
5. Find a fraction such that when 1 is added to both
numerator and denominator, it equals \, but when 3 is
subtracted from numerator and denominator it equals \.
6. Find a fraction such that when 11 is taken from
both numerator and denominator it equals f, but when 12
is taken from both numerator and denominator, it equals \.
l6o SIMULTANEOUS EQUATIONS.
7. A fraction is such that if I add 1 to the numerator,
the fraction equals unity ; but if I double the first frac-
tion and add 1 to the denominator, the value of the
fraction equals unity. What is this fraction ?
8. A number is formed of two digits of which the dif-
ference is 3. If the digits be reversed, a number is
obtained which is 7- of the original number ? What is the
original number ?
Let X represent the digit in tens' place, and y represent the digit in
units' place.
9. A farmer sold to one person 9 horses and 7 cows for
$1500, and to another, at the same price, 6 horses and 13
cows for the same money. What was the price of each?
10. The sum of two digits which form a number is 9.
If we subtract 3 from each digit, the result is 6 more
than half the original number. What is the original
number ?
11. Two masons, A and B, are building a wall, which
they could finish, working together, in 12 days. A works
3 days and B 2 days, when the wall is \ done. How
long would it have taken each to have built the wall ?
Let a;=number of days it would take A,
and y=number of days it would take B.
In one day A builds — of the wall, and B — . But together they build
^ of the wall in one day. Therefore,
3 2
In 3 days A builds — of the wall, and in 2 days B builds— of the
X y
wall. Therefore, from the problem,
l+f i <^)
Solving the simultaneous equations (1) and (2), we can find the
values of x and^.
PROBLEMS. l6l
12. A and B can together do a certain work in 30
days ; at the end of 18 days, however, B is called off and
A finishes it alone in 20 days more. Find the time in
which each could do the work alone.
13. A cistern holding 4500 gallons is filled by two
pipes. If the first pipe be opened 3 minutes and the
second pipe 1 minute, 400 gallons will run into the cis-
tern ; but if the first pipe be opened 1 minute and the
second 7 minutes, 600 gallons will run in. How much
water does each pipe carry in one minute ? How long
will it take both pipes to fill the cistern if they are opened
together ?
14. A cistern can b-^ filled by two pipes. If both pipes
be opened for 15 minutes they will fill ^ of the cistern ;
but if the first pipe be opened for 12 minutes and the
second for 20 minutes, A of the cistern will be filled.
How long will it take each ot the pipes to fill the cistern
when opened alone ?
15. A man receives $2160 yearly interest on his cap-
ital. If he had loaned the same capital at ^ per cent,
higher interest he would receive $240 more interest each
year. Find the amount of his capital and the rate per
cent.
16. A man has two sums of money at interest, the
first at 4 and the second at 5 per cent. Together they
bring in $3000 interest annually. What is the amount
of money loaned at each rate ?
17. A man has two sums, one of $10000 and another
of $15000, at interest, and receives therefrom $1200 an-
nually. If the first sum had been loaned at the rate that
the second bore and if the second sum had been loaned
at the rate that the first bore, he would have received
$25 less per year. At what rates were the sums loaned ?
l62 SIMULTANEOUS EQUATIONS.
i8. A and B can do a piece of work in 12 days ; B and
C can do it in 20 days ; A and C in 15 days. How long
will it take each to do the work alone ?
ig. A cistern is filled with three pipes. The first and
second will fill it in 72 minutes, the second and third in
120 minutes, and the first and third in 90 minutes. How
long would it take each one of the pipes to fill it ?
20. Three cities, A, B, and C, are at the corners of a
triangle. From A through B to C is 82 miles ; from B
through C to A is 97 miles ; from C through A to B is 89
miles. How far are the cities A, B, and C from one
another ?
21. A certain number consists of three digits, whose
sum is 15. If the first two digits be reversed the num-
ber becomes 180 larger, but if the last two digits be re-
versed, the number becomes but 18 larger. What is the
number ?
22. The sum of three numbers is 70. The second
divided by the first gives 2 for the quotient and 1 for the
remainder, but the third divided by the second gives 3
for the quotient and 3 for the remainder. What are the
numbers ?
23. There was a family of boys and girls and when
they were asked how many there were in the family one
of the boys said: "I have just as many brothers as
sisters." But one of the girls said: " I have twice as
many brothers as sisters. ' ' How many boys and girls in
the family?
24. It takes 72 English and 51 German yards together
to make 100 meters. Also 48 English and 84 German
yards make 100 meters. How many inches (English) in
the meter ? How many inches (English) in the German
yard ?
CHAPTER XII.
POW'ERS AND ROOTS
EXERCISE 77.
Powers of Monomials.
133. The process of raising a number or expression
to any power is called Involution. We have already
learned that «"«''= ^"+'', (1)
where ii and r are any positive whole numbers, but where
a may be either a whole number or a fraction, and either
positive or negative.
Multiply both numbers of equation (1) by a' and we
obtain a" a'' a' == a*'^'' a\ (2)
But the right side of equation (2) is the product of two
powers of the same letter, and hence from what we have
learned before, the right side of (2) equals a"+''+^.
Hence, «"a''«^=a''+''+\ (3)
1. In the same w^ay show that
2. Can you find in the same way the product of more
than four factors, each one of which is a power of <2 ?
3. Can you find the product of any number of factors,
each one of which is a power of the same number ?
4. State then what the product of aiiy number of
powers of the same number is equal to.
5. In formula (1) can the exponents n and rbe equal
to each other or must they be different ?
If you cannot readily answer this question, turn back to page 55
and sec how this formula was proved, and then the answer will be clear
164 POWERS^ AND ROOTS.
6. What does a'^a^ equal? What then does {a'^Y equal?
7. What does a^a^ equal ? What then does (a^)^ equal ?
8. What does a'^a'^ equal? What then does («*)^ equal ?
9. What does^:''^^ equal? Whattlien does (a ^)''^ equal ?
10. What does a"a" equal ? What then does (a")^ equal ?
11. In formula (3) can the exponents n, r and ^ be
equal to one another ?
12. What does a'^a'^a^ equal? What then does (<^^)^
equal ?
13. What does a^a'^a^ equal? What then does (a^)^
equal?
14. What does a^a^a^ equal? What then does {a^Y
equal ?
15. What does a^a^a^ equal? What then does (a^Y
equal ?
16. What does a"a''a" equal? What then does (a"Y
equal ?
17. What does «'V^"a" equal ? What then does (^")*
equal ?
18. What does a"a"a"a"a" equal ? What then does (a") ^
equal ?
19. What does the product of r factors each of which is
a" equal ?
20. What then does {a"y equal ?
134. These questions lead to the general formula,
[a" Y -W''.
The truth expressed by this formula n:a}- be ex-
pressed in words thus: The rth power of the n th power
of a number is equal to the nr th power of that yiiiviber.
POWERS OF MONOMIALS. 165
135. Let us now seek a power of the product of dif-
ferent numbers.
Consider, first, the square of the product of several
numbers.
(abY =abab=aabb=a'^ b"^ ',
also (abcY ^abcabc^aabbcc^a"^ b'^ c*^ ,
21. In the same way show that
also that (abcde')'^ = a-b'^c'^d'^e'^.
22. Would a similar result hold if there were more than
five factors in the product to be squared ?
23. Justify the following statement :
T/ie square of the prodiid of any number of numbers is
equal to the product of the squares of those yiumbers.
Consider, next, the cube of the product of several
factors.
(^abY=ababab=aaabbb=a^b^ ;
also (abc) ^ = abcabcabc=-aaabbbccc= a'^b'^c^.
2/[. In the same way show that
(abedy=aH^c^d^;
also that (abcde)^=a^b^c^d^e^.
25. Would a similar result hold if there were more
than five factors in the product to be raised to the third
power ?
26. Justify the following statement:
The cube of the product of ajiy number of numbers is
equal to the product of the cubes of those numbers.
27. The fourth power of the product of any number of
numbers is equal to what ?
1 66 POWERS AND ROOTS.
28. In the n th power of the product of any number of
numbers the first number appears how many times as
a factor ? The second number appears how many times
as a factor ? Any one of the numbers appears how many
times as a factor ? Therefore, the n th power of the
product of any number of numbers equals what ?
G)
136. IvCt us now seek a power of the quotient of two
numbers.
a a aa d^
, (a\ ^ a a a aaa a^
^''° Kb) =-b ' -Tfib^b^-
29. In the same way show that
,4
©'
a
A4"'
also that (-3'=|t:
also that
\b) ~~¥'
137. Thus we may raise to any desired power,
First, a power of a number ;
Second, the product of several numbers ;
Third, the quotient of two numbers.
These three cases include every monomial that can be
proposed. Hence any monomial can be raised to any
required power by the methods already given.
138. As any even power of a monomial having a —
sign is the product ot an even number of subtractive
terms, it follows that any even power of a monomial
having a — sign is a monomial having a -f- sign or no
sign. Thus, (-2)2= +4, (-3)-^= SI, {^-lab'^y
= G4«6^i^ etc.
EXAMPLES.
167
139. Again, as any ^^z\
16.
x^-^a^-y.
3.
ab-\-c.
10.
4:x—3ab.
17-
nt'^—r'^s'^.
4.
c—ab.
II.
5ab—Sac.
18.
(a^2)2+(2x)2.
5-
x-y.
12.
10 + 5.
19.
(2xy-\-2x\
6.
2ax-^b.
13-
100-1.
20.
a-{-a.
7.
2ax-2b.
3.+f.
14.
30 + 7.
21.
34.
i-ixy^'^y.
22.
28.
uv-^ 0.
uv
x^y^
a a
23.
y b
29.
lo-fo-
35-
"■+(*)'•
24.
^x"^ 2yy
'^y'^'Zx'
30.
^n Zs
Zs 4n '
36.
Ax"
w.
31.
n'^x'^ 71-x'^
37.
-.1..
25-
2x 2n '
26.
1 1
ab ac
32.
abc
rst
38.
a
27.
a b
b ■ a
33.
?>x 2yz^
y X
39-
«2 22
¥ a'
SQUARE OF A POLYNOMIAL. I/I
EXERCISE 80.
' Square of a Polynomial.
144. It is so often necessary to square a polynomial
that it is well to have a way of writing down the result
without being obliged every time to go through all the
work of multiplying.
145. Let us first take a polynomial of /our terms and
find its square by actual multiplication.
a-\-b-\-c-Vd
«2-f ab-{- ac-{- ad
ab +<^2_j_ jjcJ^ Id
ac + be -\-c^-\- cd
ad 4- bd + cd^-d"^
a"^ -\-2ab-\-2ac+2ad+ b'' ^2bc+2bd+c^ + '2cd+d''
Slightly changing the order of the terms in this result,
we may write
{a + b+c-^dy
= a'' -\-b'' +C'' ^d"- + 2ab+'lac+2ad^-2bc-^2bd+2cd.
146. The method here used applies to any polynomial,
no matter how many terms there may be.
Suppose the polynomial written down twice, the
second time immediately below the first, and call the first
line the multiplicand and the second line the multiplier,
and let us see what terms make up the product, which of
course is the square of the given polynomial.
First y it is plain that the square of each term of the
polynomial must be part of the product required.
Second, each term of the multiplicand being multiplied
by each term of the multiplier, it will be evident, if looked
172 POWERS AND ROOTS.
at carefully, that another part of the product is made up
of twice the product of each term of the -polynomial by
each term that follows it.
For, consider how any one of these latter described
terms can arise. Consider, for example, the term bd in
the case worked out of a polynomial of four terms. The
only way a term bd can occur in the product is, first, for
d of the multiplicand to be multiplied by b of the multi-
plier, and, second, for b of the multiplicand to be mul-
tiplied by d of the multiplier, and the two terms thus
produced being combined together we obtain 2bd as one
term of the product required. Notice that d is one of the
letters \}oi2X follow b in the polynomial considered.
The reason that we have no term db in the product is
because that term has alread}^ been accounted for by con-
sidering it the same as bd.
Of course, what is here said about bd will apply to any
two different terms of the given polynomial.
147. Thus, the square of any polynomial may be
written down directly b)^ writing first the sum of the
squares of each of the terms of the given polynomial, and to
this S2im adding twice the product of each term by each term
that follows it i7i the given polynomial.
148. In applying this method it must be remembered
that the letters we have used may stand for negative as
well as positive numbers, and in those terms which are
made up of twice the product of each term by each term
that follows it, the rule of signs must be observed.
By the method just explained the square of a—x—'Z
would be a"^ -\-x'^ -{-z'^—^ax—^az-\-'lxz,
the last term having the sign -f- because the two terms
multiplied together to produce this term have like signs.
ROOTS OF MONOMIALS. 1 73
Write down the square of each of the following
polynomials:
1. a-\-b—c. 15. a-\-b-\-c—d—e.
2. r+s—x. 16. 2a—2b—c—d—e.
3. m-\-r—x-\-y, 17. x-\-y+2—a—b—c.
4. a + 2b—c—d. 18. u-\-2v+Zw~4:X.
5. a—2b—Zc+d. 19. w2 4_2z;2 4_a;_|-^2
6. 2a—2n + ^x—y, 20. ^^2 4.(^2z/)2-f (3ze')2— j»r.
7. ;t:4-jJ^— 2z^— 3z/. 21. m + 2r—bs-\-t,
8. j»;2+;tr+l. 22. ;r>/-f-a<^— ^2_|_^2^
9. x^-^2x+l. 23. a^<:— .ry^+;t:2-f_y2^
10. ;r^— Sjt^H-Sjr— 1. 24. mrs'^ — uv-{-st—w,
11. ;i:3+3;i:2j^4-3ji:j/2+y. 25. («^)2-;t2j/ + (2x)2-j-jj;.
12. ;»;2+jj/2— 2'2. 26. « + 2jr+«;tr+2^;t:2.
13. a^— ;i:^— jK^.' 27. m-\-s-\-xyz—ti'^v.
14. r2--«rH-52— ^.y. 28. «<^;i:2— «z;^2^^^^)2^
EXERCISE 81.
Roots of Monomials.
149. We have learned how to find any power of a
monomial, the square and cube of a binomial, and the
square of any polynomial. The reverse process of going
back to the number or expression, when the power is
given, is called Evolution.
150. The number or expression found by evolution is
called a Root of the number or expression given.
Note that this is an entirely different use of the word root from that
of Art. 103.
151. As there are square, cube, 4th, 5th, etc., powers,
so there are square, cube, 4th, 5th, etc., roots.
174 POWERS AND ROOTS.
The square root of a given expression means that ex-
pression which squared ^\W produce the given expression.
The cube root of a given expression means that ex-
pression which cubed will produce the given expression.
The 4th root of a given expression means that expres-
sion which raised to the fourth power will produce the
given expression, etc.
For example, 2^ = 8; therefore, the cube root of 8 is 2,
24 = 16; therefore, the fourth root of 16 is 2, etc.
152. A root is indicated by the sign i/, called a
Radical Sign. A horizontal line usualh^ extends from
the upper end of the radical sign over the expression of
which the root is to be extracted. See Art. 80.
To indicate what root is to be extracted, a small
figure, called the Index of the root, is placed in the angle
of the radical, excepting in case of the square root, in
which the index is almost never used.
Thus, the square root of 16 is indicated by 1^16, the
cube root of 8 is indicated by # 8, the fourth root of 16
is indicated by 1/16.
A letter may be used as the index of a root. Thus,
V a means the nWi root of a, that is a number which
raised to the 7i th power will produce a.
153. We must notice one important distinction be-
tween raising to a power and extracting a root. If we
have an expression given to be raised to a given power,
we obtain only one result, but if we have an expression
given to extract a given root we may sometimes obtain
more than one result.
For example, 5^ = 25, hence we say l/25=5; but also
(—5)2 = 25; hence we say l/25=— 5.
ROOTS OF MONOMIALS. 1/5
It appears thus that there are two numbers +5 and —5,
either of which is a square root of 25. The two results
are often written together by means of the double sign it.
Thus, 1/25= ±5.
1. What are the two square roots of 100?
2. What does 2^ equal?
3. What does (—2)'* equal?
4. Are there two 4th roots of 16 ?
5. Are there two 4th roots of 81 ?
6. If n stands for an even n amber is there more than
one n th root of a given number ? Why ?
7. A sixth root of 64 is 2, give another number that is
also a 6th root of 64.
8. Any even root of a positive number may be either
positive or negative. Why ?
164. If any number be raised to any even power, that
number is used an even number of times as a factor, and
therefore the result must be positive w^iether the number
given was positive or negative. In other words, there is
no positive or negative number which raised to an even
power will give a negative result, so we cannot find an
even root of a negative number.
155. If an expression be raised to an odd power, that
number is used an odd number of times as a factor, and
therefore the result is an expression of the same sign as
the one given. Therefore, any odd root of an expression
has the same sign as the expression itself.
176 POWERS AND ROOTS.
156. To find any root of any expression we naturally
look to see how the corresponding power was obtained,
and then go through the work backward if possible,
thus returning to the expression from which we started
in the case of involution.
We have found that {cCy^a"'' that is, «" is an expres-
sion which raised to the r th power gives a'""; therefore.
Hence, to extract the n th root of a power of an expres-
sion, we divide the exponent of the given power by the
index of the root, but in order to perform the division,
the exponent of the power must be a multiple of the
index of the root.
We cannot extract the square root of «^, because 5, the
exponent of the power, is not a multiple of 2, the index
of the root.
157. To find the n th root of the product of two factors.
We know that a"b"={aby\
therefore, V a''b''-==-i/\aby'=ab.
In this result the first factor, a, may be found by taking
the n th root of a'\ the first factor of the given expression,
and the second factor, b, of the result may be found by
taking the n th root of b'\ the second factor of the given
expression. Therefore, the n'Oa root of the product of
two factors is found by taking the product of the n th
roots of those factors.
158. Of course the same argument may be used with
more than two factors, and hence, evidently, the n th root
of the product of several factors is fotmd by taking the
product of the n th roots of those factors.
EXAMPLES. 177
169. To find the nth. root of the quotient of two
expressions.
We know that ^""(^j
therefore ^|,= ^g) =^
In this result the numerator, a, is found by taking the
n th root of the given numerator, and the denominator, d,
is found by taking the 71 th root of the given denominator.
Therefore, ^^e nth root of the quotieyit of two expressio7is
is found by taking the quotient of the n th roots of those
expressions.
EXERCISE 82.
Examples.
Find the square root of each of the following twelve
expressions :
^2^2 4
5.
I.
a'^x".
2.
9a6;t2.
9^2
3-
16^*'
4.
r^^V*.
o 10 ; —
• 49«2;t:2* • y^ zX
49j;2y4 ^ ^
.T-^
8. '-^~. 12. wV*-^— .
Find the cube root of each of the following ten ex-
pressions :
13. a^xK 15. -|^. 17. 3sZT.--
14. 8«»^«^^ 16. -64a»;»;i2^ 18, 77^ Ti"-
12
178 POWERS AND ROOTS.
19.
27 x^
21.
^6,j,6^6
-8
-Sx' 64'
U'^V'^W^
• x'-y^'
-64a^'d^---Sa\
22.
a^x^
r^r«
20.
23. Find the fourth root of ^^^^^r*.
24. Find the square root of X'oa^x^y^'^ , and then the
square root of this result.
25. Find the fourth root of IQa^x'^j/'^^.
26. Find the cube root of a'^ ^ d'^ ^ c'^ "^ , and then the
fourth root of this result.
27. Find the sixth root of a^^d^'^c^*, and then the
square root of this result.
28. Find the twelfth root of ^'^^^^ V^*.
29. Find the square root of <2^^<^^2^24^ then the cube
root of the result, and then the square root of this second
result.
30. Find the fifth root of —32^5^-1 ^y\
31. Find the seventh root of 128^7^7^1*.
X V
32. Find the square root of ' " ^ ^ -.
160. Before leaving the subject of the roots of mono-
mials, it is well to notice that what we have learned
may be used to find the roots of arithmetic numbers,
when the numbers given have exact roots.
We resolve the number into its prime factors, and ex-
press it as the product of various powers of these prime
factors, then divide each exponent by the index of the
required root. When the resulting factors are multiplied
together the required root is found.
SQUARE ROOT OF POLYNOMIALS. 1 79
Suppose we wish to find the square root of 53361.
3
53361
3
17787
7
5929
7
847
11
121
11
Hence, 53361=3^ x 7^ x 11^;
therefore, 1/53361 = 3 x 7 X 11=231.
33. Find the square root of 5184.
34. Find the square root of 43204.
35. Find the cube root of 85184.
36. Find the cube root of 32768.
EXERCISE 83.
Square Root of Polynomials.
161. To find out how to extract the square root of a
polynomial we must see how the polynomial was pro-
duced b}^ squaring. We know that
Therefore, w^e know that the square root o{ x'^ -\-2xy-\-y^
is x-}-y.
Our problem then is this: Given the expression,
x'^-^2xv-\-j'^, to find from it the expression, x+j.
The first term, x, of the root is the square root of the
first term, x'^ , of the given expression.
Let us set down this term, x, already found, and sub-
tract its square from the given expression. There re-
mains of the given expression 2xj/-{-y'^ or (2x-\-y)j/.
From this we see that the second term, j', of the root
will be the quotient when the remainder just found is
divided by 2x-{-j/.
l80 POWERS AND ROOTS.
This divisor, 2x-{-y, consists of two terms, the first of
which is t^ice the portion of the root already found, and
the second is the new term, jy, itself.
The work may be arranged as follows:
Given Expression. Root.
'I'X.-^y
2xy-^jy^
1xy-\-y'^
After the first term x, of the root has been found, its
double, 2x, is used as a trial divisor by which to divide
the remainder, ^xy+y"^. We see that the first term, 2xy,
of this remainder when divided by the trial divisor, 2x,
gives y, from which we judge that y is the next term oi
the root. When the y is thus found it is added to the
trial divisor, 2;r, giving the complete divisor, 2x+y, and
this is multiplied by ^, giving the expression, 2xy-\-y'^.
162. Of course we may obtain by this process the
difference of two numbers for our square root as well as
the sum, as in the example just given. This will be
plain by working out another example.
To find the square root oi \a'^'-\2ab-\-^b'^ ,
Arrange the work thus :
4^2
4.a-Zb
-12ab+db^
-12ab+db^
Here the first term of the remainder, —12ab, when
divided by the trial divisor, ia, gives the quotient, —3^.
Hence we judge that —Sb is the next term of the root,
and upon trial this proves to be right.
SQUARE ROOT OF POLYNOMIALS. l8l
Find the square root of each of the following ex-
pressions :
1. «2+4«<^+4^^ II. x'^—^x-^+x*.
2. 4a''-iab+b\ 12. dx^-lSx"- +d.
3. da'^-lSab+^d^. 13. a^-2a^x^-\-x^.
4. 4a2_16«-fl6. 14. aH^-h2adcd-{-c^d'^.
5. a^—2a'^d'^-{-d*. 15. a^x'^ — Iadx^ + d^x'^,
6. a^- — 2ad-^-\-d\ 16. 4x^—4nx'^j'-{-?i^j\
7. ;t-4-f22_>'2_^y. 17. 9a4<^2_1^^3^3+9^2^4^
8. .r* + 2;r2 + l. 18. «4^'*-6«2^2^9.
9. X^-h2x^+x'^. 19. «*/^6_2^2^3^5_^^10^
10. ;»;«— 2;t-*H-:r2. 20. «2^2jr2_>'2— Sa^^r^r^j^+lGr^jtr^
163. Thus far the polynomials of which we have ex-
tracted the square root have been in ever\' case those of
three terms.
The above process, however, can be extended so as to
find the square root of any polynomial which is a perfect
square, no matter how many terms the polynomial con-
tains. For example, to find the square root o^
a^-^d'^-hc'--{-2ad-}-2ac+2dr.
First arrange the expression according to powers of
some letter, say a, and write
a^ +2ad-{-2ac+ d'^ +2dc-^c'^ .
The first term, a^, of this polynomial is produced bj^
squaring a. Therefore, the first term of the root is a,
and the wko/e root is « + something, and this something
is what we wish to find.
Proceeding as before with the first term of the root, a
part of the process may be arranged thus :
a'- -^2ad+2ac-^d'' -\-2bc-\-c'' ( a
2ah-^2ac-^b'^-\-2bc+c'^
1 82 POWERS AND ROOTS.
Now twice a used as a trial divisor would suggest b for
the next term of the root.
Call the next term b and proceed as before, and the
work would stand thus:
a"^ ■^lab^lac^b'' ^-Ibc^-c'' ( a^b
"la^-b
2ab-^-2ac+b'^+2bc+c'^
lab +^2
lac -\-2bc-i-c^
There is sfzll a remainder, so we have not yet found
the entire root, but the root is a-\-b-^ something, and
this something is what we wish to find.
Now let us consider a-j-b, the part of the root already
found, as a single term, and use it as we have before used
the first term of the root. We must then take twice
a-hb and use it as a trial divisor by which to divide the
last remainder, 2ac-\-2bc-\-c'^, from which we judge that
c is the next term of the root. When the c is thus found
it is added to the trial divisor, 2a-\-b, giving the complete
divisor, 2a-\-2b-i-c, and t/iis is multiplied by c, giving the
expression 2ac-\- 2bc+ c"- .
The work from the beginning will now stand thus:
a"" -^2ab-\-2ac-\-b'' ^■2bc-^c'' ( a-^b^c
a""
2a + b
2ab+2ac+b^-\-2bc+c'^
2ab 4-/^2
2a+2b-\-c
2ac -\-2bc-\-c^
2ac ^-2bc^c'^
As there is now no remainder the process is ended and
the root \s, a-\-b-\-c. If after finding the third term of
the root there were still a remainder, we would group the
three terms thus found into a single term, and use this
group as we have always used the first term of the root.
CUBE ROOT OF*' POLYNOMIALS. 1 83
The process may evidently be extended to finding the
square root of any polynomial that is a perfect square,
no matter how many terms the polynomial may contain.
164. From what has been said we see that the Method
of finding the Square Root of any Polynomial is as
follows :
/. Arrajige the terms according to the powers of some
letter.
II. Find the squa7'e root of the first term, zvrite it as the
first term of the root, and siibtraci its square from the give?i
expression.
III. Use twice the portion of the root already found as a
trial divisor, and divide the remainder just found by this
trial divisor. Add the quotient to the root and also to the
tria divisor.
IV. Multipiy the complete divisor by the term of the root
last obtained and subtract the product from the rej?tainder.
V. Repeat III and IV until there is no remainder.
Find the square root of each of the following poly-
nomials :
11. x'^-\-'^xy-\-2xz-\-\y'^-\-A:yz-\-z'^ .
12. ;»;26jrj/— 4;i:^+9>'2^12j/^-f4^^
13. 4.r2^16.rj/— 4x^-f l(y/2— 8j'^H-^2^
14. ^•*-2a2<^+2a2r-f^-_2^^_}_^2^
15. 4«4-12«2^2_g^2^_f_9^4_^12^V+4^2^
16. «4+2«2^2_^2«V2-f^*+2<^2^2_j.^4^
17. a-+2«^ + 2a<:+2a^+^2^2^^+2^i/+^2_^2r^4-^^
18. a"- ^^ab—^ac^-W" ^-Vlbc-^Sic"- .
19. a«-f-2a5-}-3a*-f4a'^4-3a2 + 2^ + l.
1 84 POWERS AND ROOTS.
20. 9x* + lHa^x'^-{-6x'^-\-9a^-i-Qa^-\-l,
"V^ -y^ 'Y*'" 'V*
21. ^ + 2^+S~ + 2--hl.
a* a^ a^ a
22. ji:8 4- 2;t:«+ 3:^4 + 2.^2 + 1.
23. a^b^+2a''b''cd-4:a''b'^e+c'^d'^-Acde-\-Ae'^,
24. aH'^-^2a^bc'^d+2a^be-\-c'^d'^'\-2c'^de+e'^.
25. aH''c''-]-2abcde+2abcf^-d''e''+2def^p,
EXERCISE 84.
Cube Root of Polynomials.
165. To find how to extract the cube root of a poly-
nomial we must see what the cube of an expression is
and then return, if possible, from the cube back to the
expression from which this cube was obtained.
166. We know that
{x+yy=x^-^Zx'^y + 2>xy'^-\-y^.
Our problem, then, is this : Given the expression,
x^-{-Sx'^y-\-Sxy'^-{-y^, to find from it the expression,
x-^y, which is the cube root of the given expression.
167. We see that the first term, x, of the root is the
cube root of :r^, the first term of the given expression.
Set down the x as the first term of the root and sub-
tract its cube from the given expression, and we have a
remainder,
Sx^y+Sxy'^+y^, or (Sx^-\-Sxy-{-y^)y,
We see from this that the second term, y, of the root
is the quotient obtained by dividing this remainder by
Sx^ + Sxy+y'^.
Now this divisor is composed of three terms, of which
the first is 8 times the square of the first term of the root.
CUBE ROOT OF POLYNOMIALS. 1 85
the second is 3 times the product of the first term of the
root and the new term, y, and the third is the square of
the new term of the root.
The sum of these three terms constitute the complete
divisor, while the remainder found b}^ subtracting x-^ from
the given expression is the complete dividend.
This complete divisor contains two terms which involve
the jK, which is not yet supposed to be known. However,
we may get something of an idea of what the second term
of the root must be by using the ^rsf term of the above
remainder as a trial dividend and 3 times the square of
the first term of the root as a trial divisor, and then if the
number we get by this division is correct the complete
divisor (which can then be found) when multiplied by
the new term of the root must give the complete divi-
dend, i. e., the remainder.
168. The work may be arranged as follows :
Zx'^^-Zxy^y''
3.r2_>/-J-3;r|/2-hj^/3
Sx'^y-^Sxy^-\-y^
169. The remark made under square root (Art. 162)
about the — sign applies here as well, as is illustrated in
the following example :
Find the cube root of x^—Qx'^y+12xy'^ -\-y^.
The work is as follows :
X
Sx''-Qxy+4y^~
x^ — 6x'^y-{-12xy-—Sy^ ( x—2y
3
—(dx'^y+12xy—Sy^
-6x''y+12xy''-Sy^
1 86 POWERS AND ROOTS.
Find the cube root of the following expressions :
3. x^v^ + 12x-v'^-j-4Sxy^+64y\
4. Sa^ — 60a'^x-\-loOax'^ — 12DX^.
5. 27.r« — 54x*r2' + 36x2j,/2^2_3^3^3^
6. 8jr« + 12a'4_^6;t:2 + l.
7. a^x^-SaHx^'y^+Sad^xy^-Py.
8. 8a•'^^^-24a2^V+24«^^•^-8^^
9- ^+3^+3x24-^^
10. 4+34+3^+^.
a^ a^b b-^ b^
170. So far all the expressions of which the cube root
was required were polynomials of four terms, but we ma}^
have a polynomial of more than four terms of which the
cube root is required. In this case the process already
given may be extended as in the case of the square root,
viz.: Find two terms of the root, as already explained,
and then consider these two terms as a single term and
use their sum the same as a single term was used before.
Find the cube root of each oi the following three ex-
pressions :
11. ««+3«-5+6«4 + 7^3_{_(3^2_^3^^1_
12. Ji;6— 6;t:^4-9,r4 + 4;f»-9jf2 — 6.r— 1.
13. G4.r« + 192.;r-^ + 144;r^-32.T=^-36.r^ + 12.r-l
CHAPTER XIII.
HARDER FACTORS, MULTIPLES,
AND FRACTIONS.
EXERCISE 85.
Factors Common to all the Terms of an Expression.
171. In Chapter IX the subject of factors, multiples,
and fractions was treated to some extent, but the work
there was confined to the case of monomials. We now
resume the same subject, but treat of more complicated
expressions than before.
172. The simplest case of factors of pol3momials is
where the same factor is seen to be common to all the
terms of the polynomial. In this case the polynomial nia}^
be written in a simpler form, by dividing each term by this
common factor, enclosing the quotient in a parenthesis,
and writing the common factor outside the parenthesis as
a multiplier.
Examples.
1. Factor ax'^-\-ax-{-a.
Here ti is seen to be a common factor of each term ; therefore, re-
moving this factor we have
(7X--\-llX-\-a = (7{x'-\-X-]-l. )
2. Factor oax^ •i-loax'^ -j-20ax-\-50a.
Here 5^' is seen to be a factor of each term; therefore, removing this
factor, we have
Find the factors of each of the following expressions :
3. x'^-hx'^-j-x^. 5. a-dc-ha-dd'^+a'^de.
4. m'^x'^-{-n'^x^-\-r^x'^. 6. rsLr'^-\-rsf-j/'^-\-rs'^^^2'^.
1 88 HARDER FACTORS, MULTIPLES, ETC.
7, auv^ -\-buvx-\-uvw.
10. 10^2^2 _]5^^3_25^^2^
11. ^m"^ x"^ —hm'^y'^ —^m"^ x'^ — Ihm'^y'^.
12. Z?>r'^x^-^bbr'^x^y—mr'^x'^y^.
173. Sometimes 07ie factor is common to some of the
terms of an expression, and aiiother factor common to
other terms. In this case you can so^netifnes, though not
always, simplify the expression by taking out the common
factors where they can be taken out.
13. Factor ax-\-ay—a2-{-bx-\-by—b2.
Here the first three terms have a common factor a, and the last
three terms have a common factor h. Taking out a from the first
three terms and b from the last three terms, we may write
ax-\-ay — az-\-bx-\-by—bz=a{x-\-y—z)-\-b{x-\-y—z.)
Now it is plain that in this expression the factor {x-\-y—z) is com-
mon. Hence, we may write
a{x^y-z)+b[x-^y-z) = [a+b)[x^y~z).
Therefore, putting the expression we started with equal to this last,
we get ax-\-ay —az-\-bx-\-by — bz = [a-\-b){x-\-y —z).
Factor each of the following expressions :
14. ax + bx-\-cx-\-2ay-\-2by-\-2cy.
15. ax+2ay-\-Zaz—2bx—Aby—^bz.
16. ax'^y-\-bx'^2-\-cx'^u-\-axy^bx2-^cxii.
17. x^y'^—x^y-{-x^-^y'^—y-\-l.
18. abxy2-\-abx2iv-{-abxu'-\-aby2-\-ab2iv-{-abw.
19. abxyz"^ -{-abxy2-^abxy-\-abx2'^ +abx2-{-abx .
20. a7ivwx-^buvzi>x-j-auvwy-{-b2n'wy,
21. xy -\- ax -\- ay -\- a'^ . v
22. xy — ay — bx-\-ab.
23. x'^y—ay—1b'^x'^-\-1ab'^.
FORMATION OF CERTAIN PRODUCTS. 1 89
EXERCISE 86.
Formation of Certain Products.
174. In order readily to factor expressions, we must
know what expressions were multiplied together to pro-
duce the expression which is given us to factor. This
knowledge is gained by practice in writing down the pro-
ducts of a few forms of expressions. Those we take are
binomial factors. We, have already found out in the
chapter on multiplication how to multiply two binomials
together, but it is very important that the student should
be able to write down rapidly certain products by inspec-
tion, that is, without actually going through the work of
multiplication.
175. The first case to consider is that of the product
of the sum of two numbers by the difference of those
numbers.
Thus, suppose it is required to write out the product
{a-^b){a—b~). By multiplication we find that
{a^-b){a-b)^a''-b''.
That is, the differe^ice of the squares of a and b.
Now as a and b may stand for any numbers whatever,
we may state that
The siini of any two 7iumbers niiiltiplied by the difference
of those numbers is equal to the difference of the squares of
those numbers.
Examples.
Write down by inspection each of the following products:
1. {x \-a'){x—a). 4. {fn-\-n~){7n—7i).
2. {x^-\){x-\). 5. (a^ + bXa-'^^b).
3. (jc-^2y)(^x-2y). 6. (^2+4)(^2_4),
190 HARDER FACTORS, MULTIPLES, ETC.
9. (x2_^4)(;r2-4).
10. {x^—2ab){x^+1ab).
176. The second case of the multiplication of two
binomials is that where the first term is the same in
each binomial.
Thus, suppose it required to. write out the product
{x-\-a){x-^b). By actual multiplication we could find that
(x-\-a){x-\-b)=x'^-{-ax-\-bx-\-ab
=x'^-{-(a-\-b)x-j-ab.
Notice that the first term of the product is x'^, the
second term is x with a coefficient equal to the sum of
the second terms of the given binomials, and the third
term of the product is the product of the second terms of
the given binomials.
As X, a, and b are not in any way restricted, but may
stand for any numbers whatever, the result reached is
perfectly general, and therefore
The p7^o duct of any tzvo bi7iomials in which the first terms
are alike, is eqnal to the square of the first term, plus the
first term with a coefficient equal to the sum of the seco7id
terjns, phis the p7vduct of the second terms.
Of course, due attention must be paid to the signs of
the terms in writing out such products. Thus,
(:t--5)Cr + 2)=.r2 4.(_5^2Xr+(-5x2)
= ;t-2-3jt:-10.
Write down the products in each of the following :
11. (x+«)(x+2). 14. (x-f-5)(x+6).
12. (^-2)(^^+l). 15. (^--5)(x-6).
13. (:f+3)(jr-5). 16. (;r+2)(x-15).
EXPRESSIONS OF THE FORM x^ — a^. 191
17. (;^4-l)(-r+5). 34. (^-6)(;t-+5).
18. {x-\){x-y). 35. {x-^1a){x-Vlb).
19. (;f+l)(;r-5). 36. (;r-8^)(;r+^2)^
20. (jr-f2)(.r+4). 37. (x-«2)^^_^^2>)^
21. (^-l)(^+7). 38. (;t:2-l)(.r2 + 2).
22. (jr+10)(.r— 11). 39. (a + ^^)(«— r^).
23. (j>;+25)(.r— 4). 40. {a—?nr^)(a—7ris'^).
24. (a-— ^)(.i- + 8). 41. (jt:+/;2;/)(jt; + wj/).
25. Cr-10)(;i--10). 42. i^x'^-a^-y'^Xx'^-ay).
26. (:r+10)(-r— 5). 43. (2.a-b')(2a+Zb).
27. (a'-+5)(;»;+12). ' 44. (^2a-b^){2a-c^),
28. (a'-4)(jtr-lo). 45. (;«; + 3)'^)(;r-2«z').
29. (x+8)(:t- + 20). 46. (;t:2+«2)(^2_^l)_
30. (A'-f2)(^+80). 47. a'2 + 2^^)(;»;2-3«j/).)
31. (jr-l)(j»;-5). 48. {ab^-irc:){ab''-2).
32. (jc— 2)(;f— 3). 49. (/;^^^ + o)(;;^«— 4).
33. (a- + 2)(:ir+3). 50. (^2^+;ir>/2)(^.2^_^_^2)^
EXERCISE 87.
Expressions of the Form x~—a^.
177. In this form, x-—a'^, the letters x and a can of
course stand for any two numbers whatever ; so to speak
of expressions of the form x"^ — a^ is the same as saying,
an expression composed of the difference of two square
numbers.
178. We have learned how^ to write down at once the
product of the sum of two numbers multiplied bj- the dif-
ference of those numbers, and we now take up the reverse
process of finding the two factors when their product is
given.
192 HARDER FACTORS, MULTIPLES, ETC.
As the sum of two numbers multiplied by the dif-
ference of those numbers is equal to the difference of the
squares of those numbers, it follows that the difference of
two numbers, each of which is a perfect square, is equal to
the sum of the square roots of those numbers multiplied
by the difference of the square roots of those numbers.
Examples.
Find two factors of each of the following expressions :
I. ^2_4. 7. a^^2_25^2
13-
4a^-Wa^x\
2. j»;2— 9. 8. 4a2;t:2— j/4.
14.
64;;^8-81;^^
3. :r2-l. 9. ^b^-^a\
15.
100-25.
4_ x'^-a''. 10. 167^2 _8Gr2;t-2
.16.
2500-16.
5, ;t:2-4y2. II. a-2-4^2^'^
17-
121-81x8.
6. ^2_4^2^2 12. a8— «6.
18.
625-625^2^2
ig, H\x^ — Sly\ 20. 49a2.r2__49^2^2^4
The following expressions require a factor to be re-
moved from each term before they are in the form of the
difference of two squares :
21. 5^2 _ 5^2^,4 23. 10<2 — 10^7-2 ji-2. 25. oax'^j'—r^ax'^ .
22. 20x^ — 20a'^. 24. a^'x'^ — a'^x-j^. 26. 7?ix^ — 7u?i^x^.
27. tV''j'-ti)-^-^1>'*- 34. 50?i'y-dS?iy.
28. m'Kr-'—J7t^x'y'^. 35. 2Hxy'—QHx^j.
29. 2x^ — d0xj^. 36. 9?ix'^ — o()?iy'^.
30. 27a'^—27ad'^. 37. ax'^—ax.
31. x;>/^2_9Yy3^ 28. ?^z^z£/2— 4?^z''^ze/*.
32. x-^2^—4y'^2\ 39. 24«2^6_54^2^6
33. 4dTiy—196fiy^. 40. 50ax"y—lSaxy.
EXPRESSIONS OF THE FORM :i;^-{-ax+d. 193
EXERCISE 88.
Expressions of the Form x'-^ax-\-d.
179. We have learned how to write down the product
of two binomial factors, when the first terms of the two
factors were alike, and found the ^result was always a
trinomial. We further saw the relation this trinomial
sustained to the two binomial factors which were multi-
plied together to produce it.
180. We now take up the reverse process of returning
from the product to the two factors which were multi-
plied together to produce it.
We can factor any expression of the form x'^-\-ax-\-b if
we can find two numbers whose sum is a and whose pro-
duct is b.
It will assist some in finding the two factors, if the student will re-
memjDer, that when the third term of the given expression is positive,
the second terms of the two factors will have like signs, and when
the third term of the given expression is negative, the second terms
of the two factors have unlike or opposite signs.
Examples.
Find the factors of each of the following expressions :
1. Jtr24-8.r + 7.
The first term of each binomial factor will, of course, be x, and we
are to find the second terms of the binomial factors. To do this we
must find two numbers whose sum is 8 and whose product is 7. Obvi-
ously 7 and 1 are the only numbers that have 8 for their sum and 7
for their product. Therefore, we conclude that the two factors sought
are .v-j-7 and A--f 1.
2. X'^—IX-Z.
Here we must find two numbers whose sum is —2 and whose pro-
duct is —3, Obviously —3 and 1 are the only numbers whose sum is
— 2 and whose product is —3. Therefore the factors are .i*— 3 and .r-j-l.
13
194 HARDER FACTORS, MULTIPLES, ETC.
3. ^2^5^+6. 15. x^+4x-\-4.
4. x'^-j-5x+4. 16. j»;2 4-12;t'+36.
5. ^2^8;tr+15. 17. jt-2 + 12^-+35.
6. jt:2 + llx4-28. . 18. ;r2-10;tr4-16.
7. x''+b5x-\-250. . 19. x^-\-2x-S.
8. .r2+20ji:+100. 20. jt-2 + 13;»;+42.
9. jr2 + 10;»; + 21. 21. :i:2-4;t;--60.
10. ;i;2 + ll;t:4-18. 22. ;r2-f22;t:4-120.
11. x'--dx+S. 23. x2_|_iio.r+1000.
12. jtr2_i(3;t:+48. 24. ;ir2 4-52;»:+100.
13. ji;2^_6^^9, 25. ;»;2-48;i:-100.
14. x^-+8x-\-16. 26. ;t;2 + 15;»;+36.
In the above examples the first term is x"^, and the
second term is some multiple of x. Of course some
other expressions than x^ and a multiple of x might be
used for the first and second terms respectively, provided
only that the first term is the square of something, and
the second term is a multiple of the square root of the
first term, as in the following examples:
27. a^-x'^-4ax-12. 34. ;i:2^2_i0;rjj/-200.
28. x^-Ix^'-SB. 35. a^-17a^-^70.
29. r^^x^-16rx--\-m. 36. 4;t:4 + 20x2;i;2-f 36.
30. n^a^ -^ SW^ a'' -\- SO. 37. ix^-^40x-j-SQ.
31. aV*;i;8 + 17«r2jtr4 + 16. 38. a'^b^ — Ua'^d' + ll.
32. w* + 17w2 + 30. 39. a'^x^—5axy—o0y,
33. n^x*+70n''x^+Q00. 40. ^--204-f-100.
EXPRESSIONS OF THE FORM a^ —b^ , 195
EXERCISE 89.
Expressions of the Form a^—b^.
181. By trial we find that a^ — b"^ can be dividad by
a—b2A follows :
a-b ) a^-b^ {a'^-^ab^-b"-'
a^-a'^b
'• a'^b-b^
a'^b-ab''
aF'-b^
ab^'-b^
Now remembering that the divisor multiplied by the
quotient equals the dividend, we learn from this division
ihat a-'-b^=^{a-b){a''^ab-\-b''-y,
and as a and b stand for any numbers whatever, we can
make the following statement :
The difference between afiy tivo numbers, each of ivhich is
a perfect cube, ca7i be expressed as the product of tivo factors,
one of which is the difference between the aibe roots of the
^lumbers giveyi, and the other is the sqtiare of the cube root
of the first nmnber plus the product of the cube roots of the tivo
mimbers plus the square of the cube root of the second 7iumber.
Examples.
Factor each of the following expressions :
1. ««-8.
Here we have the difference between two numbers, each of which
is a perfect cube, ^^a^ —a- and \'/8:=2. Hence, we write the two
factors (««-2)(^*+2^2-|-4).
2. x^—y^. 5. a^b^—c^d^.
3. x^ — a^y^ . 6. n^r^ — 71^ r^.
4. x^-n-iy^^ 7. ^x^-Tia^y\
96 HARDER FACTORS, MULTIPLES, ETC.
8. 64:x^-125x^y^. 15. 1000-64.
g, Sa^x^ — 27a^x\ 16. ji^v^w'^—xy^z^.
10. 27a^x^-27a^x\ 17. 12o-aH'^c\
11. x^ — Sa^b^c\ 18. 12Da^~64a^x^yK
12. jc6_)/9-l. 19. 1000;i:'^-64:j/3^^
i^. 1 — a^jr^. 20. Jtr^j/^ — ?/''*' 27 ^ze/^.
14. 125-27. 21. 8a^r«7«— 27«9a-6j/\
^^' 1^ "U^' ^^" 'c'^d-' 27j/s''
Ux^y^ u^v^ 1000 27;i:'^
^^' 27^'^ 27^3"* ^^' a^fy" a
3A3-
Sometimes an expression assumes the form of the dif-
ference of two cubes, after a factor has been removed
from each term, as in the following examples:
26. a^x'^ — a^.
If we take out the factor a' there will remain x^—a^ of which the
factors are x—a and x^ -\-qx-\-a^ . Hence, the factors of the given ex-
pression are «^, x—a, and x'^ -\-nx-\-a'^ . Hence the given expression
equals a^[x—a)[x^-\;-aX'^^-al).
27.
x^ z'^ —y"^ z"^ .
32.
Zfrr^fi^ — Sm^n^
28.
aH^x'^-c^x'^.
33.
5«3_^3_40^3^3
29.
a'^b''^x'^y—c^x-y.
34.
2a'^xy^—2a^.
30.
r\,U'-27r'sUK
35.
2^2
2«2j;6^3
a^y^
31.
27^3 ^^^'^''
36.
5a^ 6x^
b^ 8>/3-
EXPRESSIONS OF THE FORM a^ + d^. 197
EXERCISE 90.
Expressions of the Form a^-\-l>^.
182. By actual division we find that a^ + 5^ can be
divided by a-}-d as follows:
a-\-b ) a^-^-b^ (^a'^-ab+b'^
-aH+b^
-aH-ab^
ab'^ + b^
ab^ + b^
Now remembering that the divisor multiplied by the
quotient equals the dividend we learn from this division
that a^ + b^ = (a-hb)(a^-ab-\-b^);
and as a and b may stand for any numbers whatever, we
may make the following statement :
Tke sum of any two numbers, each of which is a perfect
cube, can be expressed as the product of two factors, 07ie of
which is the sum of the cube roots, of the 7iumbers giveyi, and
the other is the square of the cube root of the first number
ttmius the product of the cube roots of the tivo nuynbers plus
the square of the cube root of the second number.
Examples.
Factor each of the following expressions :
I. ««+8.
Here we have the sum of two numbers, each of which is a perfect
cube ^/a^=rt' and -^8=2. Hence we may write
2.
x^-\-y^.
6.
aH^-hSaHK
3-
a^x^ + b^y^.
7-
Sx^y^-^27y^.
4-
7?i^ -\-x^y^.
8.
64;t;3^« + 125^«
5
m^+a^b^.
9-
64x^-h8x^y\
HARDER FACTORS, MULTIPLES, ETC.
10. m^-\-n'^. 15. %a^b^x^^-^a^b^y^,
11. r^-{-x^. 16. x^y^-\-Tlx'^z^.
12. u^v^x^^-u^v^x^, 17. 64;t-6 + 125^3^6^9.
13. x^y^z^-^\. 18. 1000+64.
14. a^b^c^-^x^y^z^. 19. 64 + 8.
20 ^V-^'
- IU
27*
, , 64^5
^4- l + -27^-
^3 ^6^,3
• r« ' 27 •
+ 64
^27'
^^' a^ ' 27^3
26. -^f- + 125.
29.
aH'' ' 64a« '
^Zy.,.
/^•^Z^^^Ziy^
30.
''^- aH^ '
a'-^b'^ '
28 ""'■'.+
1000a 3^
1000
31-
1000 1000
Sometimes an expression assumes the form of the sum
of two cubes, after a factor has been removed from each
term, as in the following examples :
32. ^2-^-3 +^5 37 5^^2^9^-3^6_^40^^•2^.6^.6^
33. x^'y^z'^^-a^b^z'^. 38. 2a2r6^6_^2a2.
34. a'^bxy^ ■^-a'^bx^z'^. 39, zcv^x^y^ ■\-2iv'^w^ ,
35. ax'^yz^-^-axy. 40. 128xy^ -j- 64x* z\
- 16;rV^ . bAxy^z'^ ba^ ox^
EXERCISE 91.
Expressions of the Form x^-{-a^x^-\-a'^.
183. Some expressions that do not appear to come
under any case thus far considered may be made to do so
by adding and subtracting some expression. Such is the
EXPRESSIONS OF THE FORM ;ir* +^^^- H"^*. 199
case with the expression we are dealing with in this ex-
ercise. We notice that the expression here considered,
viz.: x'^ + a^x'-'i-a'^, is ahiiost of the form of a perfect
square, viz.: the square of x'^-\-a-, and, indeed, if the
middle term were only 2a'^x^, instead o{ a-x'-, the expres-
sion here considered would be a perfect square. So we
make the expression a perfect square by adding a'^x'^,
but if we add a'-x'^ we must subtract it in order not to
change the value of the expression. We may then write,
x^ -\-a''x'- -ha^=x* + 2a''x'- -ha^-a^x'^,
or, using a parenthesis, =(x* -i-2a'^x'^ -{-a'^)—a'^x'^ .
Now it is easy to see that we have the difference of two
squares, which we have already learned how to factor.
Thus we have
X* -^a\x'- +a* = (x^ +2a'^x'^ -ha*')-^a'^x^ ,
= (ix'-+a^y-a'^x\
= (x'^ -\-a^- +ax)(x'^ +a'^ -ax-).
Examples.
Find the factors of each of the following expressions :
1. x^-\-dx''+Sl.
To make a perfect square of this expression we must add 9^^*, and
if we add 9.v* we must of course subtract 9x^ afterward. Hence we get
We now have the difference of two squares, which, as we have
learned before, is equal to the product of the sum multiplied by the
difference of the square roots of the two squares. Therefore, we have
= (.'c'+9-f-3x)(x8-|-9-3.r),
or, as is perhaps a more natural arrangement of the terms of the two
factors, (;r2-i-3-v-[-9){.r3-:U-f-9).
2. x* + ^x^- + 16. 5. l + a'^+a*.
3. x^ + 4xy^-\-16y. 6. x^+x'^-i-1,
4. a*x*+4a^x^-j/^-hl6jy\ 7. Ux* -j- 4u'^ v^-'' + ziH^K
200 HARDER FACTORS, MULTIPLES, ETC.
8. 16jt-4-hl6.r22'4 + 16^8 14. Slx^-{-SQx^y^ + lC\v^,
9. 8Li:4+36.r2-j-16. 15. x^''-^4x'y'^-}-l(yj'\
10. 16jt:*+36jt:>2 + 8iy. 16. Wx'^ +4x^ + 1.
11. 256 + 144 + 81. 17. x^y-hx-y^-l-y^.
12. 81 + 9 + 1. 18. jt-y+x^j/fi+jK^.
13. 81 + 9«2-j-«4^ . ig. x4jj/%-i+4.r2^''^2 + 16.
20. :^ + .-4- + 81. 21. :^ + 9.r2 + 81«^.
184. Sometimes an expression assumes the form treated
in this exercise, after a factor is removed from each term,
as in the following examples :
22. lOmx^ + lOa^'mx^ + lOa^m.
23. ax^-j-a'^x^-^-a^x.
a^x^ , a^x"- , «
24. — — - + — -- + ^6^
25. 10jry + 10.;i:2j/2 + 10.
26. _^-+--_^_H-2^8.
27. 48 + 12^2^2+3^4^4^
EXERCISE 92.*
Expressions of the Form a^^—b'K
185. The two general expressions a"—b" and a"+^",
where n stands for any positive whole number, are of so
much importance that, although somewhat difficult to
factor because of the fact that n is such a general symbol,
still we must try to discover under what circumstances
expressions have factors of the form a-^b or a—b. We
begin with the first of these two expressions, viz. : a"—b".
* This exercise and the next one are more difficult than most portions of the
book, and may, at the discretion of the teacher, be taken now or postponed till
some later time in the course.
EXPRESSIONS OF THE FORM a'^—b''. 20I
1. Divide a^ — b^ by a—b and carefully note the form
of the quotient.
2. Divide a^ — b^ by a — b and carefully note the form
of the quotient.
3. Divide a^ — b^ by a — b and carefully note the form
of the quotient.
4. Divide a^ — b^ by a—b and carefully note the form
of the quotient.
5. Without trj^ing, would you think that a'^ — b^ could
be divided hy a—b}
6. Could you guess the form of the quotient of
(a'-b')^{a-b)}
7. Go through the work of division of (a'^ — b'^') — {a — b)
and see if you have guessed the right form of the quotient.
186. Now you would probably guess that a"—b"
could be divided hy a — b and that the quotient would be
«"-i_(_^"-2^_l_^"-3^2_^^«-4^3 .pother terms,
where the exponent of the power of ^ continually dimi-
nishes and that of the power of b continually increases
by 1 in each succeeding term until you arrive at b"~'^ .
A result of this kind where a series of terms are written
in regular order but where the number of terms is not
definitely known is sometimes written like this :
a"-'' -\-a"-H-i-a"-H^- -ha'-H^ -\- . . . +b"-K
The dots (read afid so on) are called the Sign of Con-
tinuation. They serve to show that the intervening
terras are written one after another according to the same
law as the terms already written.
8. What would be the term just after a"-'^b^ ?
9. What ^vould be the term just before b"'^ ?
202 HARDER FACTORS, MULTIPLES, ETC.
187. If the student will carefully get fixed in mind
the meaning of these dots, we can doubtless go through
the work of multiplying and see if this expression mul-
tiplied hy a—b will produce a''—b"; the7i, if this should
be the case, what was before a guess becomes a certainty.
For the student will remember that the expression we
are here talking about, viz. :
was a giiess at the quotient of dividing a"—b*'- by a~b,
and therefore if this quotient multiplied by the divisor
equals a"—b" the guess was right. Now let us multiply
a—b
a"+a"-H+a"-''b''^a"-^b-^-{- .... ■^ab"-'^
-a"-H-a"-H~-a''-H'^-a"-H''- . . . -b"
Let us stop a moment before adding up these partial
products to notice that there is a sign of continuation in
each partial product.
10. In the first partial product what is the term just
after a''-'H-> ?
11. What is the term just before ab"~^ ?
12. In the second partial product what is the term just
after — a"-^^^^ ?
13. What is the term just before —b"}
Now adding the above partial products w^e get the
product, which is seen to be a"—b". Therefore,
a-^b"=^(^a-bXa"-^+a''--b-^a"-'H''-\- . . . +b"-^).
Therefore, a—b is a factor oi a"—b", whatever positive
whole number is represented by ?i.
EXPRESSIONS OF THE FORM a'' — b"". 203
*
187. As the sign of continuation usually offers con-
siderable difficulty to the student, let us study a little
further the expression a:'~^ -\-a''~'^b-\-a"~^b''- -\- . . 4-^""^
in which this sign first appeared.
14. Is «"-i-fa"-2/^+«"-3^'+ . . . -f/^"-i a factor of
a'^-b"-^. Why?
15. Is a"—b" 2. multiple oi a''-"" ^-a"-'b-\-a"-^b'^ -\- . .
. . +^"-1?
16. By what expression will we have to multiply
«"-i+«"-2^+a"-3^--f . . . +b"-'^ to produce ^"—<^"?
17. Can a"—b" be divided by a"~^-{-a"~''b-\-a"~^b^-\- . .
. . -\-b"-'?
18. What is the quotient in example 17 ?
19. Will any factor of a"-i +a"-2^+a"-3^2^ . . +b"-^
be a factor of a" — b" also ? Why ?
20. In a"-'^-\-a"--b-i-a"-^b'^-{- . . . -j-b"-'^ how many
terms contain b ?
Write down 5 or 6 terms and then begin at the second term and
count the successive terms, looking at the exponent of /^ while count-
ing, and you can doubtless answer this question.
21. In a"-'^+a"-'^b+a'*-^b^-i- • . • -f ^"~^ how many
terms are there altogether ?
Notice that each term except the first contains /> and you can prob-
ably answer this question.
22. The first two terms contain the common factor
a"~'^. Can their sum be written a"~'^{a-\-b) ?
23. Do the third and fourth terms have a common
factor? What is their H. C. F.? How, then, can their
sum be written ?
24. Do the fifth and sixth terms have a common iactor ?
What is their H. C. F.? How, then, can their sum be
written ?
204 HARDER FACTORS, MULTIPLES, ETC.
188. If n=b, the expression we are considering is
a^-\-a^b-\-a'^b'^ -\-ab^ -^b^, and if the terms are grouped
together in sets of two as far as possible, we may write
(a*+<2^^)-f(^2^2_|_^^3-)_j_^4^ where there is one term left
over at the end which cannot be put in any group because
there is no term to go with it. But if ?i — Q, the expres-
sion we are considering \s a^ -\- a"^ b -{- a'^ b' -\- a'^ b^ -\- ab"^ -f- b^' ,
and if the terms of this are grouped in sets of two as far
as possible, we may write {a^ -{-a'^b) -\- (^a^b"^ -\- a-b^)
+ (ab'^-\-b^), where there is ?io term left over at the end.
25. If in a"-'^-j-a"-H + a''-^b'^-\- . . . +^"-1 the terms
be grouped together in sets of two as far as possible, when
will there be one term left over at the end and when will
all the terms be thus grouped ?
Ans7ver: All the terms can be thus grouped when there is an even
number of terms, i. e. when n stands for an even number, and one
term will be left over when the number of terms is odd, i. e. when ;/
stands for an odd number.
26. Suppose n an even number, group all the terms of
^,,_i_j_^„_2^_l_^.-3^2_}_ ^ _ _ -f ^"-1 in sets of two, take
out the H. C. F. from each group, and then tell what is
a factor of the whole expression.
Notice that the factor which you have just found is a factor of
rt"-i-|-rt"-2(^-|-(7«-3^3_j_ _|_^«-i ^w/v when n is an even number.
27 . \sa-\-b2i. factor of «" — b" when « is an even number ?
See question 19.
189. Now the student can probably understand the
following demonstration :
a—b
Hence, «"-^"=(«-^)(^"-i+^"-2^+^"-^^2+ . . +/^"-').
The number of terms of this expression, a''~^ -\- a''~'^ b
-fa^-s^a.^ . . . +^"-1 is n, for there are n—\ terms that
EXPRESSIONS OF THE FORM rt" — ^^ 205
contain b (as can be seen by beginning at the second term
and counting, noticing the exponents of Awhile counting,)
and one term that does not contain b, so there are ii terms
in all. Therefore, when n stands for an even number
the terms of a''-^ -\-a''-''-b-\-a''-^b'^ -\- . . . -^b"-^ can be
grouped in sets of two, and in each set we can take out
the H. C. F. and express the result as follows :
a:'-''-{a-\-b)^-a"-''b''-{a^b)-\-a"-'^b\a^b)-]r • -\-b"--{a-^b)
Evidently <^-f^ is a factor of this expression, i. e. a + b
is a factor of «"~^-f«"~^^ + «"~^t(^-4- . . . +^"~^.
But any factor of «"-i+a"-2/^+a"-'''<^2^ . . . 4-^"~Ms
also a factor of a"—b" because a"—b" is a multiple oi
Therefore ^ + <^ is a factor of a" + b" when 71 stands for
any eve7i number.
Notice, this demonstration will not hold if ;^ is an odd
number because the terms of a"~'^-\-a"~'^b-\-a"~^b'^-\- . .
. . +b"~^ cannot then a/l be grouped in sets of two, for
one term will remain over.
190. We have so far reached the following results :
a"—b" can a/ways be divided by a—b, and a"—b" can be
divided hy a + b when 71 is aTiy eve7i 7iu7nber, or stated in
another way, a—b is always a factor of a"—b" and a-\-b
is a factor of a"—b" whe7i n is a7iy eve7i Tiumber.
191. Thus it appears that the difference between like
powers of two numbers can always be factored, but it
must not be supposed from what has been said that the
easiest and best way to factor d" — b" is always to take out
the factor a—b first, for it may be that the remaining ex-
pression after this factor has been removed will be harder
to factor than the one we started with would be, if we
proceed to take out some other factor first. This will be
fully seen in the examples which follow.
206 HARDER FACTORS, MULTIPLES, ETC.
Examples.
Factor as far as j^ou can each of the following expres-
sions :
I. a^-b"^.
This we know has a factor a—b, and also because the exponent is
even a factor a-\-b, and if we take out each of these factors in turn
we would have left a^-^b^.
Therefore, a^~b^ = [a-b){a-{-h){a^-[-b^).
Or we might proceed thus: a'^—b^ may be regarded as the differ-
ence of two squares and factor accordingly.
Therefore, a^-b^={a^Y -(b'^Y = [a'^ — b^){fl^-\-b''^)
z=z\a-b){a-^b){a^+b^),
the same as before, as it ought to be.
2,
x^^-l.
6.
16^'*— 81.
10,
i6x^-ie,j^\
3.
x^-y.
7.
l-jr*.
II.
.^*J/*— ?/*Z'*.
4.
X^J^-2\
8.
l-x'^yK
12.
5.
r^s^-t\
9.
16a^-x^y\
13.
u"" x'^
14. a^ — b^.
From the general discussion which precedes we know that the fac-
tors of this are a—b and a^-^a^ bA-a~ b'^ -\-ab'^ -\-b^ , and this is the only
way we have at present to factor this expression.
15. x^' — \. 17. x^y'^—z'"". 19. \—x^\
16. ^5_^5 18. 2/5z/5 — 32. 20. 243—32.
21. Zlx^y^ — ii^v^w^. 23. a''::f^y^ — ?)'^lu^v^w^.
11. — < ~. 24. -^. ~.
z^ w^ y^ y^
25. a^ — b^.
This may be regarded as the difference of the two cubes, and
hence we may write
EXPRESSIONS OF THE FORM rt" — <^«. 20/
Each of these factors is a form already treated, and so we know the
factors of each factor, viz:
a^ — (,i — {a-h){a + l>)
and a* -^a^h^ ■^h^ = {a^ +b^ +ab){n^ +h^ -ab).
Therefore, a^ ^ h^ =(a-b){n+b){a^ +b^ -^ab){a^ +/>^ -ab).
Or, if we prefer, we may regard a^—b^ as the difference of two
squares, and hence may write :
a^—b^={a^)^-{b^Y-{a^-b^){a^ + b-^).
Here again each factor is a form already treated, and so we know
the factors of*each factor, viz:
a^-b^ = {n—b){a^ + ab + b^)
and ,r^ + b^=:{a + b){a^-ob + b*).
Therefore, a'^-b^ = [a — b)(,a-irb)[a^-i-ab + b'^){a*—ab + b^).
This agrees with the result obtained before, as it ought to do.
26. x^ — l. 28, — g— w'^.r^. 30. x^y^2^ — \.
27. x^y^—2^. 29. 7i^x^—r^y^'2'^ 31
X^ |, 6
32. a'^ — b'^.
From the general discussion which preceeds we know that the fac-
tors of this ^rea — b and a^ +a^b ^ro^b- -\-o^b-^ +a-b* \-ab^ +b^, and
these are the only factors of a"' —b' that we can find now
33- x'^y"'—!. 35. x'^y'^—z\
x"^ 11^ ^ ^ ?/77;7
34. -y--?- 36. 1
yi 1}i - W'^X'^'
37. a^ — b^.
This may be considered either as the difference of two fourth
powers, or the difference of two squares. Taking it in the latter way
we may write,
a^ — b^—(a^-b^){a'^A.b^^.
The first of these factors has been considered before, so we know
how to factor it. Therefore we have,
a^-b^-{a^-b^){a^Jrb^)
={a-b){a^-b){a'^^.b^'){a*' +-<5*).
208 HARDER FACTORS, MULTIPLES, ETC.
38.
x^y^ — 1.
40. X^—7C^V^.
39.
x^ u^
y% ^8*
4^- .<- .s-
42.
«9-
-dK
This may be regarded as the difference of two cubes, viz;
(a3)3_(<^3)3.
Therefore, a^ -d^={a^—d^){a^ +a^d^ + 1?^)
43. x^y^-z\
45. u^v^-sH\
x^ .
44- y, 1.
71^ ^9
EXERCISE 93.
Expressions of the Form a^'-\-b'K
192. In the previous exercise we found that a''—b"'
was always divisible hy a—b. Now this dividend, d"—b'\
means that the 71 th power of b is to be subtracted irom
the 7ti\i power of a, and the divisor, a—b, means that
the number represented by b is to be subtracted from the
number represented by «, and in both dividend and
divisor a and b may stand for either positive or negative
numbers. We may then write a7iy numbers or letters we
like in place of either a or ^ or both.
Suppose, then, we take the case where n stands for an
odd number and write —b in place of b in both dividend
and divisor. Now, because 71 is an odd number,
(^—by= — b'\ and if this expression, —b", be subtracted
from a" we get a''-\-b*' for a dividend, and if — ^ be sub-
tracted from a we get « f ^ for a divisor.
Therefore, a-\-b is a divisor, i. e.y o. factor , of a"-\-b"
when n is a7iy odd 7iU7nber,
EXPRESSIONS OF THE FORM «" + ^«. 209
193. To find out whether a-\-bova—dis a factor of
a"-{-d" when 71 is eve?i, a little preliminary stud}^ is
necessary.
1. Is 5 a factor of 55 ? Is 5 a factor of 15 ? Is 5 a
factor of 55 + 15 ? Is 5 a factor of 55—15 ?
2. Is a a factor of ax? Is a a. factor of ajy} Is a a
factor of ax-\-ay ? Is a a. factor of ax— ay ?
3. If a number is a factor of each of two other num-
bers, is it a factor of the sum of those numbers ? Is it a
factor of the difference of those numbers ?
4. If one number is a factor of a second and not of a
third, can the first number be a factor of the sum of the
second and third numbers ? Can the first number be a
factor of the difference of the second and third numbers ?
5. When n is even is a-\-l? a. factor of a"—b"? When n
is even is a-\-d a. factor of 2d" ? When ?i is even is a-\-d
a factor of (a"-d")-i-2b"?
6. Since (ia"—b") + 2b"=a"-\-d"isa + bsL{2iCtovofa"-\-b"
when n is even ?
7. Is a—d a factor of a"—d" ? Is a—d a factor of 2d" ?
Is a-d a factor of (^a"-d"-)-\-2b" ?
8. Since {a"—d") + 2d"=a" + d" is a—b a factor oi
a"-\-d"J
194. Questions 6 and 8 if rightly answered and un-
derstood, give us the fact, that a" + 5" is never divisible by
a—d and a"-\-d" is not divisible hy a + d when n is even.
196. We previously found that a"-\-b" is divisible by
a-\-d when ?i is an odd number.
14
2IO HARDER FACTORS, MULTIPLES, ETC.
196. What has been found in the previous exercise
and in this one, may now be stated as follows:
a"—b" is always divisible by a—b.
a" — b" is divisible by a-\-b whefi n is an even number^
but not when n is an odd number.
a"-\-b" is divisible by a-{-b ivhen n is an odd number, bia
not when n is an even 7iumber.
a"-\-b" is never divisible by a — b.
197. Although a''-^b" cannot be divided by either
a-\-b or a—b when n is even, yet it is not stated that in
this case a''-\-b" has 710 factors, for sometimes other factors
can be found, as will be seen by the examples to follow
and the explanations accompanying them.
Factor each of the following expressions :
9. a^-\-b^.
By the general discussion we know that this has a factor (X-\-b, and
another factor, found by division of a^-\-b^ by a-\-b, is
10. x^-\-\. 12. a'^^x'^-^-b'^'y^. 14. n^x^ -^^^x^y'^'z^ ,
11. x'^ y''' -\- z^ . 13. l + 32;»;^. 15. a^ -\-^lic^ v''' w^ .
^ ;r^ . 32 „ a^x'" , b'^z^
16. -5- + -^. 18. — 5-- + --^.
yO ^O yb yb
a^ b^ . , a^
17, ^~, + -,. 19. 1 + 32.
20. a^-\-b^.
This may be regarded aS the sum of two cubes, and therefore we
may write a^ ^b^-{a^Y +{b^Y-{a^ ^b^){a^-a^b'^ ^b^).
21. x^-\-y^. 23. «6^^ + 64^^j^. 25. 64:-\-x^y^z^.
22. x^-\-l. 24. «^z^^4-G4. 26. x^y'^z^-^x^y^w^.
27. ^ + 1. 28. -'4-4-
' y6 ^6 ' ^6
EXPRESSIONS OF THE FORM a^' + d". 211
29. a'^-hd'^.
By the general discussion we know that this has a factor a-\-d, and
the other factor obtained by division will be found to be
30. x'^-j-y. 32. 71'^v'^+x'^y^. 34. l-{-x''yz'^.
31. u-' + l. 33. aH'^c'^ -i-128. 35. 128 + 1.
36. - -J ~. 37. -7 + —-.
v^ y^ y^ x^
38. a^ + b\
This may be regarded as the sum of two cubes, and hence we may
write rt9+(^»=(«3)3+(/;3)3_(^3_,.^3)(rt6—^/3/'3 +/;«),
= (^a + l>){a^-al>-\-l>''){a^—a^b'^+i)^).
39. x'^ + l. 41. 21^+vHv^xK 43. ^ + -9.
40. x^y^+z^, 42. a9;i:9+<^9j/9^9.44. ^4-^.
45. a^« + ^^«.
This may be regarded as the sum of two fifth powers, and hence
we may write,
46. x''-{-y^\ 48. 1024 + .r^0j/io.5o. |J-^4-^o-
47. a^'+l. 49. 1024 + 1. 51. ^5 + 1.
198. In the expressions considered in this exercise
and the preceding, the exponent in each term is the same,
but the methods used enable us in some cases to find fac-
tors of an expression in which the exponents are not the
same in each term, as for instance in the expression
^5_|_^io '^\^^ exponents are 5 and 10 respectively, but
we may consider b'^ ^ =^ (^b'^^^ and, therefore, a^ + b^^ may
212 HARDER FACTORS, MULTIPLES, ETC.
be considered the sum of two fifth powers, viz: a^-\- {b'^')^ ,
and may be factored as in example 9, using b"^ the same
as b was used before.
We add a few miscellaneous examples on the last two
exercises, where in some cases a factor must be removed
from each term before the expression is in the form
treated in either of these last two exercises.
52. 3^4-48. 61. x^-a^y^"^.
53. 2a^ — Z2a''x^. 62. r^x^'^ — bx^y^'^.
54. 3;i:*-3yi^ 63. lOa^-\Qb^c^.
55. x^-\-a^y^\ 64. {x+yY—i^x—yy.
56. ^u'^v^-Zu^x^\ 65. Z{x^ +y^y -Z{x^ -y^y
4 4 ^^ {x^-y^y 64
{x'^-^y^y {x^'-y'^y ' 64 (^3_y^6-
58. {a + by-{a''-b''y. 67. «i8-ai2.
59. («2_^^2-)4_(^2_^2)4_ 68. 3;«.;ci4_3w8j/7^
32 32 • ^ «6-
EXERCISE 94.
Miscellaneous Factors.
199. Some expressions are quite difficult to factor un-
less one sees how the expression may be changed in form
by rearranging or grouping of the terms, or by adding
and then subtracting the same expression, or by some
other device to change the expression into a form that
will be recognized as coming under some case already
treated. To help the student see some of the devices,
we take some of these irregular expressions and work
MISCELLANEOUS FACTORS. 21 3
them out. The student is advised to go through the ex-
planation given, to see just what is done, and if possible,
ivhy it is done, and after two or three cases to cover up
the explanation given and see if some method suggests
itself; if not, see how the work is started and then try to
complete it without looking at the rest of the explana-
tion. If still unable to do the work, study the whole
explanation given.
1. Let us factor a3 + <53-f ^3 ^3<^2^+ 33^2^
We may arrange the terms thus :
where the last four terms form a perfect cube, viz: the
cube of b-\-c. Therefore, the given expression may be
written a^ -\-{b-\-cY .
We now have the sum of two cubes, and therefore the
factors are ^ + ^-f-^and a'^ — a{b-\-c)-\-{b-\-cYi ox a-\-b-\-c,
and a'^^-b'^-\-c^—ab—ac-\-1bc.
Hence, a^^-b"" ^-c^ -^Zb'^c-VUc'^
= (a-\-b-i-cXa''-^b^-+c'--ab-ac+2bc),
2. l^t us (sictor a'^ + b^+c'^+3a'^b-\-Sab^.
We may arrange the terms thus :
a^-\-SaH + Sab''+b^-hc^,
where the first four terms form a perfect cube, viz: the
cube of a + b. Therefore, the given expression may be
written (a-^b)^+c^.
We now have the sum of two cubes, and therefore, the
factors are a-\-b-\-c and (a-\-b)^^{a-^b)c-\-c^ , or a+b-\-c,
and a"^ -^b^ -{-c"^ -j-2ab—ac—bc.
Hence, a^-hb^+c^+SaH+Sab^
^{a-\-b-{-c)(a'^ ^b"^ ^c"^ +2ab-ac-bc).
214 HARDER FACTORS, MULTIPLES, ETC.
3. Let US factor the expression a'^-\-2a^d—a'^'-2a5^
This can be arranged thus :
where it is plain that a'^^b'^ is a factor of each of the
three parts into which the expression is grouped.
Taking out this factor, the expression may be written
or (^2_^2)|-(^2 4.,2^^_|.^2)_i]^
or {a''-b''-)\{a-Vby-\\
Each of these factors is itself the difference of two
squares, and hence each may be further factored thus :
Therefore we may write,
a^-^la'^b-a'^-lab^-^b'^-b^
= {a-b^{a^-b){a^b-V){a-Vb-\-V),
4. Let us factor the expression a2^2_^2^2_^^2_^2^
This may be written thus :
(«2_32)^2.^^2_^2^
or (^?2_^2)^2_^^2_^2)^
or («2__^2)(_^2_1)^
or {a-b){a-^bXx-r){x-\-r)^
5. Let us factor the expression a^ — a"^ -\- b^ — h'^
—2a'^b''--2ab.
This may be written thus :
{a^-2aH''-^b^)-{a''-^2ab + b-'), '
or {a'-b''y--{a-\-by.
MISCELLANEOUS FACTORS. 21$
This is the difference of two squares, and hence may be
written as the product of the sum into the difference of
the two numbers which are squared. Hence,
= [(«2_^2)_(^ + ^)][-(^2_^2)_^(^_|.^)]
= {a+b){a-b-l)(a + b){a-b+V),
6. I^et us factor a" + /^^+<:^— 3«<^<:.
This may be written thus :
a^ + {b^ -irWc+'^bc'' +c''')-Wc-Uc''-Zabc,
or a^ + {b-{-cY~Uc{a-\-b+c).
The first two terms are now the sum of two cubes, and
hence contain the factor a-\-b+c, and this is also a factor
of the last term, as is very evident. We may therefore
write the expression considered in the form
ia + b+c)[_a''-a(^b+c)^-(^b+cy^-^bc{a + b+c),
or (a-{-b-^c){a'^—ab—ac-{-b- — bc-\-c''-).
7. lyet us factor a^-\- b^-\- c^-\- ab'^-\- ac'^-\- a'^b+ a'^c
^-bc-'-^-b'^c.
This may be written thus :
In the first group ^'^ is a factor of each term, in the
second group ^^ is a factor of each term, and in the third
group c^ is a factor of each term. Hence, we may write
a''{a-Vb^-c)-\-b''{a^-b^c^-\-c''{a^-b^-c).
And now evidently a-\-b-\-c\s> a common factor through-
oat. Hence, the original expression equals
2l6 HARDER FACTORS, MULTIPLES, ETC.
8. 12;t:2-36.r4-24.
9. 5;r2+5;i:— 10.
10. x'^j/-i-xj}^—x'^—x—2y-\-2.
11. (a + b + cy-(a-d-cy. 14. x^-^-Sj"^ -\-x-\-2y.
12. (c-^dy-^(c-dy. 15. «5_8^2^3_
13. x''-iy^+x-2j^, 16. 500a'2j/-20j/\
17. 1— ^2^-2— /^2^2_j_2^^^_y^
18. a^-j-x''-(^y''-i-2^-')-2(ij^2-ax^.
19. 5.:^;4- 15.^3 -90;i;2^
EXERCISE 95.
H. C. F. OF Expressions which Can be Factored.
200. The highest common factor of several expressions
has already been defined, (see Art. 82,) and a method of
finding the H. C. F. of several monomml expressions has
been given. We now take up the H. C. F. of poly-
nomials.
201. The first method to consider is exactly the same
as that presented in the case of monomials, viz. :
/Resolve each expression into its priyne factors ajid take the
product of all those which are common to all the expressions.
By this method we can readily find the H. C. F. of any
expressions which can be readily resolved into their
prime factors.
Examples.
Find the H. C. F. of each of the following sets oi
expressions :
1. x"^ •\- xy 2,wA x"^ —y"^ . 3. j»;^— j/''^ and ^2— jj/2.
2. 2.^2 — 2;rK and 3.;t2—3>/2 ^ x^ -\-y^ 2.viA x'^—y'^ .
SECOND METHOL OF FINDING H. C. F. 21/
6. na'^x'^y-Aa^xy'' 2inA^0a''x^y'^-10a''x''y^ .
7. 8«3^V-12^2^^3 and 6«^V+4^<^V2.
8. ;r2-2;ir-3and Jt-2H-ji--12.
9. 2a{a''-b'') 2.n&Ab{a-by.
10. 3:i:=^+6.r2-24j»r and Gx^-QG;*;.
11. j»:2— 9;»;— lOand ;r2+4ji-+3.
12. jt:2— 4 and ;f2— 5;i:+6. 14. a'^+x— 6 and x-— 3a'4-2
13. rt!6 — ^6 and^* — ^4. 15. -r^+.r+l andjt-'^ — 1.
16. 2:t=^+2, 3a'2-3, and x''-{-Zx+1.
17. jf2— 3^+2, x-—Q>x + S, and ;r2+ji-— 6.
18. ;i;2j/-2-^^ x''-y'^-dxyz-^22'',2indx^y^-y2^.
19. ;t:4-8;»;2 + 16and%r3j/3 4.4_;^2y_|.4^-^3
20. ;^;3— ;i:2-f jtr— 1, x^ ^x^—x'^—x, and ^t.^ +2;r— 3.
EXERCISE 96.
H. C. F. OF Expressions not easily Factored.
202. The method used in the preceding exercise for
finding the H. C. F. is not appropriate when the expres-
sions given are not easily factored. In case the expressions
given are not easily factored, the method to be pursued
depends upon a few principles which must be understood
before the method is given.
1. If an expression be multiplied by some number or
expression, does the product contain all the iactors the
original expression contained ?
2. Does the product contain any factors the original
expression did not contain ?
2l8 HARDER FACTORS, MULTIPLES, ETC.
3. Are the factors of an expression also factors of any
multiple of that expression ?
4. If one expression is a factor of two other expres-
sions, will it be a factor of the sum of those expressions?
5. Will it be a factor of the difference of those ex-
pressions ?
6. Will it be a factor of any multiple of the first ex-
pression plus any multiple of the second expression ?
7. Will it be a factor of any multiple of the first ex-
pression minus any multiple of the second expression ?
203. If these questions are understood, w^e may state
the two principles to be used in finding the H. C. F. as
follows :
/. A7iy factor of an exp?'essio7i is a fad or of any miiHiple
of that expression.
II. A7iy coimnon factor of two expressions is a factor of
the sum or difference of those expressions or of the S7im or
difference of any multiples of those expressio?is.
204. Our problem is to find, not merely a co7nmon
factor, but the highest common factor of two or more ex-
pressions. This problem can often be considerably sim-
plified by first taking out from each of the given
expressions all moyiomial factors and finding the H. C. F.
of these factors, if there be any, and then find the H. C.
F. of the remaining portions of the given expressions.
After this has been done w^e must multiply the H. C. F.
of the monomial factors by the H. C. F. of the remaining
polynomial factors, when we will have the whole of the
H. C. F. of the given expressions.
I.et us find the H. C. F. of
and 3^r3_^;t;2_4^_20. (2)
SECOND METHOD OF FINDING H. C. F. 219
Neither of these expressions have atij^ 7nonomial factors,
therefore the H. C. F. of them will have no monomial
factors.
8. Will any factor of expression (1) be a factor of 3
times this expression, i. e., oi
3x3-3;c2_i2? (3)
9. Will any common factor of (1) and (2) be a common
factor of (2) and (3) ?
10. Will the H. C. F. of (1) and (2) be the H. C. F.
of (2) and (3) ?
11. Will any common factor of (2) and (3) be a factor of
their difference, which is
12. Will the H. C. F. of (2) and (3) be the H. C. F.
of (3) and (4) ?
13. Will the H. C. F. of (1) and (2) be the H. C. F.
of (3) and (4) ?
Now remember that the H. C F. of (1) and (2) can
have no monomial factors, and therefore the H. C. F. of
(3) and (4) can have no monomial factors, and the next
question can be answered.
14. Will the H. C. F. of (3) and (4) be the H. C. F. of
and x'^-x-l't (6)
15. Will the H. C. F. of (1) and (2) be the H. C. F.
of (5) and (6) ?
16. Will any factor of (G) be a factor of x times this
expression, which is
;j;3_^2_2jr? (7)
17. Will the H. C. F. of (5) and (6) be the H. C. F.
of (5) and (7) ?
220 HARDER FACTORS, MULTIPLES, ETC.
i8. Will any common factor of (5) and (7) be a factor
of their difference, which is
2x-A ? (8)
19. Will the H. C. F. of (5) and (7) be a factor of (8) ?
20. Will the H. C. F. of (5) and (7) be the H. C. F.
of (7) and (8) ?
21. Will the H. C. F. of (7) and (8) be the H. C. F. of
x^-x-2 (9)
and x—2? (10)
22. Will the H. C. F. of (9) and (10) be the H. C. F.
of (5) and (7) ?
23. Will the H. C. F. of (9) and (10) be the H. C. F.
of (5) and (6) ?
24. Will the H. C. F. of (9) and (10) be the H. C. F.
of (1) and (2) ?
25. What is the H. C. F. of (9) and (10) ?
26. What is the H. C. F. of (1) and (2) ?
205. I^et us now look over and see how these various
expressions came about. Expression (1) was multiplied
by 3 to get expression (3). Where did this multiplier 3
come from ? We notice that if we divide (2) by (1) the
quotient is 3 and the remainder is 4x'^^4x—8, which is
numbered (4). This expression (4) contains a monomial
factor 4, which when rejected leaves x'^—x—2, the ex-
pression numbered (6). The work so far can be arranged
as follows :
;^3_^2__4 ) 3^34. ^2_4^_20 ( 3
Zx^-Zx'' -12
4 ) 4.^2 -4.y-- 8
x'^— x— 2 V
SECOND METHOD OF FINDING H. C. F. 22 1
We now deal with (5) and (1), or what is the same
thing, (5) and (6). These two expressions have the same
H. C. F. as (1) and (2) and are easier to deal with be-
cause the highest exponents in these two expressions are
2 and 3 instead of 3 and 3 as before. Thus our problem
is a little simpler than before.
Expression (6) was multiplied by x to produce expres-
sion (7). But where did this multiplier x come from?
Notice that if we divide expression (o) by expression (6)
the quotient is x and the remainder is "Ix—A:, which is
the expression numbered (8). This expression (8) con-
tains the monomial factor 2, which when rejected leaves
.r— 2, the expression numbered (10). This second part
of the work can be arranged as follows :
x'^—x—^ ) x'^—x'^ —4 ( X
x^—x'^ — 2x
2y2^^
x-'Z
We now deal with (9) and (10) instead of (5) and (6).
These two expressions, (9) and (10), have the same
H. C. F. as (5) and (6) and are easier to deal with be-
cause in these the highest exponents are 1 and 2 respec-
tively instead of 2 and 3 as before. We can even find the
H. C. F. of these expressions, (9) and (10), by the
method previously used, or we can keep on with the
process so far employed and divide expression (10) by
expression (9). This third part of the work can be
arranged as follows :
x-2 ) x''- x-2 ) x+1
x^--2x
x-2
x-2
As this division is exact, the H. C. F. of (9) and (10)
is x-2.
222 HARDER FACTORS, MULTIPLES, ETC.
206. Thus it appears that the process of finding the
H. C. F. of two expressions, when they cannot be readilj^
factored, is to divide one expression by the other, the divisor
by the remai7ider, the last divisor by the last reniai7ider, and
so on until there is 7io remainder. The Last divisor is the
H. C. F.
It must be remembered that this process is not to be
employed until all the monomial factors are first taken
out of the given expression, so that- the H. C. F. of the
remaining portions of the given expressions contains no
monomial factors. This being done, we may, at any
stage of the process just described, remove from any
dividend or divisor any monomial factor we please ; and
it will usually be best to remove them whenever we can.
All the work of the preceding example may be arranged
as follov/s :
x^-x'^—^ ) 3;^^+ x2_4^__20 ( 3
^ x^-Zx'' -12
4 ) 4;^2_4-^;_ 8
xi- x— 2
x'^—x—^ ) x^—x'^ —4 ( X
x^—x'^—2x
2 ) 2.r-4
x-2
x-2 ) x^- x-2 ( x+l
x^-—2x
x-2
Ivet us find the H. C. F. of
4.x^-^x^-2ix-9
and Sx^-2x^-5Sx-S9.
SECOND METHOD OF FINDING H. C. F. 223
As there are no monomial factors, we may arrange the
work thus :
4x''-Sx^-2ix-9 ) Sx^-2x''-5Sx-S9 ( 2
Sx^-(3x''-4Sx-18
4:x'^-5x-21 ) 4^3_3^2_24;ir-9 ( x
2;»;2- 3;»;-9
2;ir2-3;r-9 ) 4x'^-bx-21 ( 2
4.r2^6;tr-18
^- 3
x-^ ) 2:r2-3;r-9 ( 2;t:+3
2x^—^x
Zx—9
Zx-9
207. It will sometimes happen that the numerical
coefficients are such that the first term of the expression
used for a dividend is not exactly divisible by the first
term of the divisor, in which case we multiply the divi-
dend by the smallest number that will make the division
exact. This peculiarity is illustrated by the following
example :
Find the H. C. F. of
3^3-4^-4-3^-2
and 2a^ — Za^-{-a''--\-a—l.
Here we use the first expression for a divisor and the
second for a dividend, and evidently the first term, 2a^,
of the dividend is not exactly divisible by Za^, the first
term of the divisor, so we multiply the dividend by 3;
and the work may be arranged thus :
224 HARDER FACTORS, MULTIPLES, ETC.
3^
3a3_4a2+3a-2)6«4_9^3_|_3^2_^3^_3 ^2a
— a^ — 8<^^ + 7« — 3
In a case like this, where the first term of the remain-
der has a minus sign, we change all the signs before
using it as a divisor. TRis is equivalent to multiplying
by —1. Making the change, we continue as follows :
a^-\-Sa''-7a + S^Sa^- 4a'' -h Sa— 2(3
3^^+ 9^^-21^+ 9
-lSa''+24:a-n
Here again the first term of the remainder having a
minus sign, we change all the signs of the remainder
before using it as a divisor ; but even then, the first term
of the expression we are to use as a divisor not being
divisible by ISa'^, we multiply the dividend (which was
the divisor in the operation just performed) by 13, so that
the division will be exact, and continue as follows :
a^-j- 3«2_ 7a-\- 3
13
13a2_24«-fll ) 13^34-39^2_ 91^-1.39 ( a+i
13^3-24^2+ 11^
63a 2 -102^4-39
52a''- 96^ + 44
11^2- 6a- 5
Again, when we use this remainder for divisor and this
divisor for dividend, the first term of the new dividend
is not exactly divisible by the first term of the new
divisor, so we multiply again by such a number that the
division will be exact and proceed as follows :
SECOND METHOD OF FINDING H. C. F. 225
11
11^2__e^_5 ) 143«2_264^ + 121 ( 13
143^-- ISa- 65
-186a + 186
Before using this remainder as a divisor we will take
out the factor —186, leaving a—1, and then proceed as
follows :
a-1 ) 11^2- 6a-5 ( ll« + 5
11^2-11^
5a— 5
5«— 5
Since this division is exact, we conclude that «— 1 is
the H. C. F. of the expressions we started with. This
is an unusually hard example, and the student who can
follow this will not find any trouble with any example
given.
Examples.
Find the H. C. F. of the following expressions :
27. 5x'--{-4x—l and 20.:r2 4-2U-— 5.
28. 2jt:a-4;i;2-13;t:-7 and 6x^-llx'--S1x-20.
29. x^-i-4x^-5x-20 and.r3 + 6ji:2-5j»;-30.
30. 2;«;3__8^2 4.8^ and Sx^-Qx^-dx-' + lSx.
31. 3a2-22a-15 and 5a*+a^-'54a^-\-lSa.
32. x^—2x'^—x-r2 and x^—Qx'^-j-llx—Q.
33. Sxjy{x^-x''-\-x-{-d) and4j^/2(;t;4_^^3_3^2_^_^2).
34. x^-2x'^-\-2x-l andx^-Sx^-\-2x^--^x-l.
35. x^ —Sxy^ —Sy^ and x^—Ax^jz-^dj/^.
36. x^-Sx+Sandx^-{-Sx^+x-^S.
226 HARDER FACTORS, MULTIPLES, ETC.
37. x^-\-10x''-{-S3x+SQ and x^+dx^-{-2Sx+15.
38. x'<'-x''-4:X+4.and2x^-x^-6x+S.
39. 2x^—5x'^—x+6 and 4x^-x^ — llx-Q.
40. a^+bab-^W" and ^s4-4«'^+5a^2_p23^
41. 4:fn^—Sm-\-l and 8/;z^ + ;;e— 1.
42. 2«3 4.^2_2^_6 and Qa^-a'^-Ua + S.
43. 8«3-8«2_4^_3 and 2«4 + 3«3_3^2_7^_3^
44. x^—x'^—x—land 2x^+x'^—2x+l.
45. 2jt:^^+;t:2 +2^-12 and 2jr3-7;t:2 4-14^-12.
46. «4_|.67^2^66 and «4 4-2^''^+2«2+2«+l.
47. 28^2 + 37^-21 and 35«2+62«-33.
48. 7;;^^-13w2 + 34;;^-72and 7m'^-6m'^-\-S5m-S6.
49. 2x^-Sax^- —7a^x-{-4:a^ and 6x^ —lax'^ —Aa^x+Sa^
50. 2;»;3_^5:r2_)/+2^^2_^3 ^nd dx^+2x''y+xy-{-2j\
51 2;*;3-6-r2-2^+6 and 3;i;4-9;t:24-6.
52. x^-x^—7x'^-\-x-{-6 and ;t-*+^'*-7^2_^+6.
53. 3^=^-4jt:2-;t;-14and Qx^-lW-lOx-^l,
54. ^5_^3_^^_i and Jt-7— ;t:6— ;t:4 + l.
55. 10;t-=^+25^.r2— 5^3 and 4x--{-9ax'^—2a'-x—a^.
56. 3;»;4-8;r-V + 5jt:2y2_2.rjj/3 and 9x^ + 2x''y-\-y\
208. When we wish to find the H. C. F. of more than
two expressions, we first find the H. C. F. of any two of
them and then find the H. C. F. of this result and the
third expression, and so on until all the expressions are
used. The final result is the H. C. F. of all the given
expressions.
FIRST METHOD OF FINDING L. C. M. 22/
Examples.
Find the H. C. F. of the following expressions :
57. 2x'' + Sx-5, Sx^-x-2, and 2x''-{-x-S.
58. a^-Sa-2, 2a^-\-Sa^-l, and a^-\-l.
59. nCa^-d^), 10(«6-/^«), and S(iaH-ab^).
60. x''-x-12, x''-j-(jx + 8, and x^-4x^-x-{-4.
61. x^—6x^-^nx—Q, x^-\-4x'^-{-x—(), x^ — Zx-k-2.
62. x^-lx''^-\4x-%, x^-^x''-\-hx-\-\2, ;f2-5j»;H-4.
63. x^-\-2x''-—x—2, x'^+x^—x—l, and x^—x^.
64. .^-*+^J^^, x"'v+y^, and x'^-{-x'^y'^ + v^.
65. ;r»+3A-2-jr-3, ;»;5-^-^ and ji;2_3^+2.
67. x3 4.2;c2— ;»;— 2, ;i:'+;»:4, and x'^-\-Ax-{-'6.
68. ;r2-2;»r--3, ;»;2-7^+12, and x^+x''-%x-^.
69. ;i:-^ + 3;»r2_^_3^ ;»;4_2;r2-;t:+2, and x'^-x"^.
70. .r^+5;r-— ;tr— 5, ;»;^— ^tr^, and x^^x^+x—1.
EXERCISE 97.*
L. C. M. OF Expressions that Can be Factored
209. The lowest common multiple of two or more ex-
pressions has already been defined (see Art. S^) and a
* The order of subjects given in this and the following six exercises is the
one usually given in text books on Algebra, but some authors, notably Hall
and Knight of England, prefer to take up the subjects after H.C.F. in the fol-
lowing order : I, Fractions Reduced to Lowest Terms ; II, Multiplication of
Fractions ; III, Division of Fractions ; IV, I^. C. M.; V, Addition of Fractions ;
VI, Subtraction of Fractions. This order has the advantage of wjzw.^the H.C.F.
immediately after it is treated, and also using the L.C.M. immediately after it
is treated. At the option 01 the teacher the order here spoken ol may be sub-
stituted for the order gfiven in the text.
228 HARDER FACTORS, MULTIPLES, ETC,
method of finding the L. C. M. of monomials has been
given. We now take up the case of the L. C. M. oi
polynomials. First we will consider the case oi ex-
pressions which can be readih^ lactored. The method ol
finding the ly. C. M. in this case is exactly like that given
for monomials, viz.:
Resolve each expression iJito its prime factors a7id form a
jyroduct in which each of thetn occurs as inany times as ii
occurs in that 07ie of the given expressio7is in zvhich it occurs
the greatest number of times.
Examples.
Find the L. C. M. of the following expressions :
1. x^ — 1 and jr^ — 1.
2. x^ — 1, x"^ — !, and x—1.
3. ji-2 + 7jf+12, x^'+Sx+Id, and x^'-j-Sx-j-G.
4. x'^-x-(^, x'^+x-'l, and x'^-'ix—VZ.
^ x-ij^x-AI, x2-lLr-f 30, and .^2+2:^-35.
b. Aabia'^-Zab+lb'') and ba\a'' -\-ab-W).
7. x'^y—xy'^, Zx^x—yY, and 4y(x—y)^,
8. x—y, x-\-y, x'^—y^, and x'^—y'^.
9. a—b, a + b, a'^ — b'^, and a'^-{-b^.
10. x^—4x'^-\-i^x, x'^+x-' — VIx'-, x''+Sx^ — 4x^.
11. 20(.r'-^-l), 24(jr2-x-2), and 16(.r2+:r-2).
12. x'^ — 7x-^6, x'^ — ox—6, and x^ — 1.
13. 2{x-yy, 8(x-j/), 3(;r+^), and GCr^+j^^).
14. x'^-\'7x-}-Q, x^--^Qx—7, and x^ — 6x—7.
15. .:r2-;i;-42, x''-9x + U, and j»;2+4j»;-12.
SECOND METHOD OF FINDING L. C. M. 229
i6. 5(«2_2«^), 10(«/^+2<^2-) and 15(aH^-U^),
17. x^-5x-{-6, .r^-S, and x^-21x''.
18. x''+x-2, x^-\-S, and x^-x\
19. (2;;24-2)2, (w + l)^ 5;« + 5, and ;^^2_i^
20. (^— 1)^ (2;tr— 2)2, and ;i:*-2;»;2 + l.
EXERCISE 98
L. C. M. OF Expressions Not Easily Factored.
210. When we wish to find the L. C. M. of two ex-
pressions not easily factored, we first find the H. C. F. oi
the two expressions by the method of exercise 96. This
H. C. F. is of course 07ie factor of each of the two given
expressions, and the other factor is obtained by dividing
each expression in turn by the H. C. F. In this way we
factor each of the given expressions, and then, when the
factors are known, we proceed as in the previous exercise
to find the L. C. M. of the two expressions.
Let us find the L. C. M. of
6:r3-lLr-'r+2y and ^x^ —llxy'^ —'^y^ .
Wq Jirsi find the H. C. F. of these two expressions as
follows :
6;t:3-lLrV + 2y^
3 ■
9ji;3— 22;9/2_s^3 ) iSx''-ZZx''y -f 6^ ( 2
18^^ _44^^2_ie>/g
— llj ) -SSx'^y-hUxy'' + 22y^
3:^2 — 4xy — 2y'^
Sx^—4xy'-2y'^ ) 9x^ —22xy''-Sy^ ( Sx+4y
9x'^—12x'^y— 6xy'^
12x^y — 16xy'^ — 8y*
12x^y—Uxy'^—Sy^
230 HARDER FACTORS, MULTIPLES, ETC.
From this work we see that Sx'^—4xj/—2y^ is the
H. C. F. of the two given expressions, and if we divide
each expression in turn by this H. C. F. we obtain the
other factors as follows :
Sx''-ixy-2y ) Qx^-llx'^y H-2j/» ( 2x-y
— Sx'^y-\-4xy'^ -^2y^
— Zx'^y-i-4xy'^-^2y^
Sx^—4xy—2y'^ ) 9x^ —22xy'^—Sy^ ( Sx+4y
9x^—12x'^y— Qxy'^
+ 12xy—16xy'^—Sy^
■i-12x^y—26xy'^—Sy^
From the first of these two divisions we have
Qx^ -llx''y-\-2y^ = (Sx'' -4xy-2y'')(2^-j').
and from the second of these divisions we have
9x^-22xy''-8y^ = (Sx^-4xy—2y'')(Sx-\-4y).
Now we have the two given expressions factored, and
from these factors we can readily write down the L<. C. M.
of the two given expressions. Plainly, this I^. C. M. is
{Zx''-4xy-2y'''){2x-y')(^Zx-^4y).
Examples.
Find the L. C. M. of the following expressions :
1. x^-\-8x''-^V^x-\-\2 and Jt;^4-7:r2+7jr— 15.
2. .;»;-^ + 8;i;2 + 19.r + 12 2.n^ x^ -{■Zx'^—4x—\2.
3. j»;3 +70.-2+7.^-15 andj»;3 + 3.r2-4.r-12.
4. 5.r24-ll..;t+2 and 15.;i:4+48ji;3+9ji:2.
5. 4.r'^-10j»r2+4.r4-2and Zx^-2x^-Zx-V2.
FRACTIONS REDUCED TO LOWEST TERMS. 23 1
6. (yx^'-^-llxy-^Ay^- and 4x^-Sxy-5y^.
7. x^+x^—6x-hSandx^ — Sx'^-i-Sx—l.
8. x^+xy'^-i-2j'^ and x^-\-x'^y-\-4y^.
9. x' + 2x^+x-''+8x''-i-Wx+Sandx'-4x^+x'^-4.
10. x^ — ix'^+x^—A and x^ +2x'^—ox—12.
211. If we wish the L. C. M. of more than two ex-
pressions, we first find the L. C. M. of any two of them
and then the L- C. M. of this result and the third ex-
pression, and so on until all the expressions are used ;
the result is the L,, C. M. of all the given expressions.
Examples.
Find the L,. C. M. of the following expressions :
11. 2;»;2-f2;t:— 1, 3.;<;'^— 4r-f 1, and 2x^—Sx-\-l.
12. x^+2x-+9, x^-Sx-{-S, and x^-Sx+1.
13. .r2_2;t_2, x^-4x~+S, and x^ -Sx'^ -\-2.
14. 9x^-j-2x-^l, 3;r3— 8.:r2 + l, and x'^—Sx-\-l.
15. x'^—Sx-\-2, A-3— 6.t:2 + lU— 6, and x^—5x-{-6.
EXERCISE 99.
Fractions Reduced to Lowest Terms.
212. The general properties of fractions have already-
been given, and all the operations connected with frac-
tions have been given in the case that both numerator
and denominator are monomials. It only remains now
to apply the general properties and methods to fractions
in which the numerator and denominator may be poly-
nomials as well as monomials. The first thing v.e have
232 HARDER FACTORS, MULTIPLES, ETC.
to consider is the reduction of fractions to their lowest
terms. This is done exactly as before, b}^ dividing both
numerator and denominator by the H. C. F. of the
numerator and denominator.
Examples.
Reduce the following fractions to their lowest terms :
2a''-^al?-P 4^4 4- 11^- + 25
x"^ -\- ax -\- ex + ac ^a^ -\-2a'^ — 15a— 6
x'^ -]- dx -j- ex + dc .7^^ — 4^2—21^ + 12'
(Sx-2yy -{2x-i-2j 'y ^-"^ +6^- + 11^ + 6
£3__39^70 36^^- 18 ^2 + 1
• x'^-dx-lO' ^ ' 30a=^-19«2^r
{a-^by-i^c+dy ^-5-10^2 _^ 26^-8
7- 7-T-?r^ — 7irT^T^' 17.
{a + cy — {b+dy' '' ^3-9^2^23^-12'
3.^=^-6^1:2 +.r-2 ^ 8r^-10^2_i6^_3
18.
x'^-1x-\-^ ' 6^4_22,-3_^31^2_23^_7
x^+x^xj-B 2a^-9a'"-14^ + 3
^' x^-x-Q ' ^^' Sa-'~Ua^-da-i-2'
x^+a^ l + 2«— 3^2
10. -TT^^ .— TT. 20.
x'^-\-2ax+a^' ' l—Sa—2a^+4:a^'
ADDITION OF FRACTIONS. 233
EXERCISE 100.
Addition of Fractions.
213. In exercise 08 we considered the subject of the
addition of fractions in which the numerator and denom-
inator were each monomials. We now extend the subject
there treated to the case in which the numerator or de-
nominator or both are polynomials. Of course the method
is the same here as in exercise 58, viz. : first, reduce the
fractions to a common de^iominator, and then add the
numerators. Here, as before, we take the L. C. M. of
the given denominators for a common denominator.
Examples.
Add the following fractions :
I. — -, — and —zrz. — . 3- ^ and
6 12 ^ x—\ X-—1
4a— d , ba—d w^ , nr
2. — r-^ and ., „ . 4. 7 — — ^o and
Sa-4 5a-6 ^ 7a-S
5. -TTX' -:ri:T' ^^^
a-\-d' a-b' a'^-b'^'
6. „ . ,^ and
x'' + 2x x''-2x
x—y x-\-y x--\-y^
„ u-\-v-\-w u-\-v-{-w . 1
8. ; , , and — —
n-\-v u—v U- —
n r . \
9. — ; — , , and —:^. — ^.
' n-\-r 71— r w-fr-
a-^b a—b . ab
10. „ . ,^ , -r^ T^, and
^24.^2' a-i-b"'' «=^-f/^»*
234
HARDER FACTORS, MULTIPLES, ETC.
II
12
13
14
15
16
18
19
20
and
x(x—a)
x(x—by
x+y , 1
■^ , and
x'^-y''
1
a + b-\-c a-^b
x+1 i
x—y
and
a-\-c
and
3.^4-4
^2+5^+6' (ji: + 2)2' — (^ + 3)2*
2x+5 , 2jc-f7
and
y . 1
X
, and
;r ^ ;r2+^-2
7 and
^"— ^
x-\-a x-\-b
x^ -\-a
x'^+a^x'^+a^' x^-\-ax-\-a'-'
x"^ -\- ax -j- at
x-\-a
and
x—a
x^— ax-{-a^
be b'^ — ac
be
ae
and
1
ab
ab
1 1 , 1
EXERCISE 101
Subtraction of Fractions.
214. In exercise 59 we considered the subject of
subtraction of fractions in which both numerator and
denominator were monomials, and it only remains now
to take up fractions in which numerator or denominator
or both are polynomials. The method is of course the
same as in exercise 59, viz.: reduee the fraetioiis to a
common denominator and the7i subtract the mtmerator oj
the subtrahend from, that of the 7ninucnd.
SUBTRACTION OF FRACTIONS. 235
Examples.
2;t:+l , 3.r+2
1. From — ^ — take — 7 — .
3 4
^ ax-i-d . cx-^b
2. From take .
c a
3. From ^^ take ^^.
x—y x-\-y
4. From — ^ — ~. take ., .
5. From .. ^^ take - „-, — — — ;?.
6. From „ . r — r-77 take
^y take ^>t:^'
r'^ -\-y^ x^ —y'
7- i^^^o"^ -;:3^-;;3 take ^,_ .
o ^ x'^+xy-\-y'^ , jr^ — j/^
8. From ^ -^V - take ^ ^r.
' x'^+xy •\-y^ , jr^— .rr + y^
9 From — r - take ^'-L^-.
10. From 7 TT re take
(^x-aXa-d) {x-b){a-by
II. From —X——, — — r^ — : — 7 take
x'^-\-(a-\-b)x-\-ab x'^-\-(a-^c)x+ac
In the following examples, perform the additions and
subtractions indicated and express the result as a single
fraction in its lowest terms :
1.1 1
12.
13.
x+\ ' x-\-2 x-^^'
x+\ , x-^2 x-^B
Cr+2)(;^+3) ' (x+lXv+S) ix-\-lXx-f-2y
^6 HARDER FACTORS, MULTIPLES, ETC.
1 2 3_
Zb 5c^
( a b c\ fa 6b bc\
'5- b+s-lj + U-T-e-j-
^ a-\-b ^ b—a Aab
i6. ,4-
17-
a — b ' a + b a'^ — b'^'
2x-\- l 3x + 2 Ax-\-Z
{x-Vt^x"^^) (;t--2)(x^ (x-Z)(^x~^y
a a-\-\ a a"^ — a-\-l
lo. '--{■
19-
a — 1 a'^ -\-l a-\-l a'^ — 1
3<2— 6 4«— 5 a—1
-7« + 12~a2_8^_15— ^2_9^^20'
2^4-1 3^ + 2 2_ 3
EXERCISE 102.
Multiplication of Fractions.
215. In exercise 60 we considered the multiplication
of fractions in which both numerator and denominator
were monomials, and it only remains here to extend the
operation to the case where the numerator or denominator
or both are polynomials. Of course the method used
here is the same as in exercise 60, viz. : multiply ah the
numerators together fof" a 7iew riuTnerator and all the ae-
nominators together Jor a 7iew denominator. The result
should be reduced to its lowest terms if it is not in its
lowest terms already.
216. Instead of actually performing the multiplications
it will frequently be best to iiidicate them by using paren-
theses, for sometimes in the result the numerator and
MULTIPLICATION OF FRACTIONS. 237
denominator will contain a common factor, which can be
struck out and thus save the trouble of multiplying by
these factors. For example, if we wish the product of
a-^x b , c—x
—7--, — — , and — ; — ,
we can write the product thus :
d(a-{-x')(c—x)
dic-i-x)(ia+xy
Here the numerator and denominator contain the com-
mon factor d(a-{-x'), which being rejected from both
numerator and denominator leaves the result simply
c—x
c+x'
Examples.
Find the product of the following fractions :
b-j-c , d—c a—b ^ a"" -\-ab-\-b'^
1. — — and . 3. -.and -^^ — , , ,„ .
x-\-c x—c c—d c--\- cd-\-d^
x-\-\ ^x'^-x^-X a'^b'^ + ^ab _, 2a-f 1
2. — — - and —2 T-:r. 4- —j-^ — r~ ^^^ , , o -
x^-\-xy-\-y'^ , x"^ — xv-\-y^
5. 2—^- ^^^ Tl^~'
x^—y x^-\-y^
^ «2_i21 ^ a + 2
6. — 5 — — and — —- : .
«2_4 a-f-11
7- 4:r2~'9 2jr2 + llj»;-f5*
_ Ux-—7x , x'^-^2x
^' T2:^^M^24^ ^^^ "2:^3r-
Sa'^—4ab Uja—b) a-j-b
^' l{a + b) ' 3^M^3^' '^b'
10. -a — FT-T-r-r^, , and
«2_2«^-f^2' ^ > "- ^_|_^'
238 HARDER FACTORS, MULTIPLES, ETC.
"• -TTTi. -T^-Zh^ and
12. , 5, and
1 + ^ ' a — <2^' 1 — a
a^—x^ ^2 — ^2
^3- -^-j — ^ and — — ^ ; — -.
a^-\-x^ a- — 2ax-{-x^
^^' 2ab ' a^^b^-' Aa{a-\-b) *
a^-4^^ ab + W
^^' a^a + ^b-) a(^a-2by
^ a^ — la'^x'^-Vx^ ^ a'^+x'^
10. -—-- — and .,
a'^x-i'ax-^ a^—x-'
«^+3«-/^ + 3<2<^2_j_^3 a—b , a
^7. TV— t;.^ 5 ,0 , ,i > and
^2_^2 ' a'^-\-aV a + b'
o x^'-dx+'IO x--lSx-{-42 , ^2
lo. ^^ — ^r , ~ — , and ^.
x^—bx x^—Qx x—1
x^-xy+r^ £±?:, and to^.
x^—y^ x—y x^-\-y^
20. T— , ^ — - — , and
•2 — 9 ' jr2-4^+3'
EXERCISE 103.
Division of Fractions.
217. In exercise 61 we considered the subject O- the
division of fractions in which both numerator and denom-
inator were monomials, and we now have to extend the
subject to the case where the numerator or denominator
or both are polynomials. The method used here is of
course the same as in exercise 61, viz.: invert the terms
of the divisor and proceed as i7i multiplicatiori.
DIVISION OF FRACTIONS. 239
Examples.
1. Divide by .
a—x X
_. ., x- — lox+-^'l . x—7
2. Divide :; — by — ^-.
x'—ox x^
x'-dx+20. x-'-hx
3- Divide — ^^z:^— by -:2Zrr3^4:42-
4. Divide -^T— — z by -ly — ^ ;-— -^.
5. Divide ;^-^7;7^Yo ^^' :^+6]^T^-
6. Divide ^^3-^^^^-^ by ^3-^.
^. ., «-— 8a — 4, a^— 16
7. Divide — ^^ — 3 by — ;; .
o X.... x''-V^ax-'^-\-?>a''^x^a^ . x'^^-a'^
8. Divide w~7-.^, — r-Uy " by
9. Divide 1-- by ^^
10. Divide r — ' by
11. Divide --3^^- by ^^-^.
. .^ Cr+1)- ^ (^-1)'
12. Divide -^^ by -^^^.
13. Divide ^^-, by --.
14. Divide - ^.+2a+r-:r«^~ ^^ -^^F+r-
240 HARDER FACTORS, MULTIPLES, ETC.
1—^2 --y (l-\-a)
1'
1 6. Divide 7, — J7, by —=—.
a-' — b- a-\-b
^. ., a-' — b''' -\-a — b . a — b
17. Divide — — ^,-— --7 by — -,.
a'' — b--\-a-\-ba-\-b
18. Divide ^ ^^-^ r,- by =^— — .
19. Divide— ^^-^,— by— ^^^.
20. Divide — by
X — 1 \-\-ax'
EXERCISE 104.
Miscellaneous Fractions.
218. We may take an integral expression (/. e., a
monomial or polynomial in which there is no fraction
involved,) along with one or more fractions, giving us a
form partly integral and partly fractional. Such expres-
sions are sometimes called Mixed Expressions or num-
bers. Mixed expressions maj^ always be reduced to the
form of fractions by writing the integral part in the form
of a fraction with a denominator 1 , and then performing
the indicated operations as before explained. For ex-
ample, suppose we wish to express
in the form of a fraction. We write the expression thus :
-^1 J^yl ^y
+
X"'
MISCELLANEOUS FRACTIONS. 241
Reducing to a common denominator and adding in the
usual way, we get
x*—y^+xy
X'—jy^
This process may be called Reducing Mixed Expres-
sions to Fractions.
Examples.
Reduce the following mixed expressions to fractions :
X
I.
X
2.
x' + l+l-
3-
x—a
4.
^ X y
5.
a c
6. x^+x+1
7. a-^-d+c^
1
a-^b+c
b^
0. x'^—ax+a'^-^
9. (x-i-ay-h
10. d:;c4-^
1
x-\-a
1
ax— by
219. When the given mixed expression has only one
xy
fraction, as in the example x--hy'^-\ — tt^ — =, we reduced
^ -^ x-—y^
to a common denominator by multiplying numerator and
denominator of — y^hy x'^—y'^, the denominator of the
fractional part of the given expression. This amounts to
multiplying the integral part by the given denominator
and acjding the namerator exactly as in Arithmetic.
Since the fraction was produced from the mixed ex-
pression by multiplying the denominator by the integral
part and adding the numerator, it follows that to reverse
this process and go back from the fraction to the mixed
16
242 HARDER FACTORS, MULTIPLES, ETC.
expression we would divide the riumerator by the denom-
inator as far as possible and ivrite the quotient for the inte-
gral part and the 7'emainder over the denominator for the
. fractional part, which again is exactly as in Arithmetic.
Examples.
Change the following fractions to mixed expressions :
"• x'-^y ' ^^' iO "•
x--hax-^a'^ \2a'^J^Aa—bc
"• ,.^a • ^4. 4^ .
x'^-^a'^x'^-^a^-^x + a
15.
16.
X'-^ax-j-a'^
x^ -hSax'' -j~Sa'^x-\-a^ -ha^
x-j-a
x-—y"+z'^ ax + by+c
10. . 20. .
x-^y x+y
220. By combinations or repeated applications of the
preceding processes we are able to deal with more com-
plicated cases than have yet been given. For example,
let us take the fraction
x-\-a x—a
X — a x-j-a
x-\-a x—a
x—a x-\-a
The numerator =^-'?_£---'^=^f±^^f=f2!.
X—a x-\-a x^—a^
^ (^x"- -V2ax^a'^')-{x''-1ax-\-a'^)
x-'-a"-'
4 ax
MISCELLANEOUS FRACTIONS. 243
The denominator = 1 ; — ,
x—a x-\-a
{_x-^ay-{-{x-ay
x^—a^
(x^+2ax-\-a^)-{-{x'^-2ax-\-a^')
x'^^a^
x'-a''
Therefore the original fraction
^ 4ax , 2(x^--a' -)
x^—a^ ' x'^—a"^
__ 4ax . x'^—a'^
'^x'-a' ^ 2{x''-j-a'y
Aax 2a X
This final result is much simpler than the fraction we
started with, so this kind of work may be called Sim-
plifying Complex Fractions or Expressions Involving
Fractions.
Examples.
Simplify the following expressions :
21. ± 24. £ £ ^.
x-^Z-V- a'^'-ab + b''
X a-^b
y X
X-\ q- 1— :j— — 1— :j
x—\ l-hx 1—x
244 HARDER FACTORS, MULTIPLES, ETC.
/. ^^\ /-. ab—b^-X a* a—b
^3_^^3 ^2_^^^_|.^2 ^2_|_^^_^^2
30. -^-imT-iTX-
or (^ + ^)^- (^+^)^- ^. (^-^)2_ (^_^)2
2 • {a-\-cy-{b-^dy {a-cy-ib-dy
33 p 1-U '-
^^' \n—r 71— s) 7i'- — 7i{r-\-s)-\-rs
f a^—b^ a^^-b^ \ 4ab
s=[G«)'-(H)']*[e-^)'+(H']
38. i-2^- 40.
a — o
i-i
1 — ^
39- j^^ 41-
'' x'^ + a" ■ Va
x)
-. ' + \
«2 a;i:^;i:2
1
^ .(^-J^)^-
-Axy
1 1-1.1
2 '
MISCELLANEOUS FRACTIONS. 245
(aj' — bx) ^ -f {ax -\- by) ^
1
43. 1
;ir-l +
1+ "
Fractions like this are called Continued Fractions.
To simplify a continued fraction, begin at the lowest part
and proceed upward step by step as follows :
■*"4r^~ 4-x ~4-Jc'
Hence the original fraction may be written
But
4 — a;
1 4-aj
4 4
4-x
Hence the original fraction may be written
1
1 4- a;
L — x 4a;— 4 4-4— ^ 3aj
But x-l-f-^= ^ =— •
1 4
Hence the original fraction =.t-:=^7—.
ox oa?
4~
X
44. 1 45. — ^— 46. -— T
246 HARDER FACTORS, MULTIPLES, ETC.
j,r^ o a a a^
4u. -_ X 7 H- -s — 7^-.
a-^ ^ a—o a^
72
MISCELLANEOUS FRACTIONS. 247
But we may also multiply these expressions together the
same as integral expressions were multiplied, as follows:
2 ^3^4
a 1
2 3
«3 ^2^
a"^ a
1
G" 9
"12
a^ a a
1
T"^8 9"
"12
The terms of this result may be combined after reduc-
ing to a common denominator, 72. Comb-ning terms,
a^ a_a 1 _ 18^''-+9^-8a-G ^ 18^=^+^-6
4"^8 9 12"" 72 ~ 72
which is the same as was reached by the other process.
222. As another illustration let us divide
^^_13a2 X a__\
6 30 "^ 4 ^ 3 2
The work may be arranged as follows:
3 2>/ G 3G ■*"4V2 3 2
6 4"
9 ^4
G^4
G^4
248 HARDER FACTORS, MULTIPLES, ETC.
The terms of this quotient may be combined, giving
6 '
which is the same result as would be obtained by re-
ducing both dividend and divisor to a single fraction and
proceeding by the method already given for dividing one
fraction by another.
Examples.
By the method used in these two illustrations work the
following examples :
51. Multiply .r^H — by ;t:
52. Multiply ~+^+l by |+i
53. Multiply — +^— . by — -f •
a ab a b
54. Divide -^--^+-+1-- by 2-3.
/I 1\2 1 1
55. Divide (-+^) -lby--f--l.
56. Divide -X ^hy -r
lb a^ 4 a
57. Divide «3_fL-|-^_____ by a+g-
27 3
58. Divide 8a'^-f- 3 by 1a^ — and multiply the result
1
bya+-.
a
59. Multiply a;2 _ by .r-f- and divide the result
X X
, 1
by X
X
CHAPTER XIV.
QUADRATIC EQUATIONS.
EXERCISE 105.
Preliminary Tones.
223. The Degree of a Monomial with respect to any
letter or letters it may contain is the sum of the exponents
of the letters named; unity being always understood
where no exponent is written. Thus, hab'^x^y^ is of the
first degree with respect to a, of the second degree with
respect to b, of the third degree with respect to x, of the
fourth degree with respect to y, of the seventh degree
with respect to x and y, of the seventh degree with
respect to a, b, and y, etc.
1. What is the degree of ^a'^b^x'^y^ with respect to a ?
What with respect to .^? What with respect to _y ?
What with respect to a, b and x> What with respect to
x andjj/?
2. What is the degree of oa'^x^y^ with respect to.r?
What with respect to^ ? What with respect to x and y ?
What with respect to a and x ? What with respect to
a, X and r?
224. When the degree of a monomial is spoken of
without speciiying the letters with respect to which the
degree is taken, it is usually understood to mean the de-
gree with respect to all the letters it contains, and is then
equal to the number of literal prime factors, or what is
the same thing, the sum of all the exponents of the let-
ters in the expression.
2 50 QUADRATIC EQUATIONS.
225. The Degree of a Polynomial with respect to any
letter or letters it may cojitain is the degree of that one of
its terms whose degree with respect to the specified letters
is highest. Thus,
a^x'^ -\-abc^x-\-e'^x^y^
is of the second degree with respect to a, because the first
term is of the second degree respect to a, and neither of
the other terms are of so high degree with respect to a.
The same expression is of the fourth degree with re-
spect to Xy because the third term is of the fourth degree
with respect to x and neither of the other terms are of so
high degree with respect to x.
3. What is the degree ax''-y-\-bxy'^-\-x'^y'^ with respect
to ^ ? What with respect to _>/ ? What with respect to
x and y ? What with respect to « ? What with respect
to a and b ?
4. What is the degree of x'^y-\-xy^ -\-x^ with respect
to ;r ? What with respect to ^ ? What with respect to
x and y ?
Ans. 5, for the degree with respect to x alone is 5, and the term
that determines the degree has no y, so it leaves the degree 5.
5. What is the degree oi x'^ax'^y-Vbxy'^ -\-aby'''' with re-
spect to ;tr ? What with respect tojK? What with respect
to X andjj/?
6. What is the degree of a'^ bx ■\- b''" xy^ -{- cxy"^ with re-
spect to « ? What with respect to ;i: ? What with respect
to J? What with respect a and jr? What with respect
to a, b, Cy X and y.
226. When the degree of a polynomial is spoken of
without specifying the letters with respect to which the de-
gree is taken, it is usually understood to mean the degree
PRELIMINARY TOPICS. 2$ I
with respect to all the letters it contains, and is then
equal to the number of literal prime factors in that term
which contains the greatest number of such literal prime
factors,
227. The Degree of an Equation is its degree with
respect io the 2inknown numbers or quantities, i. e., it is the
degree of that one of its terms whose degree with respect
to the unknown number is the highest.
Remember that the last letters of the alphabet are used to stand
for unknown numbers.
7. What is the degree of /2=4? Oi ax'^-\-bxy
^cy^==ci} Of ax'' + dx-{-c=0}
228. A Quadratic Equation is only another name
for an equation of the second degree. In this chapter we
deal only with equations with 07ie unknown number.
229. Quadratic Equations are divided into two classes:
Pure or Incomplete and Affected or Complete.
230. A Pure or Incomplete quadratic equation is one
which contains the second but not the first power of the
unknown number, as ojir- = 12 and — = — =2.
5
231. An Affected or Complete quadratic equation is
one which contains both the second and first powers of
X X
the unknown number, as ox'^ + 4x=Sd and —-4-— =3.
o 4
232. A Root of an equation is any number which
substituted for the unknown number will satisfy the
equation, z. e., will cause the equation to be true. For
example, 2 is a root of the equation x'^-\-x=Qi, for if 2 be
252 QUADRATIC EQUATIONS.
written in place of x we get 2^+2=6, or 4 + 2=G, which
is true. Again, —3 is also a root of x'^-\-x=G, for if — 3
be written in place of .^twe get (—3)2—3=6, or 9—3=6,
which is true.
Although we deal in this chapter only with equations
of the second degree, still this definition of root will hold
good for an equation of any degree whatever, but it must
be understood that the word can be used only with refer-
ence to an equation of one unknown number.
The student must not confuse the word root as here used with the
square root or cube root or some other root of expressions. See
Art. 151.
233. The Solution of an equation is the process by
which the roots are found.
EXERCISE lOG.
Pure Quadratic Equations.
234. If x^=4 we know that x must be some number
which raised to the second power will give 4. Now,
there are two such numbers, -f 2 and —2. Therefore,
.r=2 or — 2. Either of these numbers will satisfy the
given equation, /. e., will render the given equation true.
Also, if x'^ = d we know that .r is a number which raised
to the second power will give 9. Either +3 or —3 will
satisfy the given equation. Therefore, x=S or —3. So
whatever number is placed equal to x'^ there are two
numbers of opposite signs, but otherwise alike, which
will satisfy the equation, or in other words, there are two
values of .^• of opposite signs but otherwise just alike.
235. To solve a pure quadratic equation we reduce it
so that all the unknown terms are on one side of the
equation and all the known terms on the other side, then
PURE QUADRATIC EQUATIONS. 253
we divide by the coefficient of the square of the unknown
quantity, and lastly extract the square root of each side
of the equation.
236. In solving a pure quadratic equation we usually
write both values of the unknown quantity at once, as in
the equation ;r^=4 after extracting the square root of
each side we would write ;tr=±2, using the double sign
db to show that either -f 2 or —2 will satisfy the given
equation x^=4.
The student may think that we should write the double sign on
doi/i sides of the equation instead of on one side only, thus, ±jc= ±2',
but this would evidently mean x— 2 or x=—2 or —x=2 or —x= — 2.
The third of these equations is really the same as the second and the
fourth is really the same as the first, so that we really get no more
values by writing ±x= ±2 than we do by writing x= ±2.
237. Whenever we extract the square root of each
side of an equation we should write the double sign =b
on one side of the equation obtained.
Examples.
Solve the following equations :
1. x^ + S=4.
2. jir2+3=7.
3. (;c24-l)-f-(^2+2) + (^2^3)=30G.
4. (>2-4)-f(j>;-+2)=^2_|_i4^
5. 2(ji:2 + l) + 3(x2 + J) = 5.i-.
6. 3(x2-l) + 4Gr2-2)=o;tr2-f39.
7. 3j«;''-4=28+^^
8. 5x2-7=293 + 2jt:2.
g. (.r2-l)-Cr2-2)-Cr2-3)=54-3;»:^
2 54 QUADRATIC EQUATIONS.
10. ^— + —77r-- + —r, — =25-x^.
O 10 o
2.4, ;»: 4
"• i^l + -5=^- '3. 4=--
15. 12.v2-75=0.
iG. 2Cr2-l)-3(;r2 + l)+9=0.
x^-\-l x'^—1 .^ o.
^7- -2 8 ^°=°-
XX x
19. (2;r+l)2=4xH-2.
20. 3;»:2-199=(;r+l)2-2.r.
238. The equation .r2=4 may be written in tlie form
;»;2_zj.— 0. Now, as the first member of this equation is
the difference of two squares, it may be factored, and
hence the equation may be written (.r— 2)(;f-f 2)=0.
239. The product of two or more factors is equal to
zero whenever any one of the factors is zero. Therefore
the equation (x— 2)(;t;+2) = can be satisfied in either
of two ways: first, when ;t'— 2=0, i. e.^ when :r=2, and
second, when .r + 2=0, i. e., when .r=-— 2.
As another example take the equation
5x2-9=2a-2 4-18.
Transpose all the terms to the first member and we g2t
5.r2^2.r2-9-18=0,
or 3a'2-27=0.
Dividing by 3, .;r2-9-=0.
Factoring, (jr— 3)(a'+3)=i0.
AFFECTED QUADRATICS. 255
This equation can be satisfied in either of two ways :
first, when x—S=0, i. e., when jr=o, and second, when
^+3=0, /. e., when :r=— 3.
240. By the method illustrated in these two examples
we get the same roots as would be obtained by the former
method. Thus we have another method of solving a
pure quadratic equation, viz.: Collect together into one
term all the unknown numbers and into another term all
the known numbers ; write these two terms on the same
side of the eq2iatio)i, maki^ig the other side zero; divide both
fnembers by the coefficient of the square of the unknown
number (remembering that when zero is divided by any-
thifig the quotient is still zero); factor the 7'esulling first
member; put each factor separately equal to zero^ and solve
the resulting simple equations.
Examples.
Solve the following equations by the method just
explained :
.21. .;c2-100=0. 25. (jir+2)2=4(;»r+5).
22. 4.r-^--100=0. 26. (.;ir+2)(.r+3)=5.r+42.
23. 5.^2=80. 27. .r2+;ir+l=jr4-101.
24. jr+-=— . 28. .;t:2-2.r-3=33-2.;ir.
' X X
29. (;e-+« + Z^)2 = 2(«4-^>r+2(«2^^2)^
30..(2.r+l)2=4a;-}-82.
EXERCISE 107.
Affected Quadratics.
241. If we have given the equation .^^=25 we solve
it by the preceding exercise, aad find x—±ih. Similarly,
if we have the equation (j;-l- 1)2=25, we find ;»:+l=±5.
256 QUADRATIC EQUATIONS.
If we take the upper s'gn we get x+l=^B, or x—i, and
if we take the lower sign we get x-\-l = —o, orjr=— 6.
Thus we find the roots of the equation (x-\-iy=25y or
what is the same, ;tr2 + 2;f+l = 25, or ^2 + 2;*:= 24.
242. Therefore, to solve x'^+2x=24:, we first add 1 to
each member to make the first member a perfect square,
and get ^2 + 2;t-+l=25 ; then we take the square root of
each member, and get ;r+l = d=5, whence a'=4 or —6.
243. Similarly, to solve x'^-\-Gx=7y we add to each
member such a number as will make the first mem-
ber a perfect square. Plainly, 9 is such a number.
Therefore jr^-f 6;r+9=16. Next, take the square root
of each member, and get ;i;+3=±4, whence x—1 or —7.
244. Similarly, to solve afty affected quadratic equa-
tion, we first reduce the equation to a form where the
terms containing x^ and x are in the first member and
the term not containing x is in the second member;
second, if the coefficient of x^ is not unity, divide each
member of the equation by that coefficient, so that the
coefficient of :i:^ shall be unity; third, add to each member
of the equation such a number as will make the first mem-
ber a perfect square, and then take the square root of each
member and solve the resulting simple equations.
245. Adding to a given expression a number that will
make the sum a perfect square is called Completing the
Square.
24G. When the coefficient of x"^ is unity v/hat number
is it that we must add to each member of an equation to
make the first member a perft^t square ? To answer this
let us see how a perfect square is produced. We know that
AFFECTED QUADRATICS. 257
and (x—a)'^=x^—2ax-\-a".
Notice here that whatever luiinber is represented by a
the third term is the square of one-half the coefficient
of Ji' : hence the number to be added to each member o:
the given equation is the square of one-half the coefficient
of X.
247. Hence, to solve any affected quadratic equation:
/. Reduce to a form wJiere boih x'^ and x are in the first
member and all terms not containing x are in the second
member.
II. If the coefficient of x^ is not already nnity, divide
each member of the equation by that coefficient, thus making
the coefficient of x- unify.
III. Com pie fe the square by adding to each member the
square of one- half the coefficient of x.
IV. Extract the square root of each member of the equa-
tion and solve the resulting simple equations.
Examples.
Solve the following equations :
1. jr2-f4.r=5. ii. x--^^x=-\o.
2. j»r2-|-G.r=lG. 12. 3a-2H-12x=-3G.
3. 2A-2 + 20jr=43. 13. 2.r2-f 10ji'=100.
4. x'^ + ox=l^. 14. x^—ax^Q^a"^.
5. x''-^bx=Z(j. 15. a"2-2^A^=8«^
G. 3.r'H-C-r=9. 16. 3A-2-12^.r==63«^
7. 4;»;2-4ji'=8. 17. 4;i'2--12^x=16a2.
8. x'^-lx=-(j. 18. 5;r2-25A'=-20.
9. ^2_10ji:=— 0. ig. x'^—x=2.
10. 2j;- — 15;t=50. 20. x'^+x=a''--\-a.
17
258 QUADRATIC EQUATIONS.
248. The method already given will enable us to
solve any affected quadratic equation that may be given,
but frequently it willoblige us to use fractions, and
imless the terms of the fractions are small numbers it
will be easier to complete the square by another method,
which we will now consider.
249. We know that
and {ax—b')-=a'^x''—2abx-\-b'^y
so that each of these two second members is a perfect
square. We therefore seek to reduce the given equation
so that the first member shall be in the form of one oi
these two second members.
250. Notice two things : first, that the coefficient of
x"^ is a perfect square, and second, that the third term
equals the square of the quotient obtained by dividing
the second term by tw'ce the square root of the first
term. Therefore, to reduce the first member of any
given quadratic equation to one of the two forms
a'^x''- + 2abx^b''-
or a''-x-—2abx-\-b''-,
I. Reduce the equation so ihat the terms containing x"^
and X shall be in the first member and all terms not contain-
ing X shall be in the second member.
II. Mnliiply each member of the equation by snch a mem-
ber as will make the coefficient 0/ x- some perfed square.
III. Add to each member the square of the quotieni
obtained by dividing the second term by twice the square
root of the first term.
The rest of the process of solution is like that already
given, viz.: take the square 7'oot of each member and solve
the resulting simple equations.
AFFECTED QUADRATICS. 259
Let us solve by this method the equation
Transpose 2x and 11 and we get
Sx-'-}-5x=22.
Multiply each member by 3 or 12 or 27 or 3 times any
square number and the coeffirient of x- will be a perfect
square. Take the first of these multipliers and we get
9A-2-fl5j«r=6G.
Add to each member (V")^' or (f)-, and we get
ar2 + 15;i;+-2/=66 + -2/=lf«.
Take the sqaare root of each member and we get
3a-4-|=±-V-.
Hence, dx=G or —11, and x=2 or — -V'-
If we had multiplied by 12 instead of 3 we would have
obtained 30x2 + G0.r=2G4.
Therefore, 3G;i-2+60j»;+ 25 =204 + 25= 239.
Hence, G.r-f-5=±17.
Hence, G.r=12 or -22, and x=2 or — U.
Examples.
Solve the following equations :
21. 3ji:2+4x=7. 31. 3^-2— 24=G.r.
22. 3jt:2-fG.r=24. 32. 2.r2-22x=— GO.
23. 4j>;--5x=2G. 33. 2;i:2 + 10;i;=300.
24. 9.r2+G.v-48=0. 34. 3.r2-10ji;=200.
25. lSx^-—Sx—m=0. 35. 2;»;2_3^-^io4.
26. 5jt:--7;r=24. 36. 3.r2+7jt:-370=0.
27. 2.r2-35=3.r. 37. 4x^^7x-{-^=0.
23. 3;»;2_50=5jtr. 38. 5x''-},x—;'^=0.
29. i;r2-3.r+ij=0. 39- |-^;-t-r=— J?.
x"^ Sx , ^ f. x"^ X I ^
3°' T-T+^='^- 40. -ir-2+G=°-
260 QUADRATIC EQUATIONS.
251. Literal quadratic pq nations may be solved the
same as numerical ones. But, after the square is com-
pleted, the second member will be a literal instead of a
numerical expression, and hence the square root usually
cannot be taken, and so will have to be indicated. Thus,
to solve x'^-j-4ax+d=0, we proceed as follows :
Transpose d, x'^-\-4:ax=—b.
Add Aa- to each member.
Take the square root of each member,
x-]-2a=-^VAa'- — b.
Transpose 2a x= — 2<2d= V \a "- — b.
One value o{ x is —2a-\-V Aa'^ — b and the other value is
Examples.
Solve the following equations :
41. x''--\-2ax=b. 46. 2x''-—Q>ax—Ab=0.
42. x'^-\-Aax=b. 47. bx'^—lax-\-b=^0.
43. 2jt-2-f3^.r=4(^. 48. ax'^-{-bx+c=0.
44. x'^—bax=1b. 49. ax'^ — bx=c.
45. x'^ — (dax—ob=0. 50. ax"^ -\- a"^ x= a^ .
252. When all the terms of an affected quadratic
equation are transposed to the left member, making the
right member zero, the equation may be solved by fac-
toring if the resulting first member can readily be ex-
pressed as the product of two factors each of the first
degree with respect to the unknown numbers. Thus, to
solve the eqr»ition x'^ — bx=—Q, we transpose —6, and
obtain x'^—bx+Q)=0. By the method of exercise 88 we
find the factors of the left member to be x—2 and x—S.
Therefore the equation may be written (x— 2)(a-— 3)=0,
which is satisfied if ;»;=2 or if ;»r=3.
PROBLEMS. 261
253. This method of solving an affected quadratic
cannot be used to advantage unless the left member is
easily factored, but when easily factored this is probably
the easiest way to solve them.
Examples.
Solve by factoring the following equations .
51. x-'-i-dx+Q^O. 56. x''-5x=U.
52. x--\-llx=—^0. 57. jr2+j»;=80.
53. x''—x=Q. 58. A'2+jr=12.
54. .r2 + 7;i:=— 12. 59. :r--7:r+12=0.
55. x2-lljt;+30=0. Go. 2.r2-fGjtr=,i;2--3.r— 14.
61. 2x--\-Sx-{-4=x'^—Sx—l.
G2. x2 + 2jr+l = 6.r-fG.
63. 3;i:2 + 12;i:-10=2;t'2^2,r-31.
G4. x2 = G.r-rx 70. 2x--40ji-=4ji--240.
65. ;r2==-4.r + 21. 71. x^--(i0=900,
ar— lo0=:±30,
rc=lS0 or 120.
Each of these answers fulfills all the requirements of the problem,
and each is admissible.
4. One of two numbers is f the other one and the sum
of their squares is 20S. Find the two numbers.
5. The product of two numbers is 750 and the quotient
of the greater divided by the less is 3^. Find the numbers.
6. Divide the number 103 into two parts whose prod-
uct is 2400.
7. Divide the number GO into two such parts that the
quotient of the greater divided by the less may equal 1
more than twice the less.
8. A merchant bought a quantity of cloth for S120 ; if
he had bought 6 yards more for the same sum the price
per yard would have been $1 less. How many yards did
he buy, and what was the price per yard ?
g. A merchant sold some goods for $39, and in so doing
gained as much per cent, a-, the goods cost him. What
was the cost of the goods ?
10. Find a number such that 3 more than twice the
number multiplied by 3 less than twice the number may
give a product of 112.
11. A man traveled 105 miles, and then found if he
had gone 2 miles less per hour he would have been 6
hours longer on his journey. How many miles did he
travel per hour ?
266 QUADRATIC EQUATIONS.
12. A man bought two farms for 12800 each ; the
larger contained 10 acres more than the smaller, but he
paid $5 more per acre for the smaller than for the larger.
How many acres were there in each farm ?
13. A merchant sold two pieces of cloth which together
contained 40 yards, and received for each piece twice as
many cents per yard as there were yards in the piece.
For the smaller piece he received 2^- as much as for the
larger one. How many yards were there in each piece ?
14. The distance around a rectangle is 200 feet, and
the area: is 1344 square feet. Find the length and breadth
of the rectangle.
15. The length of a rectangle is 10 feet more than the
breadth, and the area is GOO square feet. Find the length
and breadth of the rectangle.
16. There are three lines, the first two of which are
f of the third, and the sum of the squares described on
these three lines is 33 square feet. Find the lengths of
these lines.
17. A flower bed 9 feet long and 6 feet wide has a path
around it whose area is equal to the area of the bed itself.
What is the width of the path ?
18. A number consists of two digits one of which is
the square of the other, and if 54 be added to the number
the digits are reversed in order. What is the number?
19. Find a number such that if it be added to 94 and
again subtracted from 94 the product of the sum and
dijfference thus obtained shall be 8512.
20. A man bought a number of horses for $10000 ;
each cost 4 times as manj^ dollars as there were horses.
How many horses did he buy ?
PROBLEMS. 267
21. Find a number whose square is greater than the
number itself by 306.
22. Find a number such that if its third part be multi-
phed by its fourth part and to the product 5 times the
number be added the sum exceeds 200 by as much as
the number required is less than 280.
23. A man bought a horse and sold it again for $119,
by which means he gained as many per cent, as the horse
cost him dollars. How many dollars did the horse cost
him ?
24. Find the fortunes of three persons, A, B, and C,
from the following data : For every $5 which A has
B has $9 and C has $10 ; mo'eover, if we multiply A's
money by B's, and B's money by C's, and add both
products to the united fortunes of all three we shall get
$8832. How much money has each ?
25. The combined area of two squares is 9G2 square
feet, and a side of one square is 18 feet longer than a
side of the other. What is the size of each square ?
26. A square field conta'ns a number of square rods
equal to 2G0 more than 32 times its perimeter. How
many rods in one side of the square ?
27. F!nd two numbers whose sum is 21 and the sum
of whose squares is 225.
28. Find two numbers whose product is 480 and the
difference of whose squares is 3536.
29. What is the price of oranges when 10 more for
$1.20 lowers the price 1 cent each ?
30. The sum of tht* ages of a father and son is 80 years,
and -J of the product of their ages in years exceeds 5 times
the father's age by 200 years. What is the age of each ?
268 QUADRATIC EQUATIONS.
31. A certain number is the product of three consecu-
tive whole numbers, and if it is divided by each one of
these three factors in turn the sum of the three quotients
thus obtained is 7G7. What is the number?
32. The sum of the squares of three consecutive odd
numbers is 83. What are the numbers ?
33. The sum of the squares of four consecutive even
numbers is 120. What are the numbers?
34. Divide the number 18 into two such parts that
their product shall exceed 30 times their difference by 20.
35. In a bag which contains coins of silver and gold
each silver coin is worth as many cents as there are gold
coins, and each gold coin is worth as many dollars as
there are silver coins, and the whole is worth $525. How
many gold and how many silver coins in the bag?
36. A room whose length exceeds its breadth by 8 feet
is covered with matting 4 feet wide, and the number of
yards in length of the matting exceeds f the number of
feet in breadth of the room by 20. Find the length and
breadth of the room.
37. There are two numbers whose difference is 7, and
half their product, plus 30, is equal to the square of the
smaller number. What are the numbers ?
38. A and B start together on a journey of 36 miles.
A travels 1 mile per hour faster than B and arrives 3
hours before him. Find the rate of each.
39. Two workmen, A and B, are engaged to work at
different wages. A works a certain number of days and
receives $27, and B, who works 1 day less than A, re-
ceives $34. If A had worked 2 days more and B 2 days
less, they would have received equal amounts. Find the
number oi days each worked.
EQUATIONS SOLVED LIKE QUADRATICS. 269
EXERCISE 109.
Equations Solved Like Quadratics.
256. It is bej'ond the scope of this book to treat equa-
tions of a higher degree than the second expressed in
general form, but there are a few equations of higher
degree which may be solved by the methods of this
chapter.
257. We have had such equations as x^ — lo.i-f 3G=0,
and have seen that such equat'ons are easily solved. Now
it is plain that we couhi use some other symbol in place
of X to designate an unknown number. Thus we might
have an equation where r"' stands in place of Jtr, and then
of course jK"' would stand in place of jt-, and the equation
would be
jj/^-13)/2-f3G=0,
from which, by solving in the usual way, regarding jv^
temporarily as the unknown number, we obtain
j,2=4or9.
Hence jj/=±2 or ±3.
In a similar manner we could treat equations in v/hich
more complex expressions stand in place of x'^- and x in
the equations before used, but whatever expression stands
in place of x the square of that expression must stand in
place of x'^ , else the equation cannot be solved by the
methods of this chapter.
Examples.
Solve the following equations :
1. .r4-29.r2-f 100=0. 3. y-17y^ + lG=0.
2. .r«-35a-5 + 210=0. 4. jj/+8l/J'+15=0.
5. (.r-f3)6-28(;«r-f3)« + 27=0.
:270 QUADRATIC EQUATIONS.
6. ^2+_L_^2 + i_.
- ('+l)*-¥(-l)+'=»-
10. (;r2-5.r+G)'-^ = 14(;t'2_5.^^G)_24=0.
EXERCISE 110.
Theory of Quadratic Equations.
253. The methods a'ready explained will enable us
to so ve a quadratic equation in which letters are used
to stand for the known numbers in the equation, but 01
course we nuiit not use the same letter to stand for a
known number that we use to stand for an unknown
number.
Let us solve the equation x"- -\-ax-\-b=^^.
Transposed, x'^-\-ax= — b.
•Complete square,
a , a"- a'^ a-— 41?
x-^-hax-^--=--—d=--~-
Extract square root, .r+o==i=A/ -^
Hence, x= - 1± ^^-"Z-l..
From this we see that one root ol the given equation
a , (a^-Tb
= ~2 + A/""4""'
.and the other root of the given equation
a la-—4d
THEORY OF QUADRATIC EQUATIONS. 2/1
259. It thus appears that there are two roots to a qiiad-
rat'c equation, but for certain values of a and b these two
roots are exactly the same. This will be the case when
«2_4^=0, for then the number under the radical sign
reduces to zero, and each root of the equation reduces to
—-5. In this case there is in reality only one value of ;r
that will satisfy the equation. Still, instead of saying
that there is only one root of the equation, we say that
there are two roots but that these two are cqtial to each
other. Of course this is only another way ot saying that
there is only one root, but the advantage of this mode of
expression will be apparent as we proceed.
260. Let us now find the sum of the roots of the
equation x''--\-ax-^b=^.
One root=-|+y^-j-^
a \~^-
other root=
sum=— flj
In the quadratic equation ;r-+a.r+^=0 the coelTicient
oi x"- is unity, but a and b stand for any numbers. Hence
in any quadratic equation where all the significant terms
are in the left member and the right member is 0, and
where the coefficient of x^ is unity, the sum of the roots is
equal to the coeffident of x with its sign changed.
261. Let us now find the product of the roots of the
equation x"^ -\- ax -\- b ^ ^ .
The product required may be expressed thus :
/ a \a'—\b\( a \a'^—\b\
\r^i'\-^r)\-T-\—A-)
2y2 QUADRATIC EQUATIONS.
Notice that this is the product of the sum and difierence
of two numbers, which, as we have already learned, is
equal to the difference of their squares. Therefore the
product of the two roots equals
a- /a'-—4d\ ,
Hence in any quadratic equation where all the signifi-
cant terms are in the left member and the right member
is 0, and where the coefficient o( x^ is unity, ^/le pi'oditc,
of the roots is equal to the tcmi not containiiig x.
262. The last two articles show just the relation be-
tween the known numbers in an equation and the roots
of the equation, and from the results reached we are
enabled to form an equation which shall have any desired
roots. For example, if we wish to form the equation
whose roots are 3 and 5, we know that the coefficient of
.r^ will be 1, the coefficient of x with its sign changed
will be the sum of the roots, /. ^, and the value — 2 of x goes
with the value 7 of ^.
273. If we now look to see just what was done in the
solution just given, we will observe that from the equa-
tion of \.\\Qjirst degree the value of one unknown number
was found in terms of the other unknown number, and
this value was substituted in the equation of the second
degree. This method is capable of solving a7iy case where
one equation is of the first and one of the second degree.
Hence, to find the values of two unknown numbers
from two equations, one of which is of the first and the
other of the second degree, we proceed as follows :
SIMPLE AND QUADRATIC EQUATIONS. 279
From the equation of the first degree find the value of
either unknown number in terms of the other unknown
number; substitute the value so fou7id in place of this un-
hiown 7iumber in the equatio?i of the second degi'ee, and
solve the resulting quadratic equation. This gives two values
for one of the unknowji numbers, which values substitute in
tur7i in the equation of the first degree, and thus find the
values of the other unknowii 7iumber.
Examples.
Solve the following sets of equations :
3
6.
( ;«;2-|-3>/2 = 52.
'' \ 2x -iy =1. 9- I
{ ;»;2+.rj/-f^2 = Cl. ^^ (a'2-j/2 = -0.
( x^y=l. ' \ X —y = — 1.
• \zx -4y =10. "• \x -\-y =2.
j x+y=G. ( 5x^—Sxy-2y'^ = 12.
^• I xy=6. "• j 2x-y==S.
j x'^ + 3xy—y^=o. ( 2x'^—Sxy—4y'^ = lG.
'• I Sx-y=^l. ^^' \ 2x-4>/=4.
I ;r2-j-_>/2 + 2x=31. j x'^— xy=S20
2y-\-x= 7.
M.j
I x-y-^l==0. ^' \ Sx +12y= 12.
xy-{-x+y=:S2. j xy-\-x+y=ld.
ixy+x+y^S2. j
2x-\-oy=lS.
iSx^-2xy+y^ = 81. i x'-+y-'-\-x+y=152.
I 10x-2y=54:. \ 2x-y=2,
■H
6x'--y^-+Sx-2y=^-22D.
4x—y=5.
„ j 4j»;2_4>/2+.r-j/=82.
^''' i 4x-^4y=40,
280 HIGHER SIMULTANEOUS EQUATIONS.
ix'' + bxy-Q^y-lx-Sy^ IG.
( 4:X^ + bxy-
'9- 1 x-2yJ.
20. I
10A-2-9y2+5;»r+4;/=-5G9.
DX—dy= — lo.
i x"^ +2xy+3y^ -\-x^d(j.
^^' \ 2x+Sy=12.
22.
23.
24. J
25. ^
f.4.-
1
:2.
26. ^
>/ X
U-:r=l.
^A--2j/=— G.
fl---
27. .
^-fji^+-4--=G
[;i:+ 5^=51.
3.r-j=3.
23. .
x-y^ll
.^-J=l.
;i; 7 ;ir ^
=4.
29. .
^ 14
L 5^+ 5^=50.
. 2;t:-5y^ 7.
;ir
1- -^ -
2x-\
:-';=-3.
30. .
lx-y=
x-y
= 1.
x^-
_y-^
EXERCISE 112.
Two Quadratic Equations.
274. The Secojid Case is that in which each equation
is of the second degree and possesses this pecuHarity,
viz : that all the terms which contain unknown numbers
are of the second degree with respect to those numbers.
TWO QUADRATIC EQUATIONS. 28 1
lyCt US solve the equations
;i:2 4-^2^25 (1)
jr2+jry— j/2^19 (2)
Substitute vx in place of jk, and we obtain
^2_|_^2^2^20 (3)
x'^-^vx'^—v'^x^ = ld (4)
25
From (3) we obtain x"^—- ;• (5)
19
From (4) we obtain ^^ = -^ (6)
Therefore, : = r-
Clearing of fractions, we get
25(H-f-z'-) = 19(l-hu2),
25 + 252;- 25z;2 = i94-i9e,2^
44z/2_25z/=6,
7, 2 5 1-4 1
7, — (5 f^r 16
t/=4 or —
2
TT-
Substituting each of these values in turn for v in (5),
x^±
And since y==vx, we get
;\:=±4 or ±—7-^.
jK=±3or=F-7^.
V o
The=ie values go together in pairs as indicated in the fol-
lowing scheme, where any value given for x goes with
that value of y which is dlreclly zinderneath the value
of X considered :
^ 1 11 11
;r=4 or —4 or — >-- or
Q 2 2
_y=3 or —3 or or -r-^.
"^ v'5 1/5
282 HIGHER SIMULTANEOUS EQUATIONS.
275. If we look carefully at the solution just given
we will observe that vx was substituted for jj/ in each ol
the given equations, and from each of the resulting equa-
tions the value of x"^ was found in term-; of v. The.-e two
values of x'^ were placed equal to each other, and the re-
sulting equation solved to find the value of v.. This value
of V was then substituted in the first of the equations
which expressed the value of x" in terms of v, and thence
the value of x"^ and then the value of x was determined.
The value found for x together with the value found for
V served to determine j/.
276. The method used in solving the set of equations
just given is capable of solving a?iy two equations pos-
sessing the peculiarity mentioned in the last article.
Hence the method may be stated as follows :
Substitute vx iti place of y i7t both equations, ard from
each of the resulting equations find the value oj x'^ in terms
of V. Place these two values of x"^ equal to each other and
solve the resulting quadratic equation to determine v. Sub-
stitute the value found for v in one of the first two equations
where v was used, and thus determine the value of x. Hav-
ing both x and v, determine y^ which is the pivduct of
X and V.
Examples,
Solve the following sets of equations :
j ,^2+^ = 25. ( '■Ix'^—Zy'^^-xy^Z,
^' \xy=^Vl. ^' \Zxy-\-x'-^—'l.
j x'^—Zxy-Yy'^-=-h, ( \x''- — hxy-Vy'^=l,
^' \x''-'lxy=S. ^' (x2-5>/2 = _l.
( 2x''-Zxy= 0. ( ,r2-f 2;ri/+j/2 = 3G.
^' \Zxy-2y'' = lO. ' \ x^-2xy+y''=0.
MISCELLANEOUS EQUATIONS. 28^
I 3;»ry=60. ^^' \ x'^-Zxy- y''=Z.
i ;t2-4>/2=0. j 3x2-xj/=133.
■^ ^y
90. '• ( ;r>/-_y2 = _4
II.
12
^^^=^^^' 18 i-"-3-:>'-J- = -29.
• I x""- xy+y'' = lS.
C2x^-3xy-\-4y''=4.S.
x+by= — I —
r3
4
8
Jtr2
-7.=
=.-^'
j"*
ig.
'
1 2
3
-2i
u^
-y =
^JK
+ 3j/-
-2a'=
36
>
£0.
.
r> 4
( x+y x—y ^lO
13. ^ ^—y x+y~~ 3 * 20. ^
I \ y'^ 24
EXERCISE 113.
Miscellaneous Equations.
277. Since no general directions can be laid down to
cover all solvable cases of simultaneous quadratic equa-
tions, a few examples will be worked out as samples.
The student is advised to stud}' carefully the following
solutions, and try to see, if possible, what suggested the
various steps taken in the solution.
284 HIGHER SIMULTANEOUS EQUATIONS.
(:r2+y=20.
(1)
^- 1 ;, +^ = G.
(2)
Square (2),
x^
+ 2r;/-»-jj/«=33
(3)
Pit(l)
x^
+;'« = 20
Subtract,
2xy =16
(4)
Subtract (4) frbm (1),
X*
—2xy+y^= 4
Extract square root,
x-y=±2
But (2)
x-{-y= 6
Therefore, adding,
2x-H or 4
Hence,
:r=4or 2
Substitute in (2),
y=^2 or 4
( x-hy=0.
(1)
2. < J,
\ xj'=b.
(2)
Square (1),
x»
+ 2r-y+y^=Zfj
Multiply (2) by 4,
Arxy =20
Subtract,
x^
-2xy-^y^-U
Extract square root,
x-y=±4:
But (2>
x-\-y= 6
Therefore, adding,
2x=10or 2
Hence,
.r=5 or 1
Substitute in (1),
j=lor5
ANOTHER SOLUTION.
hety—vx in both equations;
. then x + vx=z&
(3)
Z'X2=5
(4)
(i—x
From (3),
V —
X
From (4),
5
G— x_ 5
Hence. "IT^lc^
Multiply each member by x*. 6a;— .r^ =5
a-«-6r=— 5
;p«_t;x^9 = 4
Ar-:;=±2
iC=o or I
^=lor5
(r'+y=f?L (1)
^- I ;.T=10. (2)
Add 2 times (2) to (1), a:»+2 vv-h;'»=r)4
Extract square root, »-!->'= ±8 (3)
MISCET-LANEOUS EQUATIONS. 28$
Subtract 2 times (2) from (1),
x- — 2xy-}-y^=:4.
Extract square root,
x-y-±2
But '8)
X <-j— i8
Therefore, adding,
2x=10 or H or — 10 or — C
Hence,
x=h or 3 or -5 or —3
Substituting in (2),
^ = 3 or 5 or -3 or --5
0)
(2)
From (2),
y=(S-x
Cube.
y
=' = 21fi-108r+|8^2-x3
Substitute in (1),
21G
-l(l8.r+ 18^8 = 72
Transpose 21(),
18^*-l()8r=-lf4
Divide by 18,
;r«-6r=-8
Complete square.
*«-Cjr+9=l
Extract square root,
^-3=±1
Hence,
x=4 or 2
and
y-2or 4:
ANOTHER SOLUTION.
Divide (1) by (2),
x«— xy+y» = }2
(3)
Square (2),
x» + 2xy+yi = 'dQ
(*)
Subtract (H) from (4),
'6^y^2i
or
xy=S
(5)
Subtract (5) from (3),
x*—2xy-\-y^ = A
Extract square root,
jc-y=±2
But (2)
x+y=Q
Adding,
2j: - 8 or 4
Hence,
x=4 or 2
and
;K=-2or4
( x^-+xy=S6.
5- \y-hxj'=U,
(1)
(2)
Adding (1) and (2),
x^ + 2xy+y^=49
Extract square root.
x-y= ± 7
(3)
Divide (1) by (3>,
^=±5
Divide (2) by (3),
y=±2
ANOTHER SOLUTION.
Factor each left mem
iber,
x{x-i-y)=35
(".)
v(Xi-7)=14
(4)
Divide (3) by (4),
X 3 .') 5
3^~y4~2
Multiply by y.
x=^v
:86 HIGHER SIMULTANEOUS EQUATIONS.
Substitute in (4),
iy^ = U
Hence,
y^=4
Therefore,
y=±2
and
x=±5
j x+y=(i.
(l)
(2)
Raise (2) to the fourth power,
x^+ix^y^r6x^j
r-i+Uy^+y^ = U9G
(3)
But (I)
x*-i-y^= 272
Subtract, 4:X^y + Bx^jS ^ 4^^3 _ 1Q24
(4)
Factor (4), xy 2\r
2 + 3x74 273) = 512
.(5)
Square (2) and multiply result
by 2 ry,
Ay,2x'
^+Axy + 2y^}=12ry
(6)
Take (5) from (G),
(x/)2^72r;/-
513
(7)
Transpose,
(xy)^-l2xy=-bl2
"Complete square, (-*"/)"•
-72x)/f 129(5 = 784
Extract square root,
xy-'SG= ±28
Hence,
xy = ij4: or i
i
(8)
Square (2\
x^ + 2xy+y^~=3fi
Multiply (8) by 4,
4xy-2J6 or
32
(-9)
Subtract,
x^—2xy f j- = — 220
or 4
Reject —220 because we cannot extract the square root of —220 and
extract the square root of the resulting equations.
x-y=±2
But (2) x-\-y= 6
Adding, 2x=8 or 4
Hence, x—4: or 2
.and j=2or4
Examples.
Solve the following se^s of equations :
x=^-jv^ = 63. j x4-y = G00. .
7
' X —y = o
1
ixy+xy^^^l2.
^' I x-\-xy'' = lQ.
PROBLEMS.
2^
-I
■'1
■H
-I
■H
■•■I
I
20.
21. ^
22.
x'^ -j-j/"^ -{-x—j'=78.
xy='n.
^=i— j,/3 = 19(.r-j^).
xy'^-x''y=Z2\.
2ji:y-5A-+6>/=33.
y'^+xy=\'{x-\-y).
xy+x=20.
xy—y=12.
x^y 20*
xy—x'^—y- = —dl.
x'+y^-+xy=223.
23
■I
24. I
-I
26. I
27. )
28.1
29
loji;>'=2520.
x--x-hlS8=2xy+Sy
y''-\-7y-Sx=21S.
;r— _>/=2.
^J/
-1
30.
31
xy'-{-x-y-x'y'=-^^:^-g
x-'-y^ = 21d.
x-+xy-i-y'-=d3.
x^-{-y^ = lSS.
xy(,x+y) = 70.
(^'4-j'-)Gr+j) = 272.
x^-{-y''-{-x-i-y=42.
9
.-2^2^
1.
X-
.rH2.r-2^)=i|^.
EXERCISE 114.
Problems.
1. What two numbers are those whos^ product is 24
and whose sum added to the sum of their squares is 02?
2. Find two numbers such that the square of the
greater minus the square of the less may be 5G, and
the square of the less plus ^ of their product may be 40.
288 HIGHER SIMULTANEOUS EQUATIONS.
3. A number consists of two digits whose sum is 15.
If 31 be added to the product of the digits the digits will
be in the reverse order. What is the number ?
4. There is a number consisting of two digits, which
number divided by the sum of its digits gives a quotient
2 greater than the first digit. But if the digits be in the
reverse order and the resulting number be divided by a
number 1 greater than the sum of the digits, the quotient
so obtained is greater by 2 than the preceding quotient.
What is the number ?
5. Find two numbers sucl^ that their product added to
their sum gives 62, and their sum taken from the sum of
their squares leaves a remainder of 80.
6. A certain number consists of two digits of which the
digit in ten's place is 3 times the digit in unit's place,
and if 12 be subtracted from the number itself the re-
mainder will equal the square of the digit in ten's place.
Find the number.
7. The sum of two numbers is 37 and the ?um of their
squares is greater by 9 than twice their product. What
are the numbers ?
8. The sum of two numbers is 22 and the sum of their
cubes is 2926. What are the numbers ?
9. The sum of the squares of two numbers is 410. If
we diminish the greater by 4 and increase the less by 4
the sum of the squares of the two results is 394. What
are the two numbers ?
10. A man bought some horses for $1250. If he had
bought 3 more and paid $25 less for each horse, they
would have cost him $1300. How many horses did he
buy, and at what price ?
PROBLEMS. 289
11. A stock dealer bought some horses and cows for
$1 1600. The number of horses bought was equal to the
number of dollars paid for each horse, and the number of
cows bought was equal to the number of dollars for each
cow. Had the number of horses bought been equal to
the number dollars paid for each cow and the number of
cows bought been equal to the number of dollars paid for
each horse, the stock would have cost him $8000. How
many horses and cows did he buy and what did he pay
for each ?
12. A lady bought 55 yards of cloth in two pieces, pay-
ing $17 altogether; for the first piece she paid twice as
many cents per 5^ard as there were yards in the piece, and
for the second piece she paid one-half as many cents per
yard as there were yards in the piece. What was the
number of yards and the price per yard of each piece?
13. Find two numbers whose sum is 9 times their dif-
ference and whose product diminished by the greater is
equal to 12 times the greater divided by the less.
14. Two rectangles contain the same area, 480 square
yards. The difference of their lengths is 10 yards, and
of their breadths is 4 yards ; find their sides.
15. If a carriage wheel 14f feet in circumference take
1 second more to revolve, the rate of the carriage per
hour will be 2 J miles less. How fast is the carriage
traveling?
16. A cistern can be filled with water by two pipes.
By one of these pipes alone the cistern would be filled
2 hours sooner than by the other ; also, the cistern can
be filled by both pipes together in 1|- hours. Find the
time each pipe alone would take to fill the cistern.
19
290 HIGHER SIMULTANEOUS EQUATIONS.
17. A man bought a number of shares of $20 stock
when they were at a certain rate per cent, discount for
$1500, and afterwards when they were at the same rate
per cent, premium sold all but GO shares for $1000. How
many shares did he buy, and what did he give for each
of them?
18. The small wheel of a bicycle makes 135 revolutions
more than the large wheel in a distance of 2G0 yards ; if
the circumference of each were one foot more, the small
wheel would make 27 revolutions more than the large
wheel in a distance of 70 yards. Find the circumference
of each wheel.
ig. A sets off from London to York, and B at the same
time from York to London, and each travels uniformly.
A reaches York IQ hours and B reaches London 3G hours
after they have met on the road. Find in what time
each has performed the journey.
20. A man arrives at the railway station nearest his
home I7} hours before the time at which he had ordered
his carriage to meet him. He sets out at once to walk
at the rate of 4 miles per hour, and meeting his carriage
when it had traveled 8 miles, reaches home one hour
earlier than he had originally expected. How far is his
home from the station, and at what rate was his carriage
driven ?
CHAPTER XVI.
THEORY OF INDICES.
EXERCISE 115.
Meaning of FRACTIO^^AL Exponents.
278. The exponents which we have considered here-
tofore are defined by the following equation :
a"=aaaa . . . to 7i factors.
In other words, a" is an abbreviated way of writing the
product of n factors each equal to a.
279. It has been proved that exponents follow the
five laws expressed by the following examples and
formulas :
EXAMPLES. FORMULAS.
a^^ xa'^ = a^^ Art. 39, a" x «''=«"+'' A
a^^-^a''=a* Art. 54, a"-^a''=a"-''if7i>'-rB
(«ii)7=^77 Art. 134, («")"=«"" C
{abcy=a''b''c'^ Art. 135, {abcy=a"b''c'' D
We will find it convenient to re.^er to these formulas
as A, D, C Z>, and E, respectively, and we will speak
of them collectively as The Laws of Exponents or
Indices.
280. It is very plain that the exponent n must be a
positive whole number in order that «" may have a mean-
ing by the definition in Art. 278. If, therefore, we wish
to use symbols like a^ ox a ^ we must first find a mean-
ing for such expressions.
* This symbol stands for the words "is greater than."
292 THEORY OF INDICES.
281. It is usual in Algebra to define the symbols
which are to be used, and afterwards discover what laws
these symbols follow in algebraic operations. Thus, in
the first part of Algebra we defined positive integral ex-
ponents and subsequently proved that they follow the five
laws, A, By C D, and E. But an exception to this gen-
eral practice occurs in the case of fractional and negative
exponents. Here we first state some law which ive wish
fractional and negative exponents to follow and then seek
what mea7iing must be giveji to these exponents in conse-
quence of this law. Thus, we say: All fractional expo?ients
must follow law A; required tlie meaning of fractional ex-
poneiits. We will consider a few special cases at first.
282. Given that law A must hold for fractional ex-
ponents; required the meanhig of a'^.
Since law A must hold, we have
a-ia'^z=a'i i=a.
Thus a'^ must be such a number that if it be multiplied
by itself the result is a. By definition the square root of
a is such a number ; therefore a~^ must be equivalent to
the square root of a. Thus we say
1. /—
a- = V a.
283. Given that law A must hold for fractional cx-
ponents; required the meaning of a'^.
Since law A must hold, we have
JL i i 1 4.JL ■ 1
a'^a^a^^d^^'^^'^-=a.
But, by definition of a cube root,
V a \ a V a=a.
Therefore, a'^ must be equivalent to the cube root of a;
that is, ^ii= 1^^«.
MEANING OF FRACTIONAL EXPONENTS. 293
284. Given that law A must hold for fractional ex-
p07ients; required the vieajiing of a^.
Since law A must hold, we have
8 3 3 3 il.».3,3
3
Thus we see that a'^ is one of the four equal factors
which, when multiplied together, produce a"^ . Hence,
3
by the definition of a root, a'^ must be equivalent to the
fourth root of a^; that is,
a^=V~dK
285. Given that law A must hold for f-actional ex-
ponents; required the meaning of a''- .
Let r be any positive whole number. Since law A
must hold, we have
1 ! -^ . r , ? + -4.l4-, . .tor terms
a^a'^a^ . . . to r iactors=«'' '^ '- =a.
Thus we see that a^ is one of the r equal factors which,
when multiplied together, produce a. Hence, by the
definition of a root, w is eqiiivalent to the rth root of a;
that is, ar=:^i/a. [Ij
286. Given that law A must hold for fractional ex-
n
po?ients; required the meaning of a^.
It is here supposed that n and r stand for any positive
n
whole numbers, that is, that - is any positive fraction.
Since law A must hold, we have
'J ? '-? . r X. "+"+"+• . .tor terms „
fl'-a'-a- . . . to r factors=«'' '- »- =« .
Thus we see that a^ is one of the r equal factors which,
when multiplied together, produce a". Hence, by the
definition of a root, a^ is equivalent to the rth root of the
71 th power of a; that is,
a^=^^. [2]
294 THEORY OF INDICES.
287. Thus, by requiring that fractional exponents
shall follow law A, we have found that
A77y positive fractional exponent indicates a root of a
power, where the numerator shows the power and the de-
nominator the root.
288. We will now get at the meaning of a fractional
exponent by a little different process, but still by requir-
ing that such exponents shall follow law A.
From the meaning of a positive integral exponent
(Art. 278) we know
.1-. Ill
(a^y'=a>'a>-a'- . . . to n factors.
T, 1 yf l-fl + l-f ... to « terms
By law ^, =«r^r^»--r
Adding the fractions, =«'-.
Therefore we have shown that
i ?!
(«>)«=«'•, [3]
or, since a^^l^a, by [1], we have shown that
«"=(]/«)«; [3]
That is, a^ is cqinvalent to the n ih power of the rth root of a.
289. Thus we have shown that
Any positive fi^actional exp07ient indicates a power of a
root, where the nu^nerator shows the power and the denomi-
nator the root. ■
290. Comparing this wdth Art. 287, we see we have
n
found two inea7iings for a^ \ first, the rth root of the n th
power oi a ; second, the w th power of the rth root oi a.
From equations [2] and [3] we get this statement in the
form of an equation as follows :
a7-=v^/^=(^^)'V [4]
or, writing exactly the same equation, but using frac-
tional exponents instead of radical signs,
a"=(a'0^ = («b^ [4]
EXAMPLES. 295
n
While we have found tzvo 772eanings for d^ yet these two
meanings always give the same mtmerical results, as is
proved by equation [4]. Thus: 8^=l>'82 = (f/8 ;2,
81T=^''812 = (f/8i)2, etc. The order of the operations
is different, but the results are the same.
291. The above truth is very important and may be
stated in the following manner :
The rth root of the 71 th poiver of a number is equal to the
n th power of the r th root of that number.
EXERCISE 11c.
Examples.
Write each of the following sixteen expressions, using
fractional exponents in place of radical signs :
I. l/rt. 5. V~a}. 9. l^^^ 13. f/«— 5.
2. ]/?. 6. o/ay. 10. {fxy. 14. {C^^x^yy.
3. i^7\ 7. f/^. II. 1^?. 15. ^'^^^zHi
4. #^^. 8. if ay. 12. {\^xy. 16. i'\a+xy.
Find the numerical value of each of the following six-
teen expressions :
17. 4i 21. 6254-. 25. 8li 29. 25Gi
18. 273-. 22. Gli 26. I25I 30. 64^
19. oi 23. 21Gi 27. 32! 31. 5123-.
20. IGT. 24. IGt 28. 814-. 32. 1287-.
• Write each of the following expressions /;/ tzuo ways^
using radical signs i;istead of fractional exponents:
1 2 7 «
33- dS. 37. 71^. 41. ?-8. 45. «».
34. 15". 38. b^. 42. Jtv. 46. ^^i".
S 5 1 !£±_1
35. ;;z^+". 39. e^K 43. J'^ 47- -^ '•'•
35. x^. -~a'-t^ [5]
which is what was to be proved.
293. Having found a meaning for fractional exponents
by requiring that they follow law A, we will now prove
that they also follow laws B, C, D, and E.
294. To prove that fractional expone^its follow lata B.
Let n, r, s, and / be any positive whole numbers, so
7t S
that — and - are any positive fractions. We will prove
^'•-T- «'=«'• ', if ^>7.
n s tit sr
We know a'--7-<2^=«^/-=-«'^^, by Art. 292.
= («^0'"-^(«^)'"> by equation [4j.
= (^^0"'~", by law B for integral
indices, since nt and sr are whole numbers and nt^ sr
if '^>\.
r t
PROPERTIES OF FRACTIONAL EXPONENTS. 297
ftt—sr
=a-^, by Art. 289.
But this last fractional expouent is what we would get if
we subtract - from —
/ r
Therefore, a'--T-a'=a'-~f, if ^'>7, [6]
which is what was to be proved.
295. To prove that fractional expone^its follow law C.
Let «, r, s, and / be any positive whole numbers.
n us
Case I. We will prove («^) '=«'•".
n n n n
We know {wy^a^a'^a^ ... to ^ factors,
by definition of an integral index.
=.a»>---'°''^''''\ bylaws.
= «'-, by adding fractional indices.
Therefore, {aly=a'^, [7]
which is what was to be proved.
Case II. We will prove (a^)''=d:>',
n 1 tit 1
We know («')' = («^0'", by Art. 292.
= ([«-]"')'•.
by meaning of a fractional index.
=(1[''"3"!0'.
by law C for integral indices.
Dy taking the t th root of the / th power.
ft
by meaning of fractional index.
Therefore, {a*!y=a^t^ [8]
which is what was to be proved.
298 THEORY OF INDICES.
s ftr
Case III. We will prove (a^y^ast,
Wekuow («'0^"=[(«")0'.
by llie 111 calling of a fractional index.
= l^"'Y, by Case II.
tis
— a'-', by Case I.
Therefore, (a-y^^^ar^ , [9]
which is what was to be proved.
296. To prove that fractional exponents follow law D.
n ft n M
We are to prove (abcy^a^b^c^.
We know {abcy^-liabcyy ,
by meaning of a fractional index.
by law D for integral indices.
by law C, (Art. 295, Case I).
by law D for integral indices.
« n tt
=a~^b'-c^y
by taking the rth root of the rth power.
Therefore, {abcy=a~b'^c'^, [10]
which is what was to be proved.
297. To prove that fractiorial exponents follow law E,
We are to prove (j\ ''=—^-,
WeUnow g)"=[G)7
by the meaning of a fractional index.
EXAMPLES. 299
{?)'
by law B for integral indices.
« 1
by law C (Art. 295, Case I).
=[(3)T
by law B for integral indicCvS.
n
by taking rth root of rth power.
n
Therefore, (3'=-^ ["]
which is what was to be proved.
EXERCISE 118.
Examples.
Perform the indicated operat'ons in each of the fol-
lowing examples by means of the laws of fractional
exponents just proved :
I. a^Xd^.
2 «
2. a^Xa^'.
Jxa^=a^'^hhy A)=J.
3. a^Xd^. 7. d^Xd. II. a-l?^x2a^d'S.
1 . ,36
4.
x'^ X x^.
8.
111^X111^.
12.
3jf"^>* X 2.r^^T
5.
xixxi.
9-
.ix^i
13.
a^"Xa^".
6.
1 s
c^Xc^.
10.
<^^x/^i
14.
2 r
a" X d^".
300 THEORY OF INDICES.
15.
at^at.
aK
a^=z J~\hy B)=a^=>
a^(by Art. 292).
16.
dT^di.
i*.
r-d^=d^~^{hy B)z=i^^-
-^^=^^.
17.
a^-r-ai.
21. xi^x'^.
3w M
25. « '- -^flS'-.
i5.
/^t-^/^i
22. 6A't--3j»;2-.
26. 8a5/^^~4«2^i
19-
kUkk
23. 771^-— 771^.
27. 6x^ji^Sxijy^.
20.
24. 9a^-7-a^.
28. ahTr—a^b^r,
29.
(a^)i=rz~iV(by C),
30.
(di)i.
i^h
^-<^^'"(by C) = ^^'<^(by
Art. 292).
31.
(^*)i.
35. (J'¥.
39. (-^^")^^
32.
(«*)A
36. («*)i
40. [(^-)"]^.
33-
(.4)i
37. (^¥.
41. [(^'01".
34-
(/i^)*.
38. («tV)4.
St 3
42. (;ir6''«)^
43. {(^bcy.
44. (a4.r2jK«)^.
(ff^O*=«^<^^Aby -Oj.
'2 1 .S fi
45. (a3^2^T)¥.
(aMc-^)»=it(a^>^(/5^)^(^^;°(by Z>)=aM/^(by C).
46. (aH^)i. 50. (jcl 2^6)1 54. («l^f^^)i
47. («^/^^)V". 51. (2^.r^_>/t)i2. 55. (8«6^M)I
48. (aV^)i 52. (32jt:tj/f)t 56. {S6a^x"jy^y
49. {a^b^)'^. 53. (a4^.;»;^_>/i)i*. 57. (i^^^'^r^-^s.
EXAMPLES
•
^ (if-
^^(by C).
»- G)*^
- &■
- (5f
^ (7)'
-(f
- ii)'-
"■ iff-
67. (-2V
68. (a^+fl*4-l)(«^4-^-«^).
We arrange the work thus :
a^^a^ + a^
n^
..11
J
30]
69. (;r+jt'7j'7+j')(^— -^Ir^+J').
70. Ci:+2y^-f-3)/i)(;tr-2y^" + 3>/^).
71. («t4.«i^i+^l)(ai— /^i).
^ R S R
72. (x-^-hj^'^){x^—jy'^).
73. (-r^-^+^'^-l)(-^+.'r^+l)-
75. (_x^-xY^+/Xx^+xy^-hy^).
302 THEORY OF INDICES.
76. {(^—2aP^Zc^'){^aP—aP).
77. i^abi—Za^b^"-. ^
78. (xi—xj/i-\-x^jy—yi')-^(xi—yi).
We arrange the work as follows :
i I
It is just as important to keep dividend and divisor arranged
according to the powers of some letter in case the exponents are
fractional as in the case they are integral. The fractional and
integral exponents must take the order of their respective magnitudes.
79. {x^—x'^—Axi+Qx-2x^)-^{xi—Ax^-\-2').
80. {x-\)-^{x^-\).
Si. (J,_l)-^(_yi_l).
82. {a—b''-^-^{a^-\-aib^-^a^b + bi).
83. («i— 2^?tjt-t-f-ji:3)^(^T_2«lxi+x).
84. {x+y-]-z~ Zx^j^z^^) ^ {x^ -^y^ + z^).
85. {a^-b^—c^-\-2b^c^')-^{aJ+b^-c^i).
•86. (Sx^ + ^i —z + Q>x^y^z^)-^{2x'^ + y^ —z^^.
87. Find square root of x'^ + 2x'^-\-\.
1 11 1
88. Find square root of 4a'^—4.r 3 j/T-j-j/T.
89. Factor jr— 2.1-^1^^ +j'.
90. Find square root of Jt't—4.v^+4;r+2.r6—4.r^+.r'3".
ZERO AND NEGATIVE EXPONENTS. 303
EXERCISE 119.
Meaning of Zero and Negative Exponents.
298. If the product of two numbers is unity, either of
the numbers is called the Reciprocal of the other number.
Thu^, i is the reciprocal of 2, |- is the reciprocal off, etc.
In other words, the reciprocal of a number is 1 divided
by that number.
299. Given that negative exponents must follow law A;
required the 7neaning of a'"^ .
Since law^ must hold, we have
Therefore, by dividing both sides by a',
That is, a- 2 = —^.
a"
Hence, a~'^ is equivalent to the reciprocal of a^ .
300. Given that negative exponents must follow law A;
required the meaning of a~'^.
Since law A must hold, we have
aa~^=a^~^=-d^,
_^ rt^ 1
Therefore, c a== — =_-;
^ at
-% 1
That is, a 3=-^.
'a^
_i 2
Hence, a ^ is equivalent to the I'ecipi'ocal of d^ .
304 THEORY OF INDICES.
301. Given that negative exponents must follow law A;
requii'ed the meani7ig of a~".
Let n be any positive number, integral or fractional, so
that —71 stands for a negative whole number or fraction.
Since law A must hold, we have
Let r be taken greater than n. Then we know
a" '
a"
Therefore, a^'a'" = --•
Dividing both sides by a", we obtain
Hence, a~'*is equivalent to the reciprocal of a**,
302. Since n stands for any whole number or fraction
in the above work, we may say :
Any number with a negative exponent is equivalent to
the reciprocal of that number with the same exponent taken
positive.
303. Given that zero exponents imtst follow law A; re-
quired the 7neaning of a^ .
Since law A must hold, we have
r«/T = /,"+
=a'
Therefore, a^ = l.
That is, a^ is equivale7it to unity.
304. Since a in the above work stands for any nuniDer
whatever, positive or negative, integral or fractional, we
may say :
A7iy 7iumbcr with the expo7ient zero is eqiiivalent to unity.
Thus, 20 = 1, 30 = 1, 5« = 1 100 = 1, A'0 = 1, (xr)° = l,
(2« + 4.y2)o = 1, etc.
EXAMPLES. 305
305. It follows necessarily from Art. 301 that
, s.-t_ a"b-''c' _a"c'_ a" \ _ b-''d- '
"^ ^ "" d' fd' c-b'd' a-"b''c-^d'~a-"c-'
That is : Afiy factor 7nay be transferred frorn 07ie ierni
of a fraction to the other term provided the sign of the ex-
ponent of that factor be changed.
2-Kv b\r 4r^a-H'' S^bH"" ^
Thus, rr-7i:^ = 9T-' — i ="1 » ^^C-' ^^C-
oab 2U 8-4^2^-2 4i^^2
EXERCISE 120.
Examples.
Find the numerical value of earh of the following :
I.
20.
5.
10-^
9-
2(a + by
13.
1024"^
2.
2-1.
6.
1-^
10.
16-1
14.
512-i
3.
3-*.
7.
1-".
II.
G4-i
15-
G4-t
4.
(-2)-3.
1
2«>'
8.
20.
8a^2-\
12.
81-i
16.
26.
G25-i
17.
5
23.
5-2
IG-J
(-4)-=^"
3G-^'
50 •
18.
1
1-1'
21.
2-4
4-2-
24.
27-i
2-3--
27.
9-2
81-f
19.
2
0-'>*
22.
1-8
8-1-
25.
32-^
2"! '
28.
70
49- i'
Write each of the following expressions without using
zero or negative exponents :
29. x^. 32. oa-K 35. (x+j-y. 38. 2«^jir-2^~i
30. «-^ 33- G^2^-\ 36. A-O-fj'^ 39. a^b~ie-'d'K
^ ,._!
31. x2_y--. 34. Sa-H-^\S7. i-x)-K 40. (-«-)-^
23
3o6
41. ^4-- 45. ;:33. 49. [-) • 53-
42. -TT. 4e>. ^TT-r^. 50. -0. 54.
X'
in
THEORY OF
INDICES.
45. ^-3- 49.
©•■
4^- .A2.-3- 50.
1/0'
1
«~2
3«tri
b-''
c''
a
-bi
Zx-
-Vi'
' Za'
0^-2^-4
5a-
H-^c-^'
9a
^x~iy-^
43. -;^. 47. ;^pi-r^. 5i. -— j-- 55.
^ ^ y ba ^b
44. — ,-. 48. — — r. 52. z,— 56.
Write cacli of the following expressions in one line :
„ 4 ^ 3.r^j/-3 - x''y\a-by
c a u 2 r3^-2_^, -5
(«— Q ^<; '2
jy X X'
EXERCISE 121.
Properties of Negative Exponents.
306. We have required that negative exponents follow
law A in order to obtain a meaning for them. It i.^ pos-
sible to prove that negative exponents, having the
meaning thus found, also follow laws B, C, D, and B in
the operations of Algebra.
PROPERTIES OF NEGATIVE EXPONENTS. 307
307. To prove that negative expone7iis follow law D.
lyCt n and r stand for two positive numbers, integral or
fractional; then —71 and —rare any two negative numbers.
Case I. We will prove «" -7- «-''=«" "(-'').
We know a"-^a~''=^" X -^- >
a
by properties of fractions.
=^"x «-(""''),
by meaning of negative index.
=«""("''), by law A.
Therefore, a" -^rt-''= «"-("'). [13]
which is what was to be proved.
Of course —( — /') can be written ■\-r, and n—{—r) can be written
M + r. The form n — ( — /') is kept merely to show the subtraction of
the negative index.
Case II. Wc will prove a~"—a''=a~"~'',
Wc knov/ a~"-T-a''—a~"x—^
a
by properties of fractions.
=a~"Xa~'',
by meaning of negative index.
=a~"~'', by law ^.
Therefore, a-"-^a''=a-"-''. [14]
which is what v/as to be proved.
Case III. We will prove «-"-T-a-''= «-"-(-'').
We knov/ a~"~a~''—a~" x -—>
a
by properties of fractions.
—a-"Xa-^-''\
by meaning of negative index.
— a~"~^~''\ by law A.
Therefore, a-"-^a-''=a-"-^-''\ [15]
which is v;hat v.'as to be proved.
308
THEORY OF INDICES.
308. It should be noticed in the above demonstration of
law D that na restriction whatever is placed tipnn the relative
7nagnitudes of the exponents n and r. Consequently,
Law B is proved for all kinds of exponents^ whether n is
numerically greater than r or 7iot.
Thus: a^-^a^=^a
5_/,-2
■^a^^a'
etc.
309. To p7vve that negative exponents follow law C.
Let n and r stand for any two positive numbers, in-
tegral or fractional.
Case I. We will prove {a")~''=a~"''.
We know («")"''= t—^jti'
{a )
by meaning of negative index.
by law C for positive indices.
by meaning of negative index.
Therefore, («")"''= ^""^ [16]
which is what was to be proved.
Cask II. We will prove (<2-") "=«"'"'.
We know («-")' = (^-) ,
by meaning of negative index.
1
Therefore,
by law B for positive indices.
=a'
by meaning of negative index.
(a-r^a-'"-. [17]
which is what was to be proved.
PROPERTIES OF NEGATIVE EXPONENTS. 309
Cask III. We will prove {a-")-''=a"\
We know {a-y^-^^.
by meaning of negative index.
=-— -, by Case II.
a
= «"%
by meaning of negative index.
Therefore, (0-")-"=^""' [18]
which is what was to be proved.
310. To prove that negative exponents follcw law D.
Let n be any positive number, integral or fractional.
We will prove {abc)~*'=a~"b~"c~".
We know (fl^r)-"= 7-i- .
^ ^ {a bey
by meaning of negative index.
^ 1
'^a"b"c"'
by law D for positive indices.
a" ¥ d'
by properties of fractions.
=a~"b~"c''*'y
by meaning of negative index.
Therefore, {abe)-"^a-"b-"e-\ [19]
which is what was to be proved.
311. To prove that 7iegative exponents follow law E.
Let n be any positive number, integral or fractional.
We will prove y~\ =y:,r
3IO
We know
THEORY OF INDICES.
(y ) = -^^ by meaning of negativ
index.
— ;;» by law E for positive
fL indices.
0''
b''
by reducing fraction.
b-"
by meaning of negative indices.
Therefore, G)"=i^'- PO]
which is what was to be proved.
EXERCISE 122.
Examples.
Perform the indicated operations in each of the follow-
ing examples by means of the laws of exponents, now
proved to hold for negative and fractional exponents.
1. «^ xa ^.
2. b-^xb-^.
3. c'' Xc-^.
4. d^ Xd~^.
5. ;ri2xr-i
6. ?^~^^ X7V
7. a~^xxa~''j.
_ 1- 4
8. ar^xxa^x'^. 13. ^ ^X^~^.
9. 8a
10. G^Ji:"^' x^^A'-. 15.
_ 2. _i
14. 7;e a X w ^.
3
?^
-1
Xzci.
16. Ga '■xx2^-'-x~^.
17. C-7a-H--)^r-5^Ga^<5^<:-5.
33. 6;i:^j/ ■S[y6-i-2jr ^^i/^^- s".
34. (ai(^+3fl<52_^5^f^:j)_j.(^i^)^
35. («-')-^ 44. (-^Vi 53. («--^'')i
36. {a-^y. 45. (r-«)J 54. (.r-54y^.
37.. («-')^ 46. (^-^ri 55. («2^-^-i
38. (a-2)-\ 47. («/5^)-4. 5G. (^-5/5-10)-!
39. («-')-'. 48. {a-bc--)-"-'. 57. (-Lr"j-9)i
40. (a^)-2. 49. {a-^b''c^Y'\ 58. (-Ix'')-^
41. {n-^y, 50. (7«-6x-o)-2. 59. (-a-3)4.
42. («i)-3. 51. (jrVV'^. 60. (_«-i)-3^
43. (^~V*. 52. (8i3^-«)-i 61. (-««)-!
G2. (— Srt-^/^J^:"^)"'^.
=«■ (5)"' <*■ (?)-- - C-^)*
''■("f;)" -(T^r* -er
^^ (7^) 71. (^.r^--,) 76. (3;^^)
3X2 THEORY OF INDICES.
79. (iax-^-{-dx-^+cx-^+dx-^-j-cx^)xx'^.
Co. (a-jr-i+3«";»;-2)(4a-i-5.r-i+Ga;t:-2).
Ci. (2x"'^— 3.v0 +4A-o)(A-~t— 2.r~i+3.r~i),
-3 -8 -I
X "—2-1- " ^- 3r »
— i —^ _J
2jr "— 4r »+ Gr "
-3 -3 _l
— 3x "-I- Gr "— Ox »
4.r~g— 8.r~" + 12rO
G2. (.r-^2-+_y-2)(^^-i__^-2>)^
83. (;r4jj/+j/t)(;e-i__y--r)^
84. {x-\-x^+x~^s'){x^6^x-^—x-'^).
85. (•^~*+^"^+l)(;r-i-l).
86. {x-^+x-i+l)(^x-''-x-i-l),
87. (a'~J-2jr~t;/t-fj,i)(;j;-i_^l).
88. (-r-t_x-i+,i;-i_i)(^-i^l)^
89. (2ai-Zax^)Q}>a- '^+2x-^){4a^x^-^^a-^xi).
gi. (^x-i- x-'^y^-^x-^^y-yiy^^jc-^'-yt),
x-^-yl ) x~^-x--^y^-\-x-'^y-y^[ x-^ +y
■^-X-^yh
xjy-y^
x~^y—y^
EXAMPLES. 313
93. (;»;-i— j/-i)-f-Or"^— j)/~3).
94. (;»;-3+2.r-2-3jtr-i)^Cr-2 + 3;»:-i).
Arrange terms according to powers of x, so that the exponents of
dividend will be in the descending order, 4, 2, 0, —2, —4.
96. {x~'^—x-''- — ^x'^^-^x-^ _2^-^)-i-(;r"l— 4ji:"^4-2)
97. {x^-\-1xi-\-\—x~^)-^{x^^-x~^—x-'^).
98. {x^—x ■^^-^{x'i—x 2).
99. Simplify ^^ <—^. 100. Simplify -^^ — \^ ' .
(^"')~^ («;r)-2i/7ira
loi. Simplify [(rt"J^i)--^x («~^^~^p]-2 4.
102. Simplify [^^2(;^^3)i(^2^3yl^-]5-.
103. Simplify (2i"a4-3^(3ifl-2^-GT(«2__^2)^2ia^.
CHAPTER XVII.
SURDS.
EXERCISE 123.
Definitions and General Principles.
312. From the last chapter the student has learned
that there are two methods in use for indicating the root
of an expression, one by the ordinary radical sign and
the other by a fractional exponent. While it is unnec-
essary to have two ways of writing the same thing,
yet eaob method of notation has special aciv^antages in
particular cases, which accounts for the use of the two
methods. Of course the same laws (namely, A, B, C,
D, and E of the last chapter) govern the operations with
roots, whatever form of notation be used.
313. When a root of an arithmetical numeral can onlj?-
be found approximately, that root is called a Surd.
Thus, |/2 and T^ 5 are surds. Expressions like l/4,
1/8, etc., are said to be in the form of a S7ird. Expres-
sions like l/«, 'V ab, etc., are often called surds, although,
of course, they are only such when the letters stand for
numbers whose roots cannot be exactly taken.
314. Surds are of the same Order when the same root
is required to be taken in each. Thus, l/2 and Vo are
of the Second Order, 1^ 3 and f^2 are of the Third
Order, etc. Surds of the second and third orders are
often called Quadratic and Cubic Surds respectively.
DEFINITIONS AND GENERAL PRINCIPLES. 315
315. Monomials, binomials, trinomials, etc., which
contain surds are called respectively Monomial Surds,
Binomial Surds, Trinomial Surds, etc. Thus, Sf 5
is a monomial surd, 2-f-l/5 is a binomial surd, and
1/3 — 1/2 + 1/5 is a trinomial surd.
316. A factor written before a surd is often called the
Coefficient of the Surd. Thus, in 2a]/ d, 2a is called
the coefficient of the surd V b.
317. Any expression containing surds is called an
Irrational Expression, and any expression not con-
taining surds is called a Rational Expression.
318. The operations with surds depend upon prin-
ciples established in the last chapter. For convenience
of reference, we will restate below those principles which
we shall make use of in the present chapter.
319. The rth root of the product of several numbei's is
equal to the product of the rth roots of the several numbei's.
That is, V atjc=^VaV'bV~c, [1]
1 1 1. 1
because {abcy—a'b''c\ *
by equation [10], Chapter XVI.
320. The rth root of the qitotient of two mivibers is eqiiat
to the quotient of their r th roots.
V b
because
by equation [11],- Chapter XVI.
3IO SURDS.
321. The rtth root of a nitmber equals the rih root of
the t th root of the ruimber.
That is, Va=^ ^ i/a, [3]
because a'''=^{a^y^
by equation [9], Chapter XVI.
322. The rtth root of the ntth power of a ntmibcr
equals the rth root of the nth pozver of that munber.
That is, V'^'^V^' [4]
because . a>i~a'-,
by equation [5], Chapter XVI.
323. The n th power of tJie r th root of a number equals
ihe r th root of the n th power of that number.
That is, {i/'ar=\/~a\ [5]
This is equation [4], Chapter XVI.
EXERCISE 124.
To Remove a Factor from Beneath the Radical Sign.
324. When any factor of the number under the radical
sign is an exact power of the indicated root, the root of
that factor may be extracted and written as the coefficient
of the surd, while the other factors are left under the
radical sign.
(1) Thus, l/8=l/'4x2_
= l/4l/2 by[l].
= 2l/2
<2) Also, f^81 = f/27x3
= r27)/3 by[l].
==3l>'3
(3) Also, \/\{jax^ = jySx^ x 2ax
■ =\y8x-'f/2ax by[l].
= 2x^2ax
TO INTRODUCE COEFFICIENT. 317
Examples.
Remove as many factors as possible from beneath the
radical sign in each of the following :
1. 1/12. 9. 1^5G. 17. 5tlU. 25. f/a^-'d'^
2. l/28. 10. ^TgO. 18. 1/^. 26. 1/4^.
3. 1/50. II. ^2048. 19. f/;;rU'^ 27. V2om*x.
4. 1/72. 12. 1^8645. 20. l^-^^s?. 28. l/3G«-'^^
5. #^72. 13. f'5G7. 21. 1^^. 2Q. f'a^xy.
6. 1^^500. 14. 1>''112. 22. f'c'^x\ 30. 1^81;;/«x-.
7. 1^108. 15. 21/^405. 23. f'x'y\ 31. l^G4a8/;«.
8. TKiu2. 16. 3P^8G4. 24. l/y*7^ 32. 2l/8(W.
EXERCISE 125.
To Introduce the Coefficient of a Surd under the
Radical Sign.
325. It is sometimes convenient to have a surd in a
form without a coefficient. The coefficient can always
be introduced under the radical sign by reversing the
process of Art. 324.
(1) Thus, 2l/G=l/22]/6
= V2'XG by [1].
_=l/24_
(2) Also, 501/50= 1/502 ]/50
= l/50-^xr)0 by [1].
= 1/1 250 JO
(3) Also, 4l^5=f/4^|/5
= 1/4^5 by[l],
= 1^320
3l8 SURDS.
o26. As the same process may evidently be applied
in any case, we say :
Any co'-Jjficicjit of a S2ird fiiay be introduced as a factor
tinder Ike radical sign, pi ovided the coefficient be first
raised to a fewer equal to the index of ike surd.
Examples.
Place the coefficient of the surd beneath the radical
sign in each of the following :
1. 2l/2. 8. 41/4. 15. xt'\ 22. ■-2#^J.
2. 51/7. 9. 2l''3. 16. y-f/~Q,. 23. aV~b.
3. 61/5. 10. 3)' 3. 17. 2-1^7. 24. w^i/J^.
4. 2l/'21. II. 101/4. 18. (-<^)i/g 25. \VZ.
5. 3l>^2. 12. 9^10. ig. (-^)^7 26. 1^^.
6. 7T>^3. 13. 81/4I. 20. -;^l> 10 27. fl^T?.
7. 81^5. 14. aVb. 21. —hV a. 28. |V18p.
EXERCISE 126.
To Integralize the Expression under a Radical Sign.
227. The expression under the radical sign of any
surd can always be made a whole number.
(1) Thus, i/f=i/|^=l/j
= l/ix2
= F iy2 by [1].
= ^1''2 because V \=\.
>(2) Also, ^'|=l^'|xt=#'|J
= f/-2^7-Xl8
^^/J,f'Yz by[l].
=if/18
TO INTEGRAUZK A SURD. 3 19
(3) Also, i?/|=i'-'|x|=rij
= 1^3^x14
= rT>8#14 by[l].
(4) Also. V|=Vf x^=V^
=v
1
b
n
=.^-t-J/ab"-\ by[l].
^^i^ab"-'
328. As the same process can evidently be applied in
any case, we may say :
The expression under the radical sign in any surd can
be made integral by vinltiplying both numerator ajid
denominator by such a number as tvill render the dowm-
iiiator a perfect power of the indicated root, and theti taking
the root of tJie denominator thus found.
Examples.
Integralize the expression under the radical sign in
each of the following :
I.
I'^i-
6.
1/*.
II.
1/1
16.
151/^.
2.
l/i.
7.
V-i,.
12.
\yj.
17-
fJ^K-
3.
v\
8.
Vi\.
13-
^'h
i3.
4.
v'i.
9.
rl
14.
ru^
19-
l/2f.
5.
Fa
10.
^'l
15-
l/i0 8_
20.
tl/f
«\ r'
11.
Vr
23-
25-
27.
!2.
3/^
24.
2\ x^
26.
1 ab^
\2x^-
28.
2.3/ 3x
320 • SURDS.
EXERCISE 127.
To Lower or Raise the Index of a Surd.
329. We can change the index of any surd in the
following manner :
(1) Thus, i/4=^ 1/4 by [3]
= l/2 since ]/4=2
(2) Also, 1^1000= 1^ 1^1000 by [3]
= l/ 10 since 1^ 1000= 10
(3) Also, 1^2oG^2;t8 = l'^ l/25(k^ by [3]
330. Since equation [3] is true in all cases, we know
T/ie index of a siwd can be lowered if the expression
under the radical sign is a perfect power corresponding to
some factor of tlie original radical index.
^ Examples.
I^ow^er the index of each of the followinT surds :
1. 173G. 6. 1^1000. II. t/a-b\ i6. fU^x-'y^.
2. 1^^2500. 7. '1^2502. 12. T255. 17. 1^'J.
3. I^IG. 0. 'i/2882. 13. '1VT28. 18. riJf.
4. FIG. 9. 1^27. 14. F 8000. 19. 3l'/4y«2/>«^8
5. f^'m^ 10. 1^27^. 15. Vis\. 20. 4|^J^^^^.
EXERCISE 128.
To Reduce a Surd to its Simplest Form.
331. A surd is in its Simplest Form when, (1) no
factor of the expression under the radical sign is a per-
fect power of the required root ; (2) the expression under
the radical sign is integral ; (3) the index of the surd is
the lowest possible.
TO REDUCE TO SIMPLEST FORM.
321
332. Methods of making the different reductions re-
quired by this definition have already been explained in
exercises 124. 126, and 127. We give a few examples.
(1) Simplify
^^' V^'
= 2-/4.^
(2) Simplify ^'/^^-.
=il/60
(3) Simplify |l^fii_
= 5VJ
= 1/10
by exercise 127.
by exercise 126.
by exercise 127.
by exercise 126.
by exercise 124.
by exercise 127.
by exercise 124.
by exercise 126.
333. In any piece of work it is usually expected that
all the surds will finally be left in their simplest form.
Examples.
Reduce each of the following surds to its simplest form:
I. V ^. ^. V ^. 5. I ^f. 7.
6. r A. 8.
2. l^'V-.
3. i^¥
4. f/u
2T-
JlL.
V
EXERCISE 129.
Addition and Subtraction of Surds.
334. Surds which differ only in their coefficients are
said to be Similar. Thus, 61^2 and 15l/2 are similar
surds ; also fl^f and }/^y ; also of ab- and nfab"^.
322 SURDS.
335. The addition and subtraction of surds involves
no principle not already used in the addition and sub-
traction of other expressions, as the following examples
show: _ _
(1) Combine the terms of 10 1/?- 31/ 74- 51/7.
101/7-31/7 + 51/7=121/7,
by the usual process of addition of terms.
(2) Combine the terms of 7K 2 — l/l8 + 2l/8.
Putting each surd in its simplest form, we have
7l/2_i/l8+2l/8=7l/2-3l/2+4l/2
=81/2
by the usual process of addition of terms.
(3) Combine the terms of 5l/4 + 2l/32-l/l08.
Putting each surd in its simplest form, we have
6l/4 + 2l/32-l/l08=5l/4+4l/4-3l/J
=61/4
(4) Combine the terms of |l/|-|l/f.
Putting each surd in its simplest form, we have
|i/|-i>^f=ii^6-ii/6
=11/6
(5) Combine terms of 22f'a'd-Sa''l^QU + haf'aH.
Putting each surd in its simplest form, we have
22l/^-3a2i/64^+5«l/^
=22«2i/A-12^2^^^5«2^?
= 15«2#/^
336. We observe the advantage of reducing each of
the surds in any given expression to its simplest form,
for then it can be told whether or not some of the surd
are similar to each other, and consequently whether or
not they can be combined together; for only similar surds
can be combijied into a single surd.
MULTIPLICATION AND DIVISION. 323
Examples.
Express each of the following expressions in as few
terms as possible :
1. 21/3 + 31/3. 6. 2f/32-l/l08.
2. 71/2-3 1/2. 7. 41/2-]^" 64.
3. 11i/13-4t/13. 8. 1^243-51/48.
4. 71/2 + 1/18. 9. 1/^-1/20.
5. 4l/4-3l/4+2#^4. 10. il/245 + 5l/i.
11. i^'':t:._-,^^_3,^256+ 1/625.
12. 2v/|+l/60-l/l5 + l/S.
13. 2+Jl/|-|>/f. _
' 15. l/«^ + 2l/a— 1/4«.
16. l/^+il^V-3l/27a2.
17. l/27^*-l/8^* + |/l25f.
18. l^a*x+Vd\r-]^4aH''x.
19. 6«l/63^^='-3l/ll2«^^/^3 +5^1/28^33.
EXERCISE 130.
Multiplication and Division of Surds.
337. The product of any number of surds of the samf
index can always be expressed as a single surd by means
of equation [1]. _ _ _
(1) Find the product of l/2 x l/5 x Vl.
V^2 X 1/5 X 1/7 = l/2x5x7 by [1].
= 1/70
324 SURDS.
(2) Find the product of l/2 X "/I8.
|/'2xl/ 18=1/2x18 by[l].
=6
(3) Find the product of V^/54x 1^9.
l5^54xi/9=l/54x9 by [1].
= 31^/2
The result should always appear in its simplest form.
338. The quotient of two surds of the same index may
be expressed by means of equation [2].
(1) Find the quotient of |/28-^|/7.
V'28-^1/7=1/V- by p]-
(2) Find the quotient of 1^81 -7- l^G.
^/81^f/6=r-V--r^- by [2]
= ?)f\ by exercise 124.
=#1/4 by exercise 126.
The result should always appear in its simplest form.
339. If the product or quotient of surds of different
indices is sought, the surds may first be reduced to a
common index by exercise 127.
(1) Find the product of v'5 X P'4.
V '5 X #4= 1'^ 125 X 1/ IB by exercise 127.
= ri25xTG=V 2000 by[l].
(2) Find the quotient of ^9-^V 3.
f 9 _^ 1 3 = 1' 81 ^ V 27 by exercise 127 .
==f'|l = ,V3 by [2].
MULTIPLICATION AND DIVISION. 325
(3) Find the product of v' ad'^ xi a'-b.
|/^X t\iH= "Va'^b^-^ X 'f V7- by Ex. 127.
= 'f/^«^ by[l].
= b ^Va^b'^ by exercise 124.
(4) Find the quotient of f^'^x^'V'^.
>^ -K2nV^=r M-^2Y/; ^ by Kx. 127.
= i'>^i|x^f^ by [2].
=4Y 16xlG=^#'l6 bv Ex. 127
Examples.
Perform the indicated operations in each of the following
and leave the results in their simplest form :
1. 1/5x1/20. 18. 1/50-T-1/6. 35. ^y^xf'^.
2. f'Zxf'n. 19. l/96H-iV^3. 36. bV^xt'oH^.
3. l^6x#^32. 20. #'8i-^#'6. 37. af/Jaxf^^K
4. l/2xl/'l2. 21. l/80--1^20. 38. #'^^xl^>7^*
5. fAxf^. 22. y 8F--1/2F 39. 3l/27x5v''2?^.
6. l'/6xl'/8. 23. #'547^^1'/ 2^ 40. #9^1/3.
7. ^^'4x1^10. 24.- v'|H-]/f. 41. #^72h-i/6.
8. f/81x#^45. 25. V^if-^l/^. 42. f'l2-M/6.
9. f 7x1'/^^^ 26. V'^V'i._ 43- l^'l^-l^'^f.
10. #'50xf/40. 27. i/ff^i/^. 44. 1^/24^61^3.
II. I'/iix 1^121 28. i/5xf/4._ 45. i/^-^rej.
12. 2l/«xl/9^ 29. 1/125 xi/36. 46. 5h-i/3.
13. aVyxbVy^ 30. l^|xi/f. 47- 30-- 1/210.
14. 1/157x1/5731. l/fxi/^. 48. l/^H-l/'^.
15. 2i/^x3f/^32. i/eixi^Te. 49. i/2;^-^v^.
16. i'48--l"3. 33. #''l2x'lV97). 50. #y--|/2^.
17-
45-- 1 10. 34. f'fxl'^f 51. jV^5^«-T-l/(^M
326 SURDS.
52. (3 + 2l/5)C2-l/5).
53. (8+3l/2)(2-l/2).
54. (5+2i/3)(3-5t/3).
55. (3-l/6)(6-3l/6).
56. (i/ 5-t/9 )(v' 5+i/9 ).
57. (1/3 -1/2) (1/3 +1/2).
58. (1/ 9 _v/r7 ) (1/ 94.1/17 ),
59. (]/ 6+i/1 i)(i^ 6-t/1i ).
60. (T/ 9-|,v/lf) (i^9_v/i7),
61. (^12 +1/19) (l^'l2 -1/19).
EXERCISE 131.
Powers and Roots of Surds.
340. Any power of a surd can be expressed as a single
surd by means of the principle of Art. 323. Thus :
(1) Square ^2. _ _
(l/2)2 = #^22 by [5].
= ]/4
(2) Cube 31/2.
(3l/2)3=33(i/2)' by law Z> for indices.
=27l/'2^ by [5].
= 541/2 by exercise 124.
The result should always be left in its simplest form.
341. Any root of a surd can be expressed as a single
surd by means of the principle of Art. 321. Thus :
(1) Find square root of l/4.
y'~Ft=^yz=r';7t by [3].
=1/2
POWERS AND ROOTS. 327
(2) Find cube root of |l/3.
f'sy'S==2^iVs by exercise 124.
=2^1/^=21^1/^ by Ex. 125.
-=2^f^ by [3].
=2l/|=|l/3 by exercise 126.
(3) Find the /th root of aVJ.
l/ai/b^^ V~arb by exercise 125.
= f/^ by [3].
342. This last process is a general one, but if for any
particular values of a, b, r, and t this result should not
happen to be in its simplest form, it should be so reduced.
Examples.
Express each of the following as a surd in simplest form:
I.
(Vby.
7. (ii^«^)^
13. "^Wh
2.
(fzy.
8. {af'iY.
14. 1^ 1/16.
3-
(2l/2)8.
9. 1^21/8.
15. I^3l/9«i«.
4.
(^^2)2.
10. 1^^^36.
16. l/l28 1/243^7
5.
(1^)^
II. ^31/3.
17. l/_l_^2|K9^2^2
6.
(-1/8)^
. 12. ^vi
18. l^|l/-^.
19.
2"^ i/a-4.'^a + 2^ f'a^}
EXERCISE 132.
Rationalization of Expressions Containing
Quadratic Surds.
343. To Rationalize an expression is to perform an
operation upon it that will free the expression of surds.
Thus, the binomial quadratic surd 3 + 1/ 2 is rationalized
when multiplied by 3 — 1/2, for the product is 9—2 or 7,
which is rational.
328 SURDS.
344. Any multiplier which, when applied to an irra-
tional expression, will free the expression from surds, is
called a Rationalizing Factor. Thus, S~V2 is a
rationalizing factor for the binomial surd 3-^-1^2.
345. It is often convenient to perform an operation
upon both terms of a fraction so as to render either the
numerator or the denominator rational. It is suflScient
for present purposes to show how this may be done when
all the surds are of the second order. The following are
examples.
2
(1) Rationalize the denominator of j^. .
2+y2
Multiplying numerator and denominator by 2 — V^2,
we get _ _
2 2(2-l/2) _ 4-2y 2
2-m/2~(2 + 1^2)(2— ]/2)~"4-(l/2)2
Q
(2) Rationalize the denominator of
t/5-V'^2"_
Multiplying numerator and denominator by Vb-\-\^2,
we get _ _
3 ^ 3(t/5 + i/2)
l/5-l/2 ~(|/5-t/2 )(t/54- l/2 )
^8(1/54- 1/2)^^-_^^-
o — z
346. This work is based on the very evident principle
that any bmomial quadratic surd is made rational by
multiplying it by itself with the sign of 07ie of its terms
changed, for the product is the7i the difference of tivo squares.
RATIONALIZATION OF EXPRESSIONS. 329
347. Considerable labor is often saved in computing
the value of an irreducible fraction if we first rationalize
the denominator. Thus, to compute the value of
3
1/5-1/2
to five decimal places, the two square roots must be
taken to at least five decimal places, and the quotient of?*
divided by the differe7ice of these roots must be foimd. This
division by a five-place number will be avoided if we
first rationalize the denominator, for the result isl/54- V^2,
the value of which is found without the long division of
the former method.
Examples.
Rationalize the denominator in each of the following :
11 2v/2
" ■ -'^' ^' 1/8-1/7' ^* 81/2 -2V^3'
(5 211/3
10.
Vb V' 12 + 1/5 4V 8-31/2
9 2 1/5-1/3
7. -7^—7zz' II- -7^
^- 5-l/'l0 1/3 + 1-^2 * 1/5 + 1/3
4 o 3 ^^
4. ^- 8. —7= --• 12. -j^ j==
6-21/3 V 10 -1/6 l/3 + V'^2
13-
i-v'~i+v'^
1 + 1/2-1/3
-N2+V^ (l-V'2-i-V3) {l-f-V'2-fV'3) 1 + 2^3 + 3-2 V2+V6
l-fV2-V3 {[1+V2]-V3) ([H-V2J+V3) 1 + 2V2 + 2-3 2
30 2+1/6-1/2
^^' 2-1/3 + 1/5 ^^" 2-1/6+1/2
16. Find, in the shortest wav, the value of
to four places ; given 1 2 = 1.4142 +
3 + 21/2
350 SURDS.
EXERCISE 133.
Rationalization of Equations.
348. If the unknown number in an equation appears
under a radical sign, the equation must first be rational-
ized before the value of the unknown number can be
found. This is illustrated by the following examples.
(1) Solve 1/5^=20.
Squaring both sides, we get
5x=400,
whence .;»;=80.
(2) Solve SVix-S^VlSx-S.
Squaring both sides, we get
2(4:X-S)=lSx-3,
Transposing and uniting,
23;i;=69,
whence x=S.
(3) Solve l/^T9=5l/J-3
Squaring both sides, we get
x + d=25x—S0Vx-\-d.
Transposing rational terms to one side and irrational
terms to the other, we get
24;t=30l/^
Dividing by 6 and squaring,
16;t:2 = 25.;»;.
Solving this quadratic, ^=14 o^ ^'
(4) Solve VS2+x=16-Vx.
Squaring both sides, _
32-f:»r=256-82l/;^+.^.
Transposing, uniting, and dividing by 3i,
whence .r=49.
RATIONALIZATION OF EQUATIONS. 33 I
Examples.
Solve each of the following equations :
1. 1^2^=4. 10. l/x—Vx'— 5=1/5.
2. 1/8^=2. li. l/x—7=v'x—U + l,
3. }/x+A=4. 12. l/^'7=1 '-r+1— 2.
4. }/2x+Q=4:. 13. .r=7 — 1 x-^— 7.
5. ]/iar+l6=5. 14. V^^+20-l/.r^-3=().
/;r ;;- /- ~ X — 1 V X-\-l
6. V2x-^1=V 5x—2. 15. - = V-
V Jtr-1 X-^
7. 14+:^4.r-40=10. 16. -?II^=?±L£.
8. }/lG^T9=4l/4r-3 17. v/jt--f3+l/'3x— 2=7.
9. l/^f+^=f+V^. 18. 1/2^+1+1/^^= 2 1/^.
CHAPTER XVIII.
RATIO, PROPORTION AND VARIATION.
EXERCISE 134
Ratio.
349. The relative magnitude of two numbers or quan-
titie.s, measured by the number of times that the first
contains the second, is called the Ratio of the two num-
bers or quantities.
Thus, 12 contains 3 just 4 times, hence the ratio of 12
to ') is 4 or ^f-. And similarly if a and b stand for any
two numbers the ratio oi a \.o b is -r-
350. We may speak of the ratio of two quayitities of
the same kind as well as the ratio of two 7iumbers. Thus,
12 feet contains 3 feet just 4 times, hence the ratio of 12
feet to 3 feet is 4 or -^^.
The student should notice that the ratio of 12 feet to 3 feet is -^,
12 feet
not —7 , for division proper cannot be performed except when the
divisor is a number. Division implies separating into a certain num-
ber of equal parts. For example, a stick 12 feet long may be sawed
into 3 equal pieces, and one of these pieces is 4 feet long ; so 12 feet
divided by 3 is 4 feet. But while a stick 12 feet long may be divided
by 3 but not by 3 feet, still a stick 12 feet long may be measured by a
stick 3 feet long. This measured by is often confused with division
proper.
The ratio of 12 feet to 3 feet is not 12 feet divided by 3 feet, but it
is 12 divided by 3.
The ratio of any two quantities of the same kind may be looked
upon as the number of units in the first divided by the number of
units in the second. Plainly, quantities which are not of the same
RATIO. 333
kind cannot have any ratio, for one cannot possibly be measured by
the other, nor can one be contained at all in the other. For example,
ten miles cannot be measured by two quarts, nor can two quarts be
contained any number of times in ten miles.
361. The ratio of a to d is denoted in either of two
ways : Jij's^, by writing the a before the d with a colon
between them, thus, a : l>; second, by a fraction in which
a is the numerator and b is the denominator, thus, - •
Whichever way the ratio is written, it is read ''the ratio
of a to ^, " or simply ' 'a to b. ' '
352. In either waj^ of writing the ratio of a to b, a is
called the Antecedent or First Term, and b is called
the Consequent or Second Term.
353. Since a ratio is the quotient obtained by dividing
the number of units in the antecedent by the number of
units in the consequent, it follows that the properties of
ratios may be obtained immediately from the properties
of fractions.
354. Since a fraction may be multiplied either by
multiplying the numerator or dividing the denominator,
it follows that a ratio may be tnultiplied either by tnulti-
plying the antecedent or by dividincr the conseipient.
355. Since a fraction may be divided either by divid-
ing the numerator or multiplying the denominator, it
follows that a ratio may be divided either by dividing the
antecedent or by multiplying the consequent.
356. vSince a fraction remains unchanged in value
when both numerator and denominator are multiplied or
334 RATIO, PROPORTION AND VARIATION.
divided by the same number, it follows that a ratio temai7is
unchanged in value ivheii both antecedent and conseque7it are
multiplied or divided by the same number.
357. If the numerator of a fraction is greater than the
denominator, the fraction is greater than 1 ; therefore,
if the antecedent, of a ratio is greater than the consequent,
the ratio is greater tha7i 1.
358. If the numerator of a fraction is less than the
denominator, the fraction is less than 1 ; therefore, if the
antecede7it of a ratio is less than the consequent^ the ratio is
less than 1.
359. If the numerator and denominator of a fraction
are equal to each other, the fraction is equal to 1 ; there-
fore, if the antecedent and conscque?it of a ratio are equal to
each other ^ the ratio is equal to 1.
360. Theorem. A ratio which is greater than 1 is de-
creased by increasing both antecedent and consequent by the
same amoimt.
Let T be a ratio which is greater than 1 ; then a'^b.
Now form a new ratio by increasing the antecedent and
consequent by the same amount, x. The new ratio is
a-\-x
'bTx'
If we multiply antecedent and consequent of the original
ratio by d-{-x, we get
a_ab-}-ax
'b~ b-'+bx ^^^
If we multiply antecedent and consequent of the new
ratio by b, we get
a-{-x _ ab-i-bx
b-]-x~b- + bx ^^
RATIO. 335
Now, as a><^, it is plain that ax^bx, and hence
a.b-{-ax'>ab-\-bx. Therefore,
ab-\-ax ab-\-bx
'WVbx-^ b'^^bx
Therefore, from (1) and (2),
a a-\-x
ly^T^x
which is what was to be proved.
->1
Since a^b, it is plain that a^x^b-\-x. Therefore
b+x'
Therefore, as each ratio -y and -r- — is greater than 1 , and
o o-j-x
as -r'>T-. — , it follows that y- — ts nearer the value 1
b b-j-x b+x
a
than — is,
361. Theorem. A ratio which is less than 1 is in-
creased by increasing both ajitecedent and consequent by tlie
same amo2int.
Let T be a ratio which is less than 1; then a<^b. Now
o
form a new ratio by increasing the antecedent and conse-
quent by the same amount, x. The new ratio is
a-\-x
b+x'
If we multiply antecedent and consequent of the original
ratio by b-\-x, we get
a_ab+ax .^.
'b~b^-{-bx' ^^
* This notation denotes that a is less than b. Whenever this symbol is used
the point of the angle is toward the lesser and the opening toward the greater
number. See note, page 291.
336 RATIO, PROPORTION AND VARIATION.
If we multiply antecedent and consequent of the new
ratio by b, we get
a-[-x ^ ab-\-bx
'b+x b'^^bx ^^
Now, as a<^b, it is plain that ax<^bx^ and hence
ab-\-ax<^ab-{-bx. Therefore,
ab-\-ax ^ab-\-bx
W+bx^'PTbx'
Therefore, from (1) and (2),
a ^a-\-x
l^'b+i'
which is what was to be proved.
Since a<,b, it is plain that a+x-Cb + x. Therefore,
b+x
Therefore, as each ratio -r and -r-i — is less than 1, and as
o-\-x
a a-\-x . ^ ,, , a + x . ,, , ^ ,, ^ .
-7-<-i— — , it follows that ,— — IS nearer ilie value 1 than ,- is.
b b+x b+x b
This last statement, together with the last statement
in the previous article, shows that any ratio (except the
ratio 1) is made more nearly the value 1 by i7icreasing both
antecedent ayid conseqtient by the same a^nount.
362. If from two given ratios, ,- and -or a\b and c : d,
b d
we form another ratio by multiplying the antecedents
together for a new antecedent and the consequents to-
gether for a new consequent, we get t;^ or ac\ bd, which
is said to be Compounded of the given ratios a : b
and c : d.
RATIO. 337
363. If ^>1 it follows that xy^j/, or, expressed in
words, if the multiplier is greater than 1 the product is
greater than the multiplicand.
Therefore, if :i> 1, t ,i>t- Also, if -> 1, ^ -^>4-
a a odd
But in each of these cases 7- - or — , is the ratio com-
d cd
d c
pounded of the ratios — and -j-
Therefore, if a ratio be compounded of tivo ratios each of
which is greater tha7i 1 the result is greater than either of
the given ratios.
364. In a similar way it may be .shown that if a ratio
be compoimded of two ratios each of which is less than 1 the
result is less than either of the given ratios.
365. If xl,^^>^.
Therefore, if a ratio be compoiinded of two given ratios,
one of ivhich is greater than 1 and the other less than 1, the
result is iiitermediate in value betiveen the tivo given ratios.
Problems.
1. Arrange the ratios 2:3, 3:5, 5:8 in the order of
magnitude.
2. Which is nearer unity 2 : 3 or 8 : 9 ?
3. Which is nearer unity 2 : 3 or 2-\-x : 3+jtr?
4. For what value of x will the ratio 84--^ : 36+^ be
equal to the ratio 1:3?
22
338 RATIO, PROPORTION AND VARIATION.
5. What must be added to each term of the ratio 9 : 16
to produce the ratio 3:4?
6. What must be subtracted from each term of the
ratio 3 : 5 to produce the ratio 5:9?
7. A certain ratio will become equal to -J- when 1 is
subtracted from each of its terms, and equal to f when 9
is added to each of its terms. Find the ratio.
8. Find two numbers such that if each is increased by
1 the results have the ratio 2 : 3, and if each is increased
by 7 the results have the ratio 5:6.
9. Find two numbers such that if the first is increased
by 2 and the second decreased by 2 the results have the
ratio 6:5, but if the first is decreased by 2 and the
second increased by 2 the results have the ratio 4:7.
10. A rectangle is 39 feet long and 36 feet wide. Ex-
press the ratio of the length to the breadth.
11. What is the ratio of 12 lbs. 8 oz. to 21 lbs. 14 oz.?
12. What is the ratio of 5 ft. to 6 ft. 3 in.?
13. Two rectangular fields have the same area. The
length of the first is 180 feet and of the second 150 feet.
What is the ratio of their widths ?
14. The areas of two rectangular rooms have the ratio
2:3. The length of the first is 12 feet, and of the second
20 feet. What is the ratio of their widths ?
15. Two numbers are in the ratio of 3 : 4, and the sum
of their squares is 400. What are the numbers ?
16. Two equal sums of money are on interest ; the
first runs 8 months at 7 per cent., the second runs 9
months at 6 per cent. The interest in the first case has
what ratio to the interest in the second case ?
PROPORTION. 339
17. Divide the number 10 into two such parts that the
squares of these parts shall have the ratio 9 : 4.
18. A train of cars travels 140 miles in 3| hours, and
another train travels 240 miles in 5 hours. What is the
ratio of their rates of speed ?
19. Show that the ratio aid is equal to (a+x)-: (d+xy
\i x'^ = ab.
20. V/hat must be the value of x in order that the
ratio 6 : x may equal b—x ?
EXERCISE 135.
Proportion.
366. The expression of equality which exists between
two ratios is called a Proportion. Thus, a\b=c\d\S2i
proportion. In this case the four numbers or quantities
are said to be in proportion.
367. Sometimes the proportion is expressed as an
ordinary equation in fractions, thus, -T—-7, and some-
times by the notation a:b::c:d. However written, the
proportion is read, ' 'a is to ^ as <: is to ^. "
368. In the proportion a:b=c:d the letters a, b, c,
and d are called the Terms of the proportion. The first
and fourth terms are called the Extremes, and the
second and third terms are called the Means.
369. VL a, by c, and d are proportional, we have the
proportion a : b :: c:d, or, written in the fractional form,
CL C
T=~j- Now, multiplying each member by bd, we get
ad^= be.
That is, the product of the means equals the product of the
extremes.
340 RATIO, PROPORTION AND VARIATION.
370. If we have given the equation
ad= be,
then dividing each member by bd, we get
ad be a c . .
Hence, if the produet of two numbers equals the produet of
two other numbers, the numbers are in proportion, or, in
other words, if the produet of two numbers equals the
produet of two other numbers, the factors of one produet may
be taken for the extre7?ies and the faetors of the other product
may be taken for the means of a proportion.
371. From the equation ad=bc we may infer either
a\b=c\d (1)
or ' a : c = b : d (2)
or b:a=d:c ['-))
or b:d=a\c (4)
and therefore from (1) we may infer either (2) or (3) or (4).
The proportion (2) is said to be deduced from (1) 1)y
Alternation. The proportion (3) is deduced from {1}
by Inversion. The proportion (4) is really the same as
(2) except that the two members have changed places ;
it may be deduced from (3) by alternation.
372. If ^ : b=c : d, then writing in fractional form,
a e
Adding 1 to each member,
a\-b e+d
or in another form, —7—= — v-,
a
or in the common form of proportion,
a-{-b\ b=e-{-d: d.
This last proportion, a-\-b : b=c-\-d \d, is said to be
derived from the proportion a : b=c\ d by Composition.
PROPORTION. 341
d c
373. If « : b—c\ d, then t=;j, and subtracting 1 from
each member,
b d '
or in another form,
a — b c—d
or in the common form of proportion,
a — b: b=c — d: d.
This last proportion, a — b:b=c—d\d, is said to be
derived from the proportion a \b=c:d by Division.
374. If a : b=c : d, then by composition,
a-^b_c-\-d
b d ' ^^^
and by division, — , = — j . (2)
p d
Divide (1) and (2) member by member and we get
a-\-b _c-\-d
a—b c—d
or written in the ordinary form of a proportion,
a-\-b ■.a—b=c-\-d: c—d.
This last proportion, a-\-b :a—b—c-\-d \c—d, is said to
be derived from the proportion a\b=c\d by Compo-
sition and Division.
375. The products of conespondmg terms of two or more
sets of proportional numbers are proportio7iaI.
Let a : b= c : d
Mild e\f=h\k
and n\r=^ s:t
342 RATIO, PROPORTION AND VARIATION.
Writing each of these proportions as an equation in
fractions, we have
a_ c
f-k
n s
and from these equations by multiplication we obtain
a e 71 _ c h s
l}fr~d'kt
ae?i chs
^^ Ifr^'dki
or writing this in the ordinary notation of proportion,
we have aeyi : bfr—chs : dkt,
which is what was to be proved.
376. Like powers or like roots of proportional numbers
are proportional.
Let a : b=^c : d.
Writing this proportion as an equation in fractions, we
obtain |=|, (1)
and raising each member to the 7i th power, we have
a'' ^"
or writing this in the ordinary notation of proportion,
we have a''\ b"=c'':d''. (3)
Again, taking the nih. root of each number of (1), we
a^' ^"
have -=_ . (4)
b- d-
or writing in the ordinary notation of proportion, we have
1-1 11
Equations (3) and (5) are those which were to be proved.
PROPORTION. 343
377. In the proportion a : b=b :c, b is called a Mean
Proportional between a and c, and c is called a Third
Proportional to a and b.
If in this proportion we write the product of the means
equal to the product of the extremes, we get
b'-^ab,
or extracting the square root of each member, we get
b^V'^b,
Therefore, a mean proportional between two numbers is
equal to the square root of their product.
378. A Continued Proportion is a series of equal
ratios. Thus, a : b=c : d=e :/, etc.
379. In any continued proportion any a7itecedent is to
its corresponding consequent as the smn of all the antecedents
is to the sum of all the consequetits .
Let the continued proportion be written as several
equal fractions, thus:
ace
IT'd^J
and let r be the common value of each of these fractions,
then
--f=r or a=^br,
— =r or c=dr,
a
— =r or e=fr.
Therefore, by addition,
a + c+e^br+dr^-fr^^iib+d^fy.
Hence, by dividing by b-\rd-\-f,
a-\-c-\-e _ _a
b + d-[-f~'^~T
which is what was to be proved.
344 RATIO, PROPORTION AND VARIATION.
Examples and Problems.
1. Write two proportions from the equation 5 x 8=4 x 10
2. Write two proportions from the equation ab—cd.
3. Write two proportions from the equation ab=ac.
4. Write two proportions from the equation x'^—yz.
5. Write two proportions from the equation (a-{-by = 7ir
6. Write two proportions from the equation ?>n'^r=4:pq.
7. Form a proportion with the numbers 3, 5, 21, 35.
8. Form a proportion with the numbers 3, 20, 90, 600.
9. Form a proportion with the numbers 3, 6, 20, 40.
10. What must x stand for in order that .r : 3=6 : 9 may
be a true proportion ?
11. What is the value of x \i b \x=% : 12 ?
12. What is the value of Jt: if 12 : lb=x : 35 ?
13. What is the value of .r if 8 : 10=90 : x ?
14. What is the value of ;r if jt: : 6=jr— 1 : jir+ll ?
15. What is the value of x if
Ibia + b) lQ{aby a-b
\^{a-b')'1\{aby a^-b
16. What is the value of x if
X?
(^-)^(^+-)-
17. What is the value of x if
x\ (5;z— 6r) = (3;z2-f 2;2r— 8r2) : {bn'^ ^A.nr—Vlr'^^ ?
18. What is the value of x if
ia-^b) :x={a''-\-ab^b'') : {a^—b^) ?
19. If a : b=c : d, prove a : inb-=c : md.
20. If a : b=c : d, prove na : rb=^nc : rd.
21. li a : b=c\d, prove
{jia-\-rb) : {iia — rb)-={nc-\-rd^ : (nc—rd).
VARIATION. 345
22. If a : b=c : d, prove 7ia : rb=~ : —
r n
23. If a : b^c : d, prove that a : a-\-c=ab : a-\-b-\-c-\-d.
24. If a : b=c : d, prove that
a"^ -}-ab + b'-. a- —ab-^ b^=c'^ -}-cd-]-d^ ■ c"^ —cd+d'K
25. Prove that a : b=c:d, if
(a-^b+c+d) (_a—b—c+d) = {a—b^-c—d') {a-\-b—c—d^.
EXERCISE 136.
Variation.
380. Thus far whenever a letter has been used to
stand for a number, it has always represented the same
number throughout the same problem; or in other words,
the value of the letter has always remained the same in
the same problem. But sometimes it is necessary to con-
sider quantities which change in value in the same
problem, and then the letter used in connection with such
a quantity may stand for one number at one instant and
another number at another instant in the same problem.
For example, if a train of cars travels at the rate of 40
miles an hour, of course the number of miles traveled
depends upon the time of traveling. Now we may take
a letter, say x, to represent the number of miles the train
travels, and another letter, say jr, to represent the number
of hours the train travels; then plainly we have
x=iOy or —=40.
Here x and jj^ stand for numbers which are ?zo^ the same
at one time as at another; or in other words, x and jy
change in value, but however much or little they change,
the ratio of these remains unchajiged, or, as it is usually
exprCvSsed, the ratio reinains constant.
34^ RATIO, PROPORTION AND VARIATION.
381. One number or quantity is said to Vary Directly
as another when the number of units in the first is equal
to the number of units in the second multiplied or
divided by some constant number, or what is the same
thing, when the ratio of the number of units in the first
to the number of units in the second is some constant
number.
In the above illustration the distance traveled is said
to vary directly as the time occupied, or the number of
miles traveled varies directly as the number of hours
occupied.
382. The word directly is often omitted, and we say
simply, that one number or quantity varies as another.
The same idea is often expressed by saying that the
second number or quantity is Proportional to the first
number or quantity. In the above illustration the dis-
tance traveled is proportional to the time occupied, or the
number of miles traveled is proportio?ial to the number of
hours occupied.
383. If X is proportional to y then —=c where <:is some
y
constant, i. e. some number that remains unchanged.
Now, if we represent some particular value of x hy x^
and the corresponding particular value oi y by j^, then
— ^=^ and from this it is easily seen that when one num-
yx
ber is proportional to another, we know the constant by
which one of the numbers must be multiplied to produce
the other, if we know any corresponding /^^r/z-f/z/^^r values
of the two numbers. For example, if we know that a
traveler is walking uniformily, i. e. at the same rate, then
we know that the number of miles traveled is propor-
tional to the number of hours occupied, hence if we
VARIATION. 347
represent the number of miles traveled by x and the
number of hours occupied by jk, we have
X i
x^cy or — =^.
y
Now if we further know that the traveler walks 15 miles
15
in 5 hours, we have <:=^=3.
o
From this we see that 3 may be written in place of r, and
X
hence we know x=Sy or — =3.
384. One number or quantity is said to Vary In-
versely as another when the first is equal to some con-
stant divided by the second. Thus, 'Ji: varies inversely
as r, if X— — where c is some constant.
y
385. If in the equation x=— we multiply eac^j number
by y, we get xy=^c.
Therefore, we say that one number varies inversely as a
second when the prodtut of the two numbers is some constaiit.
For example, the time occupied in traveling a certain
distance varies i7iversely as the rate of speed, or the number
of hours occupied in traveling a certain number of miles
varies inversely as the number of miles traveled per hour.
386. Instead of saying that one number varies inversely
as another, the same idea is often expressed by saying that
one number is Reciprocally Proportional to another.
387. One number Varies Jointly as two others when
the first is equal to some constant multiplied by the
X
product of the other two. Thus, if x=cyz, or — =<:,
where c is some constant, x is said to vary jointly as
y and 2, The same idea is often expressed by saying that
one number is Proportional to the Product of two others.
348 RATIO, PROPORTION AND VARIATION.
388. One number is said to vary as the square of
another when the first is some constant multiplied by the
square of the second, or when the first divided by the
square of the second is some constant. Thus, if x=cy^,
X
or — 2 = ^j where c is some constant, x is said to vary as j^'^.
The same idea is often expressed by saying that one
number is Proportional to the Square of another.
389. One number is said to Vary Directly as a second
and Inversely as a third when the first is equal to some
constant multiplied by the ratio of the second to the third.
Thus, ii x=c~ , x' \s said to vary directly as y and in-
2
versely as z.
The same idea is often expressed by saying that the
first number is Directly Proportional to the second
and Inversely Proportional to the third.
Examples and Problems.
1. If Ji; is proportional to y, and x=45 when ^^=3, find
the value of x when j>/=15.
2. If Ji; is inversely proportional to ^, and x=l when
y=6, find the value of x whenjK=15.
3. Form a proportion with the two sets of values of x
and y in problem 2.
4. If X varies as y^, and x=l when y=5, find the
value of X whenjj/=15.
5. Form a proportion with the two sets of values of
X and y^ in problem 4.
6. If X varies jointly as j/ and z, and x=20 when jj/=2
and 2=3, find the value of x whenjK=3 and 2=6.
7. If .r varies inversely as the square of y, and x=l
when y =10, find the value of x when ^^=5,
VARIATION. 349
8. If ,r is directly proportional toj/ and inversely pro-
portional to z, and ;r=20 when y=^ and 2'= 4, j&nd the
value of X whenjK=3 and 2'= 10.
9. If X varies as^, prove that .f ^ varies 2s, y^ .
10. If X is inversely proportional to y and y is inversely
proportional to z, prove that x is proportional to z.
11. If ;t: is proportional to z and^ is also proportional
to ?, prove that xy is proportional to ^-'^ also that x'^+y'^
is proportional to z"^ .
12. If Sx+7y is proportional to Sx-\-VSy and x=r>
whenjj'=3, find the ratio of x toy and thus show that x
varies asy.
13. The number of feet a body falls is proportional to
the square of the number of seconds occupied in falling.
Knowing that a body falls 16 feet the first second, find
how many feet it will fall in 5 seconds.
14. With the same supposition as in the last example,
find the height of a tower from which a stone dropped
from the summit, reaches the ground in 3| seconds.
15. The weight of a metal ball is proportional to the
cube of the radius, and a ball whose radius is 2 inches
weighs 10 pounds, what is the weight of a ball whose
radius is 5 inches ?
16. If a heavier weight draw up a lighter one by
means of a cord passed over a fixed wheel, the number of
feet passed over by each weight in any given time varies
directly as the difference of the weights, and inversely as
the sum of the weights. If 10 pounds draw up 6 pounds
16 feet in 2 seconds, how high will 14 pounds draw 10
pounds in 2 seconds ?
CHAPTER XIX.
PROGRESSIONS.
EXERCISE 137.
Arithmetical Progressions.
390. An Arithmetical Progression is a series ol
terms such that each term differs from the immediately;
preceding term by a fixed number, called the Common
Difference. The following are examples of arithmetical
progressions :
(1) 2, 4, 6, 8, 10. (3) 2i, 3f, 5, 6i, 7^
(2) 31, 26, 21, 16. (4) (x-y), x, (x-hy-).
(5) a, (a-i-d), (a-\-2d-), (a + Sd).
391. The first and last terms of any given progression
are called the Extremes, and the other terms are called
the Means.
1. The first term of an arithmetical progression is 3
and the common difference is 2. What is the 5th term ?
What is the 10th term ?
2. The first term of an arithmetical progression is 5
and the common difference is 3. What is the 4th term ?
What is the 7th term ?
3. In the last progression, how many times must the
common difference be added to 5 to produce the 4th
term ? to produce the 7th term ? to produce the 7i th term ?
4. The first term of an arithmetical progression is 19
and the common difference —4. What is the third term?
What is the fifth term ? What is the n th term ?
ARITHMETICAL PROGRESSIONS. 35 1
392. The n th Term of any Arithmetical Progres-
sion. It is usual to represent the first term of an
arithmetical progression by a, the common difference by
d, and the n th term by /. With this notation we may
represent any arithmetical progression by
No. of term: 12 3 4 5 . . .
Progression: a, (^ + ^), {a^-'W), (« + 3^), (« + 4^)
We notice that the coefficient of d in the 2d term is 1 ,
in the od term is 2, in the 4th term is 3, and, by the
nature of the progression, the coefficient oi d in any term
is 1 less than the number of that term. Therefore, the
n th term in this progression will be
a-\-{7i—\)d,
or, representing the n th term by /, we have the formula
l=a-^[n-\)d, [1]
393. Evidently the sum of an arithmetical progression
is not changed if the order of the terms be reversed; thus,
3 + 5 + 7 + 9 + 11 may be written 11 + 9+7 + 5 + 3
2|-+3i+4 + 4f may be written 4|+4+3^+2i
in which case the first term becomes the last term, the
last term becomes the first term, and the commoji difference
chmiges sign.
394. When the common difference is positive the pro-
gression may be called an Increasing Progression, and
when the common difference is negative the progression
may be called a Decreasing Progression.
395. The Sum of n Terms of any Arithmetical
Progression. If s stands for the sum of 71 terms of an
arithmetical progression, evidently we may write the two
352 PROGRESSIONS.
following equalities, the progressions being alike except
written in reverse order :
s=:a-j-(ia + d) + ia + 2d')-\-(a-\-Sd)-{-. . .+a + (n—l)d (1)
^=/ + (/_^) + (/-2^) + (/-3^) + . . .4./_(«_iy (2)
Adding (1) and (2) together term for term, noticing
that the terms containing the common difference niilif}-
one another, we have
25=(«+/)4-(«4-/) + («+/) + (« + /) + . . . + («+/).
Since the number of terms in the original progression
has been called n, we write the last equation
2s=n(a + l),
whence the formula for s,
s=^n{a+l). [2]
396. Formula [1] enables us to obtain the value of /
when a, n, and d are given, or the value of «, when /, u,
and d are given, or the value of d when /, a, and n are
given, or the value of n when /, a, and d are given. Thus:
(1) Find the 20th term of 3 + 8 + 13 -f-. . .
Here a=3, d=5, ^^=20, therefore
/=3 + 19x5=98.
(2) Find the number of terms in the progresssion
5 + 7 + 9 + . . . + 37.
Here a=5, d=2, 1=37, whence
37=5 + (;z-l)2.
Solving for w, n=17.
(3) Find the common difference in a progression of 11
terms where the extremes are ^ and 30|-.
Here a=^, /=30^ and n=ll, whence
Solving for d, d=3.
ARITHIVIETICAL PROGRESSIONS. 353
(4) Insert 3 arithmetical taeans between 5 and 21.
Here n=5, whence
_ 13xGV) «-13
_ 13 -13000000 _12999987_ .
that IS, "^- 100000- lOOOOOO" 900000 -^^•4444o.
(3) Insert 3 geometrical means between 31 and 496.
Here a=31, /=496, and ?2=5, whence
496 = 31x^4,
or r* = 16,
therefore, r==fc2.
Consequently, the required means are 62, 124, and 248,
or -62, +124, and —248.
403. The two equations
_ar"—a
r—1
GEOxMETICAL PROGRESSIONS. 359
contain five letters. If any two of them are unknown
numbers and the values of the other three are given, the
value of the two unknown numbers can be determined
b}^ solving the system of two equations. But if r is an
unknown number, the equations of the system are of a
high degree, since n is usually a large number and
always greater than 2 at least. In this case we will be
unable to solve the system, as it is beyond the range of
Chapter XV. Also, if n is an unknown number, we will
have an equation with the unknown number appearing
as an exponent, which is a kind of equation we have not
yet considered. Hence there are a limited number of
cases in which, with our present means, we can solve
the above system. We give a few examples of the cases
readily solved.
(1) Find the sum of a geometrical progression of 7
terms, of which the last term is 128, the ratio being 2.
Here /=r28, r=2, and w=7, whence
( 128=a26 ' (1)
( '- 1
From (1) ^ = 2, whence, from (2), 5= 254.
(2)
(2) Find the sum of a geometrical progression of 5
terms, the extremes being 8 and 10368.
Here «=8, /=10oG8, and ;^ = 5, whence
( 103G8=8r^ (1)
\ r 10368 -8
From (1) r=6, whence from (2) ^=12450.
(3) Find the extremes of a geometrical progression
whose sum is 635, if the ratio is 2 and the number of
terms 7.
36o PROGRESSIONS.
Here ^=635, r=2, and n=l , whence
(/=^26 (1)
j 635=?^ (2)
Substituting /from (1) in (2), we get
635=128«— «.
Whence a=5 ; hence /=320.
(4) The 4th term of a geometrical progression is 4, and
the 6th term is 1. What is the 10th term?
Here «r^=4 (1)
and ar'^ = l (2)
Whence, by dividing (2) by (1),
^2 1
Whence, r=d=|-.
4
Therefore, from (1), ^=--=±32.
Then the 10th term is ±32(=bi)9=J^.
EXERCISE 140.
Examples and Problems.
1. Find the sum of 7 terms of 4+8+ 16+. .
2. Find the sum of 9 terms of 2 + 6 + 18 + . .
3. Find the sum of 7 terms ofl+4+16 + . .
4. Find the sum of 11 terms of 9 + 3 + 1 + . .
5. Find the sum of 10 terms of l + i+i+, .
6. Find the 10th term and the sum of 10 terms of
4-2 + 1-. . .
7. Sum the series 1^3 + 1^^6+1^12 + . . . to 8 terms.
8. Sum the series 3— 2+|— f +. . . to 9 terms.
g. Sum the series —4+8—16+32—. . . to 6 terms.
lo. Sum « + «(l+;r) + «Cl+^)2 + . . . to 8 terms.
EXAMPLES AND PROBLEMS. 36 1
11. Sum a-\ — 5^ + 7-^ — ?T2 + - • • to 10 terms.
12. Sum d(l-j-xy-^+d(l-}-xy-'^-{-. . . to ?2 terms.
13. Snmx"-^+x"-^y+x"-^y'^-\-x"-^y^ + . . to ;e terms.
14. Snmx"-'^—x"-y-j-x"-^y'^—x"-*y^ + . . to w terms.
15. Find r and s; given a=2, /=31250, ^=7.
16. Find rand s; given <2=36, /=^, n=7. ;
17. Find rand s; given a=S, /=49152, n=S.
18. Find rand s; given a=7, /=3584, 72=10.
19. Insert 2 geometrical means between 47 and 1269.
20. Insert 3 geometrical means between 2 and 3.
21. Insert 1 geometrical mean between 14 and 686.
22. Given /=78125, r=5, n=S ; find a and s.
23. Given /=^V, ^=i, 7z=5 ; find a and j.
24. Given 5=635, 72=7, r=2 ; find a and /.
25. Select 6 terms from the progression •^—2 + 8—. . .
whose sum shall equal —6536.
26. The sum of the extremes of a geometrical pro-
gression of 4 terms is 56, and the sum of the means
is 24. Find the 4 terms.
27. Insert 7 geometrical means between a^ and d^.
28. Given a—^, /=1024, ;2=14 ; find r and s.
29. Sum the series 2+22+P + 2^"^2^"^26"^' ' ' ^^ ^
' /I , 3 , ^ \ , /"l ^ ^ _!_ ^ \ 1 _L
terms, or the progression 1 2+22 "^2^/ \2 2^ '^2^/2^'^
to 3 terms.
30. Find the sum of the first 10 consecutive powers of 2.
31. Find sum of the first 10 consecutive powers of — |.
32. Sum the progression .272727 ... or TVTr+rffVTnr
+ Towtnrir+- • • to 6 terms.
362 PROGRESSIONS.
33. Sum a—ar~^-\-ar~'^—ar~^-{-. . . to n terms.
34. The 4th term of a geometrical progression is 192
and the 7th term is 12288 ; find the sum of the first 3
terms.
35. The 6th term of a geometrical progression is 150,
and the 8th term is 7644 ; what is the 4th term ?
36. Prove that if numbers are in geometrical progres-
sion their differences are also in geometrical progression,
having the same common ratio as before.
37. If a + d-\-c-{-d+. . . is a geometrical progression,
prove that (a^ + d'')-i-(id^ +c'-) + {c'^ -i-d^) + . . . is also a
geometrical progression.
38. A man agreed to pay for the shoeing of his horse
as follows : .0001 cents for the first nail, .0002 cents for
the second nail, .0004 cents for the third nail, and so on
until the 8 nails in each shoe were paid for. How many
dollars did he agree to pay ? How much did the last
nail cost him ?
CHAPTER XX.
BINOMIAL THEOREM.
EXERCISE 141.
The Lav/s of Exponents and Coefficients.
404. The Binomial Theorem enables us to find any
power of a binomial without the labor of obtaining the
previous powers; in other words, it enables us to obtain
any power of a binomial without actually performing the
multiplication.
405. Let us obtain several powers of x-\-a by actual
multiplication :
\st power ^ x+a
2d poiver^
Sd power,
4itk pozver,
x+a
x'^+ ax
ax-{-a"
x'^-\-2ax+a^
x+a
x'^ + 2ax'^+ a-x
ax'^+2a-x-{-x^
x^+Sax^ + Sa^-x+x^
x+a
x^ + Sax'^ + Sa^-x''+ a'^x
ax^+^a^-x^+Za'^x+a^
x'^ + 4ax^ + i]a^-x"-+4a'^x+a^
x+a
x^+Aax^+ {5a'^x^+ Aa^x'^
ax^+ Aa-x^+ 6a^x-
'+ a^x
'+4a^x+a^
hth power, x^ + 6ax'^ + 10a^-x'^ + 10a'Kr^ +da'^x+a^
When any power of x + a is written out in full, as in the
above, it is called the Expansion of that power of r: + a.
364 BINOMIAL THEOREM.
406. The Law of Exponents. We notice in the
expansion of the different powers of x+a that x appears
in each term except the last, and that a appears in each
term except the first, and that both x and a occur in
each of the other terms. We observe also that
THE EXPONENTS OF X FOLLOW THE EXPONENTS OF a FOLLOW
THE FOLLOWING SCHEME. THE FOLLOWING SCHEME.
In (x+ay 10 1
In (x+ay 2 10 12
In (x+ay 3210 0123
In (x+ay 43210 01234
In (_x+ay 543210 012345
It is a necessary consequence of the successive multi-
plication by x+a, that the exponents v/ill continue to
fall into the above schemes, for at each multiplication all
the power of x, likewise of a, are increased by 1 , and
there will alwa3^s be one term which does not contain x,
and always one which does not contain a ; therefore the
law of exponents :
I7Z any power of a binomial, x-\-a, the exponent of x
begins in the first term with the exponent of the poiver, afid
in the following terms co7itinually decreases by 07ie. The
exponent of a com?}tences with one in the second term, and
c'07itinually i7icreases by one.
407. Number of Terms. We observe that in the
first power there are two terms, in the second power there
are three terms, in the third power there are four terms,
and so on. In any power of x-\-a there is a term con-
taining each of the powers of x, as high as the required
power of ^4-^, and one term which does not contain x \
therefore,
The 7izt77iber of terms i7i any power of a bi7iomial is
always one greater than the ext>07ient of the poiver.
LAWS OF COEFFICIENTS AND EXPONENTS. 365
408. The Law of Coefficients. First Statement.
The coefi&cients* in any power of x+a are of course the
sums of the coefficients in the two partial products which
are added together to produce the required power. Now
the coefficients in each of the partial products are just
alike, and each partial product has the same coefficients
as the multiplicand, or next lower power of x-\-a to the
one we are finding. Thus the coefficients which are
added together to produce the coefficients in (^x-\-cC)^
are as follows :
18 3 1
13 3 1
14 6 4 1
The second coefficient in (x-\-aY is the sum of the
second and Jirst coefficients in {x+a)^, the t/ii'rd coefficient
in (x+a)* is the sum of the t/ii'rd and second coefficients
in {x-\-cC)^, i\iQ fourth coefficient in {x-\-d)^ is the sum
oi Xho: fourth and third coefficients in (x-\-aY, th& fifth
coefficient in {x-\-d)^ is the sum of the fifth (which is 0)
and fourth coefficients in (x-^a)''. Writing down the
coefficients in the different powers of :r+a, we may present
this same truth in a little different form :
Coefficients in {x-\-aY 1 1
Coefficients in {x-\-ay 12 1
Coefficients in {x+aY 13 3 1
Coefficients in \x-\-ay 14 6 4 1
Coefficients in {x+aY 1 5 10 10 5 1
By arranging the coefficients in this triangle we
may say that each coefficient is the S2im of the coefficient
immediately above it aiid the coefficient imtnediately to the
left of this last.
* By the coefficients in any power of ;r+a is meant the coefficients of the powers
ofx and. a. This lan.ernage is not exact, (see Art. 14.) but it is c<:stomary.
366 BINOMIAL THEOREM.
Since the partial products will continue to be combined
in the way we have noticed if we continue to multiply
by x-\-a, we may extend the above triangle indefinitely
and get the coefficients in any power oi jt+« we wish:
Coefficients in (x-{-a)'^ 1 1
Coefficients in (x-\-ay^
1 2 1
Coefficients in {x-\-ay
13 3 1
Coefficients in {x+ay
14 6 4 1
Coefficients in {x+ay
1 5 10 10 5 1
Coefficients in {x-^a)^
1 6 15 20 15 6
1
Coefficients in {x-\-ay
1 7 21 35 35 21
7 1
Coefficients in {x-\-ay
1 8 28 56 70 56
28 8
409. The Law of Coefficients. Second Statement.
It is more usual to use a law of coefficients different from
the one given in the previous article. It is found that
fhe coefficient in any term in a power of a binomial can be
made from the coefficient and the two exponents in the pre-
ceding term. Thus consider "Oa^ fourth power oi x-\-a,
x^-h4ax^-hQa'^x'^ -h4a^x-^a^.
The first coefficient is 1, and the second coefficient is 4,
the exponent of the power; the t^ird coefficient, 6, is the
preceding coefficient, 4, multiplied by 3, the exponent
of X in the second term, divided by 2, one more than the
exponent of a in the second term. Also, 4, the fourth
6x2
coefficient, is — o~> where 6 is the coefficient in the pre-
ceding term, 2 is the exponent of x, and 3 is one more
than the exponent of a; and so on.
lyikewise in the fifth power of x-\-a,
x^ + bax^-]-lOa'^x'^ + 10a^x^-\-bax^+x^,
the coefficient in the first term is 1 ; the coefficient in the
second term is 5, the exponent of the required power ;
the coefficient in the third term, 10, is the coefficient in
LAWS OF COEFFICIENTS AND EXPONEN IS. 367
the preceding term, 5, multiplied by 4, the exponent
of X, and divided by 2, one more than the exponent of a;
10, the coefficient in the fourth term, is similarly — ^-v-;
o
10x2
5, the next coefficient, is — ^ — J ^"^ 1» the last coefficient,
is -TT . The next coefficient would be —77-.
b
After treating any other of the first five powers in the
same way, we would find,
In any of the first Jive powers of x-\-a, the coefficient in
the first term is 1, that in the second term is the expo7ient of
the power y and if the coefficient in any term be multiplied by
the exponent of x in that term and divided by the exponent
of a increased by one, it will give the coefficient in the suc-
ceeding term.
410. The law stated in the last article may be observed
to be true in, any of the five powers of Jt-f-^ that we have
actually worked out ; it now remains to prove that the
same law holds for ^//powers oi x-\-a.
lyCt 71 stand for a positive whole number, and suppose
we wish to find {x-\-d)''. We know from Art. 406 that
the terms without the coefficients will be as follows :
x*\ ax"-\ a'^x"--, a^x"-^, a^jtr"-*, etc.
Now, if the law stated in the previous article does hold,
we would write (x-{-a)"=
i^this is true, by Art. 408 the coefficients in (x-tay^^ are
n(7i-l). ;^(;^-l)(7^- 2) n(n-l)
1, ^ + 1. 2 ^ ' 2xS 2 '
n(n-lXn -2Xn-S) n (?i-l)(7t-2)
2x3"x4 "^ 2x3
368 BINOMIAL THEOREM.
or, removing a common factor from the third, fourth,
etc., of these expressions, we would write them
nCn-lXn-2){^^+^), etc.
or, performing the additions in the large parentheses,
1, n + 1, — ^-, 2^ ,
(?g+l)^gOg — 1)0?— 2)
2x"3x4 ' ^^^•
or, supplying the powers of j«;and a for the (n+1) power
of x+a by Art. 406, we get
Z Z "K o
+ 2^3x4 « -^ +■ • • (^)
Now we know that equation (2) is true if equation (1)
IS i7'ue. But equation (2) is of exactly the same form
as (1), merely having {n-{-V) in place of n, each coeffi-
cient being obtainable from the coefficient and exponents
of the preceding term by the law of Art 409. Therefore
we have proved that the law of coefficients of Art. 409 holds
in the {7i-\-V) power of x-\-a if it holds in the nth power
of x-\-a. But we know this law of coefficients holds in the
5th power of x-\-a ; therefore it follows that it holds in
the 6th power. Now we know this law of coefficients
holds in the 6th power of x-j-a, and therefore it holds in
the 7th power; therefore it holds in the 8th power, and so
on. Therefore it holds universally.
Thus we have proven the law of Art. 409 holds for
any power of a binonial.
LAWS OF COEFFICIENTS AND EXPONENTS. 369-
412. The Statement of the laws of exponents and
coefficients for any power of x-\-a is called the Bino-
mial Theorem, and is usually given as follows :
I. Exponents. In any poivcr of a binomial, x+a, ihe
exponent of x begins in the /irst term with the exponent of
the power, a7id in the folloiving terms continually decreases
by one. The exponent of a commences with 07te in the sec-
ond term of the power, a)id continually increases by 07ie;
II. CoKFFiciENTS. The coefficient iii the first term is
one, that in the second term is the exponent of the power;-
and if the coefficient in any term be multiplied by the expo-
nent of X in that term and divided by the exponent of a
increased by one, it will give the coefficient in the succeeding
term.
413. The expansion of (jr=b«)" is usually called the
Binomial Formula.
If in equation (1), Art. 400, we substitute ±a for a, we
get the following as the expansion of (jrdra)":
[x±a)"=
., , , n{n—\) o „ , . n{n—\)[n—2) , ., ,
ac" ± nax"-^ -\ — -a-x"-- ± — — ^^ -a'^x"-^
+"-^r^^«*--* . . . t.i
Therefore, in any power of the difference of two num-
bers the sign of the first term is +, of the second —, and
so on, alternately + and — .
414. We will now give a few examples of the use of
the binomial theorem.
(1) Expand {a^-by.
We may expand this at once by the theorem as follows:
flC+6a5^+15«4^2^20«^^3 + 15^-/^^ + 6^/^^ + <^6^
or we may substitute x^a, a—b, and «=G in formula [1]
21
370 BINOMIAL THEOREM.
.0x5x4x3x2 ,., 0x5x4x3x2x1,,
2x3x4x5 "^ 2x3x4x5xG
which reduces to the same result as before.
(2) Expand (z^ + 3_y)^
Here x=il and a='^y. By the theorem we get
Performing the indicated operations, we get
?r^ + 15?^V+^0^^V^ + 270/^2^3 _|_405?/;/'*-f243j'.5.
(3) Expand (^^-2)4.
Here x^r"- , a=—2, and ;^=4. By the theorem
Performing the indicated operations, we get
5K8_8r6 + 24r-*-32r2^1G.
. (4) Expand (2^-iy.
Here x=3^, «=— |, 7^=3. By the theorem,
(3^)^-3(3<^)2(i)4-3(3^)(i)2-(i)«.
Performing the indicated operations, we get
27^^-V^-+|^-i.
EXERCISE 142.
Examples.
Expand each of the following by the binomial theorem
or formula :
1. {a+xy\ 5. {\^ay\ 9. {m'+zcy,
2. {b+xy. 6. (2+xy, 10. {a-xy,
3. {d-\-yy, 7. (2-A-)*. II. (5flr-3;0^
4. (^+;r)«. 8. (i+ji-)^ 12. (a^-b'^y.
EXAMPLES. 371
13. i-x+2ay. 17. (d'^-c^-)^. 21. (2/i''-Bx-^y.
14. (2x+Say. 18. Gr+2r)*, 22. (3;i;2_l)4.
15. (1-xy. 19. (Sa+iy. 23. (T/«+;r)«.
16. (1— rt)8. 20. (2^zjir-.r2)4. 24. (2^-i/i)5
25- G'Tj'— :«:-)^ 31. («— <^+:r— 2)3.
2 3.
26. (]/^^— l?^rt/^)». 32. (jtra+.r^)^
27. (^ + [:r+;/])3. 33. («-2_^i)4^
c8. (la-i-d]-2y. 34. (.r2 4-2«j»:+rt2)3.
29. («+^-ji')'. 35. {y'c''-2xy\
30. ([a4-/^] + [^+^])^ 36. {^-^■^)'-
THE END
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