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BIRCHARD, M.A., PH.D., Mathematical Master^ Collegiate Institute^ Brant/ord, TORONTO: WILLIAM BRIGGR, WESLEY BUILDINGS, 29 TO 33 Richmond Stkekt West. V. I \\ ■ "■3"4' J toO /■ PEEFACE. TBA0FRB8 of mathematics have for some time felt that the Algebras now in use in our High Schools and Collegiate Insti- tutes are not adapted to the wants and requirements of the present day. In these works some of the most important departs ments of Elementary Algebra, such bb Factoring. Symmetry, Theory of Divisors and Theory of Equations, are treated so briefly or so superficially that the pupil has found it impossible to obtain a satisfactory knowledge of these subjects without drawing heavily on the resources of the teacher. In the following pages an effort has been made to treat with considerable fullness the various departments either deficient or wholly absent in the ordinary t^xirbooks. While no branch has been sUghted, special attention has been devoted to the Theory of Positive and Negative Numbers, to Factoring, Surds, Sym- metry, Theory of Divisors and Theory of Quadratics. Convinced that a large and well-graded selection of problems is a desidei- atum in any manual intended for class work, we have selected and constructed with great care such as we hope will meet the wants of both teachers and pupils. An effort has been made to secure accuracy, but it is quite possible that errors may have crept in despite our vigilance. We shall be glad to have such pointed out so that they may be removed from subsequent edi- tions. Some difference of opinion may exist as to the propriety of the order in which the different subjects have been introduced. IV PREFACE. t The treatment of Symmetrical Expresrions and the Theory of Divisors has been delayed until the pupil has acquired consider- able familiarity with algebraic symbols and their manipulation. We trust that this arrangement wiU commend itself to the expe- rience of the mathematical teachers of the country. Surds and the Theoiy of Indices have been introduced before Quadratic Equations, as we did not think it possible to deal satisfactorily ^th the Theory of Quadratics without some knowledge of Surds. An effort; has been made in this work fo encourage pupils to re- sort; to factoring as much as possible in solving equations. In pursuance of this object at a very early stage equations of an easy character, capable of solution by resolution into factors, have been introduced. This Algebra is intended for all classes of pupils whose studies do not extend beyond the limits prescribed for Second Class Cer- tificates and Pass Junior Matriculation. Having in view the fa^t that pupils of very difierent algebraic attainments study within these limits we have so graded the problems that the judicious teacher can easily select such as are adapted to a junior pupil, and leave those more difficult until he becomes well ad- vanced in his work. It should, however, be understood that many of the problems in Factoring and Symmetry go beyond the • requirements for Second Glass work. Should this venture prove successful it is the intention of the authors to foUow it up with Part; II., which will deal with the subjects required for Junior Matriculation with Honors and for First Class "C» Cert;ificates. July, 1886. W. J. ROBERTSON, L J. BIRCHARD. CONTENTS. Chaptbr L Definitions and Explanattons of Signa '*** « Algebraic Notation ..'.[.. i. Quantity and Number .* ! .^ 17 Algebraic Numbers [[[][ ] Chapteb n. Addition— Subtraction— Use of Brackets on Addition ^ o.t...... .s...,, 2Q Addition with Literal Coefficients q^ Subtraction "*'" ' ^ Subtraction of Polynomials 00 Brackets 29 Craptbb III. Multiplication Multiplication of Polynomials 04 Powers of a Binomial [[[ Chapter IV. Division Homer's Method of Division f? 49 CuAxrrxB V. Factoring Monomial Factors L* Trinomials ....'."...!.......... gg Complete Squares *.!!!.!!!! ka Difference of two Squares * _- Sum and Difference of Cubes ^ Trinomials ** 61 Chapter VI. Simple Equations of One Unknown -- Problems producing Simple Equations of One Unknot .' .' .* .' * [ [ ] [ [ [ [ , -, <, stand for the words "is greater than » "IS equal to," "is less than," respectively. The signs .-. and •.• stand for the words "therefore" anr^ "because." 4. The sign +, plua, written between two symbols, signifies that the numbers which these symbols denote are to be added. Thus 7 + 5 = 12, read 1 plus 5 is equal to 12; a + b read a plus b, signifies that the numbers denoted by a and 6 are' to be added. But unless we know the numbers for which they stand there is no other way of expressing their sum. A similar remark applies to subtraction, multiplication, etc. 5. The sign -, minus, written between two symbols, signifies that the number denoted by the second is to be subtracted from the number denoted by the first 2 10 DEFINITIONS AND EXPLANATIONS OF SIGNS. IM I Thus 7-5 = 2, read 7 minus 6 is equal to 2; a-b, read a mvnua b, signifies that the number denoted by 6 is to be sub- tracted from the number denoted by a. The sign ~ is sometimes used to denote the difference between two numbers when we do not know which is the greater. 6. The sign x , called the sign of multiplication, wKtten b©. tween two symbols, signifies that the numbers which they denote are to be »jiulii].lied together; thus axb, read a into b, signifies that the number denoced by a is to be multiplied by that denoted by b. The sign, however, is usually omitted between two letters, or between a figure and a letter; thus ab, 36 mean the same as axb, Zxb. A point is sometimes used instead of the sign x , especially when several numbers expressed in figures are to be multiplied together; thus 3.4.5 means the same as 3 x 4 x 5. The numbers multiplied together are called the Factors of the product. 7. The sign -;-, called the sign of division, written 'between two symbols, signifies that the number denoted by the fonner is to be divided by that denoted by the latter; thus a-i-b, read a by b, signifies that the number denoted by a is to be divided by that denoted by b. The line between the points is sometimes emitted, thus o:6, and sometimes the points are omitted and the symbols written in their places, thus ~. b 8. When two numbers are multiplied together each is called a Coefficient of the other. When one factor is expressed in figures and the other by letters the former is considered the coefficient. Thus in 76, ai; 7 and a are the coefficients of b, A coefficient denoted by a figure is called a numerical coefficient'; by a letter, a literal coefficient. When no numerical coefficient is written 1 is always understood. NoTK.— It is customary, in order to avoid cumbrous phraseology, to use the word "number" when we mean "symbol denoting number "; also we Bay " the nuniber a " when we mean " the nM?T^bc,r denoted by a." DEFINITIONS AND EXPJJLNATIONS OP SIGNS. ll 9. A Power of a number is the product obtained by multi. plymg It by itself any number of times. Thus axaxa, or aaa, is caUed the Third Power of a. The Second and Third Powers are also caUed the square and cube respectively. 10. An Exponent, or index, is a small figure placed above and to the nght of a number to show the Power to whrch the number is to be raised. ^us a' (read a to the fourth) is a short way of writing aaaa When no exponent is written 1 is understood. 11. A Root of a number is one of two or more equal factors whose product equals the given number. 12. The Index of a Hoot is a figure which shows how many equal factors ar_e to^ found. The roots of numbers are indi- cated thus : Va,Va,^a, etc., which denote the second, third fourth roots, etc., of a. The second and third roots are usually called the square and cube roots respectively. The index 2 is usuaUy omitted from the sign for the square root. The sign v^ is a corruption of the letter r in radix, and ia called the radical sign. 13. One or more numbers represented by algebraic symbols is called an Expression. o j « Thus 5a, 3a\ 6x - 7y, etc., are algebraic expressions. Tems^^^ """'^^'' connected by the signs + and - are called 15. Like Terms are those which difi-er only in their numerical coofficien.^; thus 3a6 and 5a6, ia^y and 7aPy are like terms, but 6ax and 5ay, 3a»6 and 5a6» are unlike terms. 16. An expression consisting of one term is called a Mono- mial ; an expression conr isting of two terms is called a Binomial • an cxpreiiiou consisting of three terms is caUed a Trinomial • Si I ft! Ill 12 DEFINITIONS AND EXPLANATIONS OF SIGNS. an expression consisting of four or more terms is called a Multi- nomial or Polynomial. The word Polynomial, however, is fre- quently used to denote any algebraic expression except a Mono- mial, 17. The Dimensions of a term are the literal factors in it; the Degree is the number of such factors. Thus ia^b'o is a term of 2-i-S-!-l = 6 dimensions, or of the sixth degree. a, 18. An expression is HomogeneoUS when all the terms are of the same number of dimensions. Thus a^ + 2ab + b^ ia homogeneous and of two dimensions. 19. A polynomial is said to bo arranged according to the powers of a letter when the exponents of that letter in the various terms are in order of magnitude. Thus 3a' + ia'x - baa? + ear* is arranged in descending powers of a, but in ascending powers of x. 20. The signs (), {}, [], called Brackets, signify that the terms enclosed form a group, which is to be treated as a single term. Thus a-{b-{-c) signifies that the sum of b and c is to be taken from a; {a-{b-^c)Y sigiufies that the former result is to be cubed; [m - [a - {b + c)}]y signifies that the sum of b and c is to be taken from a, the remainder subtracted from m, and this last remainder multiplied by y. A line, called a Vinculum, drawn over a n umber of terms, is sometimes used instead of a bracket • thus a-b + c signifies tl»e same as a - (6 + c). Examples. — If a = 1 2, 6 = 4, Then a6 = 12x4 = 48; 1 = 1? = 3; 7a'=.7x 12x12 = 1008; a'- (a-6)^= 144 -64 =--80, */«> k/\AA CiCt ■ 1 ^ . //> i n ALGEBRAIC NOTATION. * 18 terms are 1. 2a + 3b-e. 3. a? + ab + b\ 6. c* + 2cd + cP. 7. 3b^-ia' + 7^. 9. B(a + by-(m + e)\ 11. (a + b){c + d). BXBROISB I. JlZ^^' *'2 c = 3. rf=4, <, = 5, m-0, find the value of the tollowmg expressions : 2. 2c + 3rf4-4e-7. 4. bc + cd+de. 6. abc + bcd + cde. 8. We^-a%~m^, 10. 4e«-{(c;-c)«+36'}. 12. «(a + 6cc? - «»)> + {e - (cf - 6)«}«. Prove the following equalities :— 13. a + 6 + c + c?=^rf«. 14. -/"^T^^a. 15. i?^?T^rr;?=^cd 16. ^io75TSi7i;»«^, 17. «' + i» + c» + e^» + e«=(a + 6 + c + ,; + g)i.. 18. S(<' + ^ + «)' = cV + e) + rf»(« + c) + e»(c + d). 20. 2a»6»4-2W + 2c»a»-a«-6*>c«=ww?. 8«' •<■ 36' 4c» + 66« c« + 21. -i^^cf*. 22. If a=16, 6:.10. a: = 5 and y=l, find tiie vaJue of {h-x){^a^b)^^{i^a-b){x^y))^ andof (« - y){ V2bx + «»} + v'{(a - aj)(6 + y)}. ALGBBRAIO NOTATION. 21. The following examples are designed to furnish additional exercise m Algebraic Notation. It is very important tW^^^^ student should be able t. express the various rthtatitu^^^ t;ons and the operations to which numbers may brstbiected either in words or in al^ebmin «^k.u ._. .. / subjected, IfttA fjio «„^ * i. "^ j-—^^<=, oxiu w acouraUiiy trans- late the one form of expression into the other. \ I 14 ALOKBRAIO NOTATION. Example L— Express the following statement in algebraic symbols :— The diflFerence of the squares of any two numbers is equal to their sum multiplied by their diflFerence. Let a and b represent the numbers, a being the greater. Then a» and 6> represent their squares; o»-6» represents the diflFerence of their squares; a + b &nd a -b represent their sum and their diflFerence. .'. a« - J« = (a + b){a - 6) is the expression required. Example 2.— A pedestrian having agreed to walk a miles in h hours travels the first k hours at the rate of m miles an hour. At what rate must he travel the remainder of the time? In k hours at m miles an hour he would travel k times m miles, t.e., km miles ; a - Am is the remaining distance in miles -, h-k is the remaining time in hours. .-. a -km divided by h-k, a - km . T _ » - M the number of miles an hour required. ».«. EXBROISB II. Express hi algebraic symbols the following :— 1. The sum of any two numbers. (Use a and b). 2. The sum of the squares of any two numbers 3. The square of the sum of two numbers. 4. Six times the product of two numbers. 5. The sum of the cubes of two numbers divided by the sum of the numbers. 6. The square root of the sum of the squares of two numbers. 7. The square of the sum of two numbers is equal to the sum of their squares together with twice their product. 8. The diflFerence of the cubes of two numbers, divided by the diflFerrnce of the numbers, is equal to the sum of the squares of the numbers, together with their product. 9. flow many cents in x dollars? How many dollars in x cents. ALGEBRAIC NOTATION. I5 10. How many inches in x feet and y inches 1 In x yards and y feet 1 11. Find the cost of 7 hats at x dollars each. 12. A man gives x dollars in payment for 5 books at a dollars each and 7 books at h dollars each ; how much change should he receive % 13. How far will a person travel in x hours at y miles an hour ? 14. How long will it take to travel a miles at x miles per hour! 15. A man works q hours a day for n days and p hours a day for m days. He receives x cents per hour; how many doUars does he receive altogether ? 16. Find the sum of a; + a: + a; + .... where x is written m times. 17. A flower bed is x feet long and y feet wide ; how many square feet in the bed ? - 18. A block is X feet long y feet wide and z feet thick • how many cubic feet in iti How many square feet on all the faces? How many feet in the sum of the lengths of all the edges? 19. A book has X pages ; each leaf is y inches long and z inches wide; how many square yards of paper in the book? 20. From a rod x inches long I cut off y inches, and divide the remamder into m equal parts; how many inches in each part? 21. A boy is X years old and his brother y years; find the sum of their ages after five years. 22. What number subtracted from x will leave 10? 23. What number subtracted from x will leave y? 24. The dividend is a: and the quotient y/ what is the divisor? 25 The divisor is x, the quotient y, and remainder r; what is the dividend ? X What value of x will make 7x eaual to .^.5 7 omiol f/% 1^0 ?nal to 3.-X X X + 2 equal to 20 ? 3a: + 5 equal to 26 ? 4a:' greater than 7 by 29 ? 16 ALGEBRAIC NOTATION. I 27. The number oj is to be increased by 3, twice the sum is to be multiplied bj a + 6, and the product, diminished by d, is to be divided by the sum of m and n. Express these operations alge- braically. ' ° 28. A grocer mixed a pounds of tea worth x cents a pound with h pounds worth y cents a pound; what was the mixture worth a pound 1 29. a: -f-y houses have each a + 6 rooms, and in each room are p-\-q persons ; how many persons in all % 30. There are a^ + y rows of trees, x + y trees in a row, on each tree x + y bushels of apples worth x + y cents a bushel; how many dollars' worth of apples in all t 31. A man having m dollars buys x pounds sugar at a cents a pound and y pounds of tea at h cents a pound ; how many pounds of coffee, at c cents apound, can he buy with the remainder of his money ? 32. A man divided x dollars among m boys and y dollars ' among n girls. Two boys and three girls put their money to- gether and bought p pounds of candy; how many cents a pound was the candy ? 33. A man lua a journey of x miles; he travels a hours at h miles per hour and c hours at d miles per hour; how long will it take to finish the journey at y miles per hour ? 34. A train, having to make a journey of x miles in h hours ran for k hours at the rate of r miles an hour and then made a stop of m minutes; how fast must it run during the remainder o'" its journey to arrive on time 1 35. Two cities are m miles distant from each other; two travellers start at the same time, one from each city, and travel towards each other at the rate of x and y miles, respectively, per hour. How long before they will meet? How far wHl Lch travel 1 QUANTITY AND NUMBEB, IT oom are QUANTITY AND NUMBER. 22. A Quantity is that which is capable of being divided into TLus Distance, Time, Weight, etc., are Quantities. 23. A Quantity is measured and its magnitude estimated by selecting some known, definite Quantity of the same kind as a Standaj-d or Unit, and then finding by trial how many times this unit must be repeated to mak. up the given Quantity. 24. Number arises from considering the repetitions of the unit necessary to make up a given Quantity. CntZf^" ^^u^"" '"^ connection with a particular unit is called Uoncrete ; without a particular unit. Abstract. Thus the maff- mtude of a Quantity is represented by a Concrete Number. 25. When two Quantities are so related to each other that being teJcen together, they cancel or destroy each other, eitho; L".h ''^ ^""'' """" "^ '^""^ ^ ^^^^^ ^ Positive Quantity, the other a Negative Quantity. ^' 26. The preceding Arts, may be illustrated as follows :— -e -6 -4 _8 -1 b I +1 +2 +8 +1 ?6 +6 1. Draw any straight hne A/; its length will be a Quantity. -. To measure that Quantity select any length Aa for a unit, and mark off portions each equal to the unit Aa 3. The numbers 1, 2, 3, 4. 6, 6, in connection with unit Aa i^present he d^'stances of the points ., 6, c, c/, ,/ from T ' 4^ Similarly, if the line be produced to/, Af may be measured n the same way, and the numbers 1, 2, 3, 4, 5, 6, will represent the distances of the points a', h', c', c/', e',/, Wm A. ^ 0. It a point move from A through any number of units of distance U> the right, and then through the'same number of uni J to the left, its distanoA fmrv, a .^u u ,, having cancelled each other ; therefoi-e f ; r I i il 18 ALGEBRAIC NUMBEBS. 6. If distance to the right be called a positive Quantity, dis- tance to the left IB & negative Quantity. 7. Positive Quantities are represented by numbers with the sign + before them, negative Quantities by numbers with the sign - before them ; hence the signs ^. and - are called the positive and the negative sign. ALaSBRAIO NUMBERS. 27. Numbers taken in connection with the signs + and - are called Algebraic Numbers— the former, Positive Numbers- the latter, Negative Numbers. Without any sign they an^ called Absolute or Arithmetical Numbers. 28. Two algebraic numbers, the one preceded by the sign + the other by the sign -, are said to have unlike signs. In practice th« sign + Is usually omitted, and the sign - is used to signify that the number before which it stands represents a Quan- tity of a nature opposite to some other Quanf y previously con- ndered, and which was represented by a number without regard to sign; hence absolute numbers are tacitly considered to be posi- tive, and when no sign is written + may always be understood. 29. Two Quantities, one positive the other negative, containincr the same absolute number of units, have the same magnitude, or m other words, a negative Quantity is as large as the correspond- ing positive Quantity: the signs -h and - have nothing to do with the magnitude of a Quantity. 30. One number is said to be algebraically greater than an- other when in a scale of numbers, as in Art. 26, it lies in a post- tive direction from the other. Thus - 2 is said to be algebraically greater than - 6, greater than - 2, etc. This is only a convenient way of speaking, and so long as the moaning of such expressions is clearly defined no confusion can arise. It would, of course, be absurd to consider either a number or A Onn.nfif.iT aa »»qo;-v» ^n^„ iU __j.i-* -- — -w J "~ »\/MwiJf xvioo Xiuuiit. uOWiUlJf, M l( ALOEBRAIO NUMBEBS. I9 31. The following examples show how algebraic nnmbers are appHed to represent positive and negative Quantities. To find the numbers which represent the lines 6/, ea, a'f, e\ in magnitude and direction. From 6 to / the distance is 4, the direction positive. ''^' " " 8, «« " positive. Hence the lines are represented by the numbers +4,-4 -5 + 8, respectively. * * * In these examples the signs + and - denote direction; in other applications their signification will be readily perceived. N0TB.-The word "quantity" is frequently used inatead of "alw. braical expression." When used according to definition, Art. 22. it will be written with a capital, otherwise without. » "' EXBIROISB IIL 1. If a line 8 inches long be the unit, what number will repre- sent 22 yards ] a quarter of a mile 1 2. The number 25 represents half a ton, what is the unit] 3. A certain distance is represented by 36 when the unit is 2 feet 6 inches; what number will represent the same distance If the unit be changed to 10 inches? to 40 inches? 4 The sum of the lengths of the edges of a cube is represented by 36, and the unit of length is 7 inches: find the numbers which represent the area of a face and the volume respectively. 5. If 5 miles to the east of any place be represented by + 6 what will - 5 represent ? ' 6. If a tree 100 feet high be represented by +20, what wiU correctly represent the depth of a well 40 feet deep? 7. If cash in hand be represented by positive numbers, what will negative numbers represent ? 8. If we denote a pound weioht bv j. 7 -i.ri,„<. „An j„_-x- ., force of a baUoon Ufting 100 pounds? r OHAPTEE II. I ADDITION-SUBTRAOTION-USB OP BRACKETS. ADDITION. 32. The Sum of two or more algebraic numbers is the single number which correctly represents the Quantity formed by com- bining the Quantities represented by the several numbers. The process of finding the sum is called Addition. 33. The signs + and -, when used to designate positive and negative numbers, are distinguished from the same signs when used to indicate the operations of addition and subtraction by enclosing them with the number in a bracket Thus ( + 4) + (-3) indicates the addition of 4 positive and 3 negative units ; ( + 4) ~ ( - 3) indicates the subtraction of 3 nega- tive from 4 positive units. 34. Let it be required to perform the foUowing additions;— 1. (f5) + ( + 2). 2. (-5) + (-2). 3. ( + 6) + (-2). 4. (-5) + ( + 2). Referring to the scale of algebraic numbers, Art. 26, we see that 5 positive units and 2 positive units make 7 positive units just as 5 inches and 2 inches make 7 inches. Similarly, 5 nega^ tive units and 2 negative units make 7 negative units. To ^d 6 positive and 2 negative units we start from A, the zero point, move 6 units to the right, and then 2 units to the left; this leaves us at c, whose position is denoted hi +3. Similarly, moving 5 units to the left and two units to the right brings us to c' whose position is denoted by - 3. We have then the following results :- 1. ( + 5) + ( + 2)=+7. 2. (-5) + (-2)=-7. 3. ( + 5^ + (-2)= +3. 4. (-6) + ( + 2)= -3. ADDITION. fl 35. To add algebraic numbers we have from Art 34 the fol lowing RuLB—ra^ the sum of the absolute values of numbers havirw hke sxgns and prefix the common sign. Take tlte difference of the absolute values of numbers leaving unlike signs and prefix the sign of the greater. ^ ^ 36. If there are several numbers to be added we may proceed m the same way until all are combined into a single number It may be assumed, however, that the result will be the same in whatever order the numbers are taken, and it will be found more convenient to add the positive and the negative numbers sep^ rately and then to combine the results. 37. If we divide each of the units of length (Art. 26) into m parts the lines there denoted bv 2 3 4 9 i I ^ :ii 1 J . , , ^ .7 I "> ^1 ■^j — o, — 4, etc., w^U be denoted by 2;^, 3m, 4m, - 2m, - 3m, - 4m, etc.; and b^ the same reasoning as before we shall get such results as 1. ( + 5m) + ( + 2m)= +7m. 2. (-5m) + (-2m)= -7m. 3. ( + 5m) + (-2m)= +3m. 4. (-5m) + ( + 2m)= -3m. From which we learn that Like terms are added by taking the algebraic sum of their coefficients and annexing the common literal factors. The addition of unlike terms can only be indicated by con- nectmg the terms with the proper signs. 38. In practice the brackets used to distinguish the different uses of the positive and negative signs are omitted, and the ex- pressions ( + a) + ( 4- J) and (+«) + ( -6) are written a + 6 and a- b The latter expression has already been defined to mean that b IS to be subtracted from a. So long as a is greater than b It IS evident from Art. 34 that the two interpretations give the same result, and it will be shown hereafter (Art. 44) that this is always true. From these considerations we deduce two conclu- sions : — 1. An algebraical expression, being the sum of the several 22 ADDrnoN. algebraical numbers oomposing its terma, is itself an algebraical number, and may be treated as such. 2. Tho terns of an algebraical expression may be arranged in Z^tlerm's!" ^"''" " "^ "^'"^ "^^^^^ '^ — -« 39. The addition of algebraical expressions is indicated by r^ 7 ^J^"^-"" '"r^^"^' '^^ ^"^^ ^y i*« P^°P«r sign ; and the addition is performed by collecting like terms, aid tiu^ reducing the expression to its simplest form. ^x— Add +3a, +26. -56, +7a, +b, ~ia. 3a+26-56 + 7a+6-4a = 3a + 7a-4a + 26 + 6-56 = 6a-26. BXBBOISB IV. Add the following : — 1. +7, +8. ' 3. +12, -3. 5. -7, -20, +3. 7. +5a6, -3a6, -7a6. 9. +4< -3n« -10n« +8m», -2w». 10. +2a, +36, -26, -5a, +7c, +rf. 11. 4a:+(-3y) + ( + 2a;) + ( + 3y) + (-7«) + (_24 12. 5aj-7y-3y-2aj + 4a;-3y + 2a. 13. +7a»+(-36») + (-4a») + (-5a6) + ( + 46'). 1885'A^n'^w^^'^'^'*^ ^'^^"'^ *^^ ^^*«« 4^ B-0. and 1885 A.D. What dates are 25 years after each of them 1 15. A person travels 20 miles and then returns 15 miles E- press his journeys algebraicaJly aad, add them : add them arith- metacaUy. What is taken into account algebraically which is neglected arithmeticaUy? ^y wnicn is he^wes $2000 to one man and $2500 to another; find^ ^et 2. -7, -8, 4- -17, +7. 6. +13, -20, -8. 8. -8< -3< +20»»«. ADDIITON, f8 W. When two or more polynomials are to be added it is oon- vement to arrange the terma in columns so that like terms shall stand m the sa^e column. The columns are usuaDy added Tn succession, beguuung at the left ; but they may be taken in any orde., care being taken to prefix the proper sign to the sum of each column as it is written dowix. Examples. 1. ic^+Zxy-2i/ 2. 2a -36 + 4c -3^+7a:y + 3y» 5a-76 + 2c 5^-2a^+ y> -5a + 36-6c + a, iotP-xy 2a-.% + x Add BXBROISB V. 1. 2a + 3J + c, ia-b+2c, -2a + 5b + io, 2. 4a + 56-6c, 46 + 5c-6a, 4c + 5a-66. 3. 7a-46-3c, 26-5c-4a, 6a-116 + 4c. 4. 2a6 - 36fl + 4ac, 56c - 3a6, 4ac - 76c. 6. Zx»~.5xi,-2y^, 2a?y + 3/, 7ar-4a^-6^. 6. a + 6-2c, b + c-2d, c + d~2a, d+a~2b. 7. 7(a: + y), 4(a; + y), -5(a; + y). 9. SaC-^-y), 4a(a:-y), -9a(a:-.y), a(a;-y). 10. (m + n)^ + a;, 2(m + n)Hy, «-5(m + n)« o(2a-36)-9c-c?. 12. 9(a:^ + y)-3xy, «:» - 7a;^ + 3,', lOory- 10(a:» + y.) 13.7a-36 + 5c-10c^, 26-3c + ^-4, 5c - 6a - 4. + 2 iq WW u *'^'^""*-^)*-(«-* + -)y-(ft-« + a>. J 3 . T" o ' ^^''^^^t^d from the sum of 4^^ + 3.rV - v» 4a^-v/-3a-. 70:^4- V_2a.V. to leave the remainder 2a:3 - 3^V + ^ fy*. BRACKETS, ■.rlt Tll^^"^'"'- '""' ^»''*'*'=«°» of polynomials fa frequently md,cated by enoloamg the expression to be added to, or sublS fron> another expression, in a bmoket, preceded by the sig^ I or^ 50. To add a polynomial to another expression we add ea.h proceed as m addition; hence a bracket p.eceded by the rion - n-il'ir^Sir ^m:;';^:""^ "'-"'^'^ '^'"^ -^ »- ^^ "■« ij I 80 BRACKBTEk 51. If two or more pairs of brackets be used in the same ex- pression they may be removed, one pair at a time, by the pre- ceding rules. It is easiest to begin with + he inside pair, but we may begin with any pair, and a little experience will enable the student to remove several pairs at one operation. At each step of the operation like terms should be combined, to save labor in writing. Ex,i. a-{b + {c-d)}=a-{b + c-d} = a~b-e + d. Ex.2. «-[6-{«-(6-a)-6}-a] = a-6^-(a-(6-a)-5}+a '=2a-b + a-{b-a)-b = 3a - 26 - 6 + a = 4a - 36. In Ex. 2 the outside brackets were removed each time and like terms combined. EXERCISE IX. Remove the brackets from the following expressions and com bine like terms : — 1. (o-6) + (6-c)-(a-c). 2. (2a-6-c)-(a-26 4-c)-(a + 6-2c). 3. 3a -{6 + (2a -6) -(a -6)}. 4. 2a + (6 - 3c) - {(3a - 26) + c} + 5a - (46 - 3c). 5. (3c-2d) - {2d -de) + {-{c-d)- {3c + 2d)}. 6. a; + [a;-a-(2a-2a;)+{a-(a-/K)}]. 7. - {(3a; - 4y) - (2x-5y)} + {4a; - {2y - 3x) - 5y}. 8. a - [6 - {a - (6 - a - 6) - a} - 6]. 9. 3a-[a + 6-{a + 6 + c-(a + 6 + c + d)}]. 10. a-[26+{3c-3a-.(a + 6)}+2a-(6+3c)]. 11. ^-l^y-(^z^x)-{2y-(x + y)-z}-{y.z) + xl. 12. ;g^[3^-(a;-y)-{a;-(y-a;)-2/|-|a;-(y-^^-a.)}]. I same ex- f the pre- ir, but Ave snable the each step 3 labor in )-ft}+a and like md com BRACKJrrs. 13. «-[56-{a-(3c-36) + 2c-(a-26-c)}]. 31 15. ((3a-2i) + (4c-.)}_|«.(26.3a)-c}^|a-(6-5c-a)}. 1 6 Enclose a-b^c-d-e+f in alphabetical order in brackets, two letters in each : three letters in each. ^ 17 Enclose all but a in an outer bracket, with c, t^ and e en closed in an inner bracket. «, o ana « en 18. Enclose h and c, c? and «, in brackets, and then enclose these groups with/ in another bracket.' 19. Enclose e and/ in brackets, then this group with d in a 21. If a = 1, 6 , 2, c = i c?= 1, find the value of «-f3a-66-i7a-<)6-llc-(13a-m-17c-19^}]. ^)}]. I CHAPTER m. MULTIPLICATION. 62. '^e Product of two algebraical numbers is formed by substitutmg one of the numbers for the unit in the other, and reducing the resulting expression to its simplest; form .J^iSr^?rv ''"™^' " "^"'^ *^" Multiplicand; the latter the Mu tipher; the process of finding the product is called Multiphcation. 53. Let it be required to perform the following multiplications: Multiply 1. +7 by +3. 2. -7 by +3. 3. +7 by -3. 4. -7 by -3. Since + 3 = 1 + 1 + 1 and - 3= - 1 - 1 - 1. Substituting the multiplicand for the unit in the multiplier in the several cases we get 7^ l-( + 7)x( + 3) = ( + 7) + ( + 7) + ( + 7)=+21. (Art.35.) 2. (-7)x( + 3) = (-7) + (-7) + (-7)=-2L (Art.35.) 3. ( + 7)x(-3)»-( + 7)-( + 7)-( + 7) = -7-7-7=-21. 4. (-7)x(-3)=-(-7)-(-7)-(-7) = 7 + 7 + 7 = 21. .-Jl Similarly, if a and 6 are any absolute numbers, ( + a)x( + 6)=+a6. (Art. 45.) (Art. 45.) '^ -r t*/ A ^ - w^ = - ao. (-a)x( + 6)= -ah. ( - a) X ( - 6) = + a6. formed by other, and the latter is called plications: Itiplier in [Art. 35.) [Art. 35.) Art. 45.) Art. 45.) !». >. MULTIPLICATION. gg 54. From Art. 53 we have for the multiplication of algebraic numbers the following nvLE.--Take the product of the absolute values of t/ie numbers andprejlx the sign ^• or - , according as the two factors hive like err ^hke stgns. This rule is often abbreviated thus : ''Like signs give plus, unlike stgns give minus," y»« x - 5xy!^ ^ a:yz. 19. 3aAx-ibxx6xyx-a^xabxy. 20. Find the value of a=' + 63 +c3-3a6c when ^•r? ^-"'^ 3. a=-6 6 = 3 , 6=-10 5^_5 21. (-l)3(6H.,)>f(_l),(,^^j.^_j^.^^^^_^g^_^^^^^ MULTIPLICATION OF POLYNOMIALS. 60. Let a, b, c, d, represent any absolute whole numbers, and 1. Let it be required to multiply a-6 by c. c(a-b)=.{a-b)-^(a~b) with (a-b) written c times. "'' + ^ c times, 'b-b- c times '=ca~ cb. 2. Lr t it be required to multiply a-bhyc~d. {c~d)ia-b)^ {a-b)Ha-h)^ with (.-6) written times -U«-6) + (a-6)+ with (a-6) written t^ times} ''ca-cb-{da-db) * "^ca-cb-da + dh. 61 The multiplication of algebraic numbers ha. been shown to differ from the multiplication of absolute numbers only in determining fie sign of the product; therefore, bv f..l.,-n„ =L„ MULTIPLICATION OP POLYNOMIALS. S6 into consideration in accordance with the rules already given, the preceding results are true for all integral values of a, b, c, d. 62. From Arts. 60 and 61 we learn that 1. To multiply a polynomial by a monomial we multiply each term m succession, and connect the partial products by the proper signa 2. To multiply a polynomial by a polynomial we multiply each term of the multiplicand by each term in tne multiplier, and con neot the partial products by the proper sign. Multiply aXBEOISH XL 1. 2a!»-3aj + 4 by Zx. 2. 5a:»-2ay by Zxy, 3. a«-a6+6» by -2ah\ 4. 2m'n-5mn»by -4mnp. 5. 1a*x - 5a6y + 1 1 6»y - 6ay» by - baby\ 6. 3a;>-5y»-7ajy-4«-3y+2by -8ay. Simplify 7. 3a;(a:»-4aj+3) + 5a:(2a;'-3a; + 7). 8. 4a(a -b)- 56(2a - Zb) + Zab. 9. 4a!{2y-3(2a:-y)}-2y{5a:-2(2y-a;)}. 10. 20a:«-24-3«»-4«»-3a;{4a:»-5aj-2(3aJ-4aj+l)}]. 11. a{a-\rb-c) + b{b + c-a)-\-c{o^a-b). 12. a{b-c)-^b{c-a) + c{a-b). 13. (a+6)(c + c0-(«-*)(c-rf). 14. (aJ + 2/)(m-n)-(aj-y)(m + n). 15. (a-J)(a + 6-c) + (6-c)(6 + c-a) + (c-a)(c + a-6). 16. 3(a-6 + c)-6(a-26 + 3c) + 4(a-36 + 2c)-2(a-76-2c). 17. («a:+%) + (a;+y) + (a-l)a:-(6+l)y. 18. (a + 6)aj + (6 + c)y-{(a-6)a;-(6-c)y}. 19. {(« + % + (a + cM + {(6-c)a; + (6-a)y}-{(a-c)a;-(c-%}. 20. (a-6)«+(6-c)y+(c-a)«-{a(aj-y) + %-«) + c(2-a.)} -(a«+6y+c«) + (a+6+c)(aj+y+«)-2(ca:+ay+i«). 86 f I I i MULTIPLICATION OF POLYNOMIALS. 63. In the multiplication of polynomials like terms in the several partial products should be collected so as to give the Z V^" 'j"^'"' '""• ^'^ '^"^"^ «-°^P^«« ^ow the best methods of proceeding:— -fi'a 7. 2a:»- «»+ 3a:-l Sa:*- aj + 2 JKb. ;». a; - 6aJ'-3«*+ 9a:»-3a:> -2a;*+ a:»-3aJ+ j. +Jf - 2a:' + 6x - 2 6a:* - 5a;* + 14x^ - 8«» + 7«c^ (a + 6)aj + ab e ^~ - ci)^ + (ac + 6c)a; - abo ^-(a + b + c)x' + {ab + bc + ac)x-abc 64. The multiplicand and multiplier should be arranged in orTr^ ''''I'^^'^S''''-^' both descending, as in the examples, or both ascending In working long examples it is frequents convenient to om:t the letters and work with the coefficient^ only. The student should work the following example in full and compare with the work given:— ^ f ^uu Multiply «» > 3a;« + 2a;« - 5a; + 7 by a:» - 2.C + 3. 1-3 + 0+ 2- 5 + 7 1+0-2+ 3 1-3 + 0+ 2- 5 + 7 -2+ 6- 0-4+ 10- U + 3- 9 + 0+ 6- 15 + 21 1-3-2 + 11- u + 3 + 16 - 29T2I Result : a;* - 3a;^ - 2a;« + 1 la;^ - 14;,4 + 3a;» + 16a;» - 29a; + 21. A cipher is introduced, both in the multiplicand and multiplier In place of a regular term in the series which is wanting, to keep the other terms in their proper columns. When the terms are MULTIPLICATION OF POLYNOMLiLS. $7 written in full this is unnecessary. ^ above process is called multiplying by detached coefficients." Multiply i. x'-Zx + i by a?+2. 3. ix'+Sx-l hy 2z-3. 5. a^ + x+l by a;'-a;+l. 7. a'+ab + b^ by a-b. BXBRCISB Xn. 2. 23s*-ix + 6 by x-2 4. ix^-3x-5hj -2a; + 3. 6. x^+2x+2hj x^-2x + 2. 8. a'-a6-i-6' by a-4-6. 9. 'j»-2a + 3 by a» + ?3-3. 10. 2a»-6a6 + 36' by 2a> + 5a6 + 3o«. 11. 3a:«-7a;2 + 2a;-5 by La;- 3. 12. 3^-x*+x+l by 2a:»-4a; + 7. 13. l-2a; + 3a*+4a:» by l + 2a;-3aJL 14. Z-x' + da^ + x* by a;»-2a?+l. 16. 4a;-3 + 2ar»-a;» by -3a; + x»~6. 16. a;«+a:*-a:*+l by ar'-a;-!. 17. a;*+2ar' + 3a:2+2a:+l by a;*- 2x» + 3a;'-2a: + l. 18. a:»+4ar'+5a;-24 by a:»-4x+ll. 19. a:»-4a;2+lla;-24 by a:»+4a: + 6. 20. a^ + b^ + c'-ab-bc-ac by a + b + c 21. a» + 2a6 + 62-c« by V-a'+2a6-6» 22. a^'-ajy+ys + aj + y+i by x + y-l. 23. a;* + a:2 - 4a; - 1 1 + 2a:» by a' - 2a: + 3. 24. 49a;« + 56a;»y + 642/2 by 7ar'-8y. 25. ^ + 4a;»y+6a;'2/» + 4ay + y* by x^-W^^+Qx'^.ia^ + ^, 26. ar» + 4y^ + ^+2a^^ + 2y«-a»bya;-2y + «. "^ ^ ^ Find the continued product of 27. z + a, x + b and x + e. 28. a; -a, x-b and a; — c, 29. a:- 2, «;-3 and a;- 4. 30. a;- 3, a:-l, a;+l and a; + 3. i 38 MULTIPLICATION OP POLYNOMIALa 31. «>+aaj + a«, aP-ax-^a^ x + a and «-«. 32. a:*- a:y + y, a^-a:y + y. and :^ + xy+y». 33. 9m»+3am+a>, 9m»-3a«» + a', 3m+a and 3m-a s.ridi:^e";:™^^^^^^^^ ^^ ^- ^~e .nd ^- « + * 2. o - 6 a'- oi - a6 + 6» a'-2a6 + 6» a'+2ab + b* Weee reault, should also be r^membe'red in wo«.s''thus-r u e^r ^ai^r: :"^^ s::r--' - «- 61x51 = (50 + l)« = 2600 + 2x50 + l = 2601 48x48 = (50-2)> = 2500-4x50 + 4«2304* 67 X 73 = (70- 3) (70 + 3) « 4900-9-4891 EXERCISE XIIL Perform the operations indicated. 1. (a;+y)« 4. {2x + 7jy. 7. (2« + 3y)>. 10. (aJ+y*)'. 2. (x-2/)\ 5. (a:-2y)l S. (2a;-3y)> 11. (x'-yy. 3- («+y)(a;-y). 6. (a;+2y)(aj-2y). 9. (2a: + 3y)(2a;-3y). MULTIPLICATION OF POLYNOJOALa ^9 16. {a'-^beXa^^bc). 18. (ax + by) {ax- by). 21. (4ar'+l)(4a,-»-l). 24. {9x+7y){7y-9x). 26. 81 X 79. 29. 257='-243». 13. (a»+Jc)«. 14. (a^-bc)\ 16. (ax+by)\ 17. {ax-byf. 19. (1 +«:»)». 20. (3ar»-4)». 22. (5m'n + 6mn«)». 23. (8m»-5m/)». 25. {x-y){x-\-y){a?+y'){x^ + y<), 27. 97 X 97. 28. 88 x 92. 30. 9 X 11 X 101 X 10001. (Use Ex. 25.) 31. 4(a - 36)(a + 36) - 2(a ~ Uf - 2(a» + 66'). 32. ar'(a;»+y'7- 2a.V(a: + y)(a:- y) - {a^.y^y, 33. 16(a» + 6')(o» - b"^) - (2a - 3)(2a + 3)(4a2 + 9) + (26-3)(26 + 3)(46'+9). 34. {a ~ 26)(a + 26) + (26 - 3c)(26 + 3c) + (3c - c/)(3c + d). 35. Show that {{<^ + W^{ad~bcf){{ac + bd)^-{ad-^bcf)=.{a<-b<){c*-d*) 68. To form the square of a trinomial. a + 6 + c a + 6 + c a?-¥ ab+ ac + ab+ 6«+6o + 00 + 60 + 0* a + b - a + 6 - a' + 2a6 + 2ac + 6^ + 26c + c* a'+ ab- ac + a6+ 6« _6c - ac - 6c + c' a2+2a6-2ac + 6'»-26c + c' = a»+6^ + c» + 2a6 + 2ac + 26c. I =a^ + 6' + c=' + 2a6- 2ac-26c. These results consist, in each case, of two sets of terms:— 1. The sum of the squares of each term of the trinomial. 2. Twice the product of each pair of terms. The sign of each of the square terms is positive. The si-n of any product is positive or negative according as the signs of the terms fro.i which it is formed are alike or different It is worthy of note that the signs of the products in the square of any tnnomial are either all positive, or two aro n--*? - one positive. n^^^v^v. auu 40 MULTIPLICATION OF POLYNOMIALS. 69. A ittle consideration will show that the square of any polynomial is fomed in the same way as that of a trinomial follo^n °^''''^'"^' arrangement of terms is according to the RuLE.~ro tJ^ sufn of the eguarea of each term add twice the ^ product of each term into each of the terms tluit follow it. 3. {x~y-¥z)\ 6. (-x + y + z)\ 9. (2a-J + 3c)». 12. {(L^ + xy+y^y 15. (4-2a:>+a:)». BXBROI8B1 XIV. Perform the operations indicated. 1. (a; + y + «)«. 2. {x+y-z)\ 4. (x-y-.z)\ 5. {-X'y + z)\ 7. (a + b + 2oy. 8. (a-2/> + 3<;)» 10. (l+aj+a:»)« n. (i.^.^^.^^,^ 13. (2ar'-3a; + 4)». 14. (a;-r'+2)». 16. (a-b+c-dy. 17. (a^+ab-ac~bcy. 18. (^~Z+Z\y 19. (^+t/+^)'+(^^y-zy-i-(x-y+zy+(.x+y+zy. 70. To form the product of two expressions which differ only m «ie sign of one or more terms, we first arrange the terms of each expression m two groups, placing those which have iiie same sign m the two expressions in the first group in each case. and those which have diflcrent signs in the second group. Th^ terms of the second group in the two expressions, having di. orent signs as they sUnd. will have the same sign after being encC n bruckets with a positive sign before one group a.d I negative sign l^efore the other. We h.ve now to find the product of the sum a^d the difference of two quantities, which, by Art 65 is the difference of their squares The work may be arranged a.' in the followmg example:— o^ «w m -ffi'ar.—Multiply a-b + o-d by a + b-c-d. (a~6+c-.;)(a-f6-o-rf)-{(a-c£)-(6-c)}{(a-d) + (6.on -(a-<30»-(6-c)' MULTIPLICATION OF POLYNOMIALS. u I BXEROI8E XV, Perform the operations indicated. 1. (» + *-«)(a+4-,). 2. («+4- «)(„-»+.). 3. (2x-„.3z)(2.-y^3z). 4. (=»-2y+3.)(3,-.+2yl .. 3a'-ai + 26.)(3a.+ai + 26.). 8. (^+2a»+2a')(».-2j+2an 1. (s^-2s,.-^Xs^+23^-^-(y.j„.2^)(y.^ 2^ 12. (a'+4'-o»+2o4)(o'-a'-6>+2ai) -(»'-4'+o'+2«)(J'-o.-„.+2a«). 13. (<»+4+ a'6 + 2ai» + 6» we get and (a + b)' = a» + 3a»6 + 306" + 6» a* + 3a»6 + 3a»i«+ ai' a'6 + 3a »6« + 3a6» + 6* (a + 6)« - a* + 4a»6 + 6a»6»T4a4»+6* J 1 POWERS OF A BINOMIAL. 43 In the above ex? mple observe * a« \Ju ^''* ^""" ^ r^ '''"^* is a raised to the same power as the Wzn.al; m each succeeding tenn its exponent is reduced 2. The second term contains the first power of b, and in each succeedmg term its exponent is increased by unity ' ^"^ "" "^^ • ^' ^^ TTi^ ^ homogeneous, and of the number of dimen- Biona indicated by the exponent of the binomial. 73. The powers of a~6 should be written out in the same wav as those of a-f i in the preceding Art The results IT^Z same except that the terms containing odd powers of 6 will hat the sign - prefixed. (See Art. 56, 3.) 74. It is sometimes convenient to write the cube of a bint, mial m the following form, which may easily be verified:- (a-6)»-a»-6»-3avXa-J). 75. The coefficiente of the teiins of the successive powers of a bmonual, up to the fifth, should be committed to memory Thev may be arranged thus:— «"i"^y. iney 1 2, 1 3, 3, 1 -*. 6, 4, 1 «*h " 1, 6, 10, 10, 6, 1 ^ This^list may be continued to a^y extent by carefully studying 1st power, 1, 2nd " 1, Srd «• 1, 4th «• 1, (4 BXBROISB XVII. Perform the operations indicated. 1. (aJ+l)» 2. (x-l)>. 4. (a:-l)«. 7. (x+2yy 10. (a -26)* 5. (aj+l)». 8. (2x~ijy 11. i2a+by. 3. (x-i-iy. 6. (x-iy, 9. (2a-3M» 12. (x~yy. I •If 44 POWERS OF A BINOMIAL. 13. (a + x^+(a-xy 15. (a:+y)»+(a:-y)« U. (a + xy.{a-xy. 16. {x^yY-{x~yY. 17. Find the value of a«+6>+c'-3a6o in terms of a and 6 (1) if c-(a + 6); (2) ifc=-(a+ft). * 18. Find the value of a^-\-h^+hj -y>. 1 9. 8a» - 1 6a»6 - 24a6' by 8a. 20. 25a'* - 30a*6 - 40a'6« by - 5a«. 2 1 . m/w:* - m'p V + Tnp" by »ip. 22. 16a^2/-28ar'2^ + 36a:*y»by-4a:«y. 23. -49a;»2/Si»+63a^y»«by -7a:»y2. 24. 52a«6»-65a*67+78a»6«by 13a»6». 25. 34a«6 + 5 la*J» - 68a%» by 1 7a%. 26. -144a:»3/+132ar»y-120Vby -12a;y. 27. -46a'6»c + 69a»6V-115a*6»crf'by -23 a«6«c 83. When the divisor consists of more than one term we pro- ceed according to the following ^ -RVLU. -Arrange tJie terms of both divisor and dividend acctyrd- %ng to the powers of some commmi letter, both in descending or both in ascending order. Divide th^ first term of the dividend by the first term of the divisor; the remit will be the first term of the quotient. Multiply each term of the divisor by the first term of the quotient and subtract tJie product from the dividend. If there be a remainder consider U a new dividend, and proceed as before. If a remainder occurs of lower dimensions, with regard to the letter of reference, than the divisor, the division cannot be exactly performed. Such examples wiU be considered in the chapter on Fractions. 84. The reasons for the preceding rule are the following:— 1. The term containing the highest exponent of the letter of reference in the dividend must be the product of the terms con- taining the highest exponents of that letter in the factors of the dividend, i.e.. the divisor a.nH flia n^r^fU^*- 48 Diyi8I0H. by each term of the quotient, if we subtract the product oflTe ^^til ' "^^ '' "^^ "^^^ -"^ ^^ «^- ^-« of ^e the to^ltl i^'^'^^-'T'" ^^" "^^ P^^P*'^ arrangement of the work in the division of polynomials by polynomials;- Divide 1. 2a:»+7a:« + 5a;+100 by x + 6. 2. a*+b* by a + 6. 3. a*-6» by a*+al> + b\ i. 16a* + 4aV+«« by ia*~2ax+a». 1. x + 5J2x'+ 7x»+ 5aj+100(^2»»-3a!+20 2aM0^ - 3a^-16aj 20a: +100 20a; +100 2. a+5;rt«+^^ (^fl'-o^ + J* o'+o'6 -a'b + b' -a'b-ab* ab*+b* ab'^+b* 3. a' + ab + b^Ja^-b' |a-6 ■ 4. 4a«-2aa:+a:«;i6a« + 4aV+ aj« [4a«+2aa5+a:« 16a*--8a'a; +4 a»a:« " 8a'a; + x* 8o^ -4aV + 2aa:» 4aV-2aa:«+aj« *aV-2aa:'+«* HOBHBB'S METHOD OF DIVJSIOS. 49 HOENBB'S METHOD OF DIVISION. order, may be ooaveniently worked by the foUowing ^ cienta of the dmdend m a horizontal line, and those of the ^™or .n a vertical line to the left of the diWdend. changing Z sign of every term in the divisor except the fl«t. ^® Divide the first term of the dividend by the first term of the d.v«or; the result will be the fi«t term of the quotient Hi. 7 »f ">« 'J-of^nt. and arrange the partial proLct, d««o„aIly under the second and following te™ of the l^Z iTq^oTent •"™°'^ "" '""" '^ ^ '^^ --<» '- o' thiM"l!!S'f ?^ ''°'°" ""* '"^^^ *■'« P»^'»' P«^»o<» "nder the th.ri and following terms of the dividend. Continue the prod untU the number of terms in the quotient i. greater by n'Z ciero;CXr ""™°" *'^'' "■"" '^ •» "» ooeiE. tiet**tf "" '^"'^' l'*!"" ''°'°" *" '^^ ooeffleients of the quo- tient, the exponent of the first term being the difference of "he exponents of the first terms of the dividend and the divS>r and the others following in regukr owier. ' The literal factor of the terms of the remainder will bo the s^e a. those of the terms of the dividend under which the; 60 HOBNER'S METHOD OF DIVISION. 87. The following examples show the arrangement of the ^«. 1. — Divide 2 -1 + 3 -4 6-1-11 + 16 + 1+8-19 + 20 -3+ 2+ + 1-5 + 9- 6+0-3+15 -12 + S + 0+ 4-20 3-2+ 0- 1+5 Quotient 3a:^-2;«»-a: + 5. Remainder. 0. since the sum of each of the last three columns is zero. Ex. j?.— Divide 2at^ + a:«y- Qs^f - lla^y + 2ar»y + Sx*/ + 16a:;/ - 12t/» by a?+2xhf~Zf. 1 -2 + + 3 2 + 1-6-11+ 2 + 6 + 16-12 -4 + 6+ + 10-6 + 0+ 0+ + 0+ + 6- 9 + 0-15+ 9 2-3 + 0- 6+ 3r"Tl^ 3 Quotient, 1x' - 3a:»y - bxif^ Z^. Remainder, xi/' - 3/. In the above example the coefficient of the first term beinff - 1 no division of the sums of the columns is necessary. From its brevity this method is also known as "Synthetic Division." "^ Divide bxbroisb xix. 1. a:»+15a: + 50bya;+10. 2. a:»- lla: + 28 by a:-7. 3. a:»-«-56bya;-8. 4. or'+aj-gO by «:+ 10. 5. «»+13a:« + 54a: + 72bya; + 6. 6. a?-^2,s?+2x+l byar+l. 7. a«-6»bya-6. 8. a»-6» by a-6. 9. a»+6» by a«-a6 + 6«. 10. a»+6» by a + 6. rornkb'8 mrhoo of division. $1 11. a»-7a!-6 hy x-3. 12. 4a*+5x+2l by 2x+3. 13. 8a:«+27y>by 4aj»-6«y+V. H. 64«»-l by I + 4«+16x«. 1 5. «»- 5a:» + 7a:" + 6a; + 1 by «> + 3a; + 1 . 16. 6««-a:«y+2aJy>+13xy» + 4y*by 2a:"+4y»-3«y. 17. a»-4a*6 + 4a^6''+4rf«6»-17aA*-126» by a«-36«-2a5. 18. o«+13aV-6a»a;+4aV-12aVby (^-3ah:+2ax». 19. a« + a'6»+6«by a»+a5 + 6>. 20. aJ»+aJ*+l by««-a:«+l. 21. aj" + y»« by x^'^-afy^ + i/o, 22. aj'^-y'* by x'-if. 23. ^•-Oar'+l bya;»-2a:+l. 24. «' + 2«»+l by «»-aj + l. 25. x'»-a:«/ + 2/"'byaj»-ajy+y». 26. 4a:»-a:» + 4a; by 2 + 2^^i i.r. 27. 2a:U2a:V-2a,y-7a:»y-y - y«. 31. 81j:«y+18«»y»-54a:«y»- 18a^y-18;V9y by 3a;< + a:y+y. 32. 2a76 - 6a"6» - 1 la»6» + 5a<6* - 26a»6» + 7a»6« - 1 2ab'' by a*-4a''^ + a«6»-3a6». 33. «'-3a;»-31«'+25«»+3ar»-22a:*+44a:»-2a:»-15a;+10 by»*-7ar' + 3«-2, 34. «"-a:'y'+aJ«y«-a:»y»+y«byaJ«-««y+aV-«2/'+y*. 88. When several terms contain the same power of the letter of reference it is usually best to collect them into one term by the use of brackets. The following are examples : 1. Dividea;»-(a + 6+c)a;»+(a5 + 6c+ca)a;-a6cby«-a. 2. Dividea»+6'+c»-3a6cby a+6+c. 1. «i-a)^-{a-^h-{-c)a?-k-{ab-^hc + ca)x-abc \a?-{h^e)x-\-be «*- oo* (6 + c)a:* + (a6 + 6c + ca)aj {h-¥c)3i?-k-{ab +ca)x bex-abe hex-abo ONTARIO COLLEGE OF EDUCATION ,f^^-0S»^ik, 68 HORNEB'8 METHOD OF DIVISION. -a'(i + c)-3o*c + 6*+c» Beginners usually find considerable difficulty in examples of brackets n.ight be removed, the subtraction performed, and the ^ms agaxn arranged in order, befon. proceeding with the d^^! Divide bxeroisb xx. 1. ^-(<.+J+«y+(aiH.J„+<„)«-„i„ by «_J and by «_«. 2. 'fH«+i+c)a^+(al,+io+ca)x+aic by :«'+(J + oW+4, bv each »f fl,. j; • _. T^'^"'^*'"''^"*''''^'^ "+«*<"* oy each of the diraora ai'-(a + c)x + ac and «•-« te+j. ■ :^ . ;'* "^ "'*-"• ^- "'-"-'^'^ "^ -*' «• 8. 8a'-6» + ca + 6a6cby2a-6 + c. 9. ^+^y + 2xz-2y'+7yz-3z'hyx-y+Sz, 10. 2^-6y'-12«» + a:3,4.17y*-2;.^bya: + 2y-3«. 11. «'(6 + c) + fi*(c+a)+c»(a + 6) + 3a6cbya + 6 + c 12. Ab + c) + b%a-c) + c'(a~b) + abc by a + 6 + c. 13. «^-83/» + 27^+18xy.bya:»+V+9.'+2«:y + 6y.-3.^^ 14- («+y)»-3(;B+y)'»« + 3(a:+y)^-^ by a;+y-«. 15. b(a^ + a') + ax(x^-a^) + a'(a: + a) by (a + 6)(a: + a) 16. 6(ar»-a») + aa.(a:^-a») + a»(a: -a) by (« + 6)(a:-.a). 17. (a:»-l)a'-(..^ + ar«- 2)a'4-(4«:» + 3a: + 2)«-3(a:+l) 'o7(*-i)a'-(aj- l)a + 3. Horner's method op division. 53 89. The foUowing cases of division are of frequent occurrence and should be carefully remembered : If n stands for any positive integer, «* - y** is divisible hj x-y always ; «"-y" " " " a;+y when n is even; «" + y" " «• « x + y when n is odd; «*+y*" " " oj-y never. By actual division we obtain «* - y* = a;' + a;y + y»; «• y -a^ + a^y + xy^ + i/; « - y - - X - y In these examples observe that 1. The first term of the quotient is obtained by dividing the first term of the dividend by the firs'- term of the divisor. 2. The exponents of the first letter decrease, and those of the second increase, by unity in each succeeding term. 3. When the connecting sign of the divis( r is -, the signs of the quotient are all +; otherwise they are + and - alternately. EXBBOISB XXI. Write down the quotients by inspection in the following examples: — 1. a;*-y«bya:+y. 2. «»-y» by ar-y. 3. x«-y«bya:-y. 4. a:^+y'bya:+y. 6. a?+\hyx^\. 6. «•- 1 bya;+l. 7. l-aj»y»byl-ajy. 8. aj»+y«2;»bya?+ya. 9. a:»4-32 bya:+2. 10. 27-a:»by 3-«. 12. 8aj»-27y»by 2a;-3y. 14. a:*- 32 by a? -2. 16. a*- 81 byaj-3. 18. 8x»-343y»by2;B-7y. 20. 64ar'-343v»bv43!-7,/ 11. 243+«»by 3+ax 13. a:« + 8byaj + 2. 15. 16aj*-l by 2a;+l. 17. 216-a:»by 6-aj. 19. 27x»+1000y»by 3a!+10y. 9] 1 Ofi-Jj. KIO^J 1 e_ . o 22. Divide (a:'+ay+y«)»+(a:>_jey + y3)8 by 2ar'+2y*. r CHAPTEB V. PAOTORINQ. 90. In multiplication two factors are given, and their product IS required; m division the product and one factor are given from which to find the other factor. We have now to coiSder how to find both factors from the product alone. 91. The only rule which can be given is to examine carefully the process of the multiplication of various kinds of factors, to note the resulte, and then learn to retrace the steps from the product to the factors which produced it MONOMIAL FACTORS. 92. The actors of a Monomial are evident by inspection. If each term of a Polynomial contains a monomial factor it may be discovered by inspection, and then the other factor may be obtained by division. ^ Ex, 1. 1 Ooc + 1 5 Jc - 5c(2a + 36). Ex. 2. 3a:»+15a;>-9a;=.3a;(a:«+6x-3). Ex. 3. Wb + 6a«6' + 2a6» = 2a6(2a' + 3a6 + ft«), 93. This principle may frequently be extended to groups of terms, thus: — ° ^ Ex. 1. ax+ai/ + bx + bi/^a(x + ,j) + b(x + tj)„(cc+y)(a + b). Ex.2. ac~bo-ad + bd^c(a-b)-d{a-b)^{a'b){c-d). Resolve into factors 1. 3a:*- 16. BXBROISB XXII. 2. 10x'-16a:y + 20/ 4. 22m»-33mn-110n». I TRINOMIALS. 55 i 6. aaf-abaP + ace. 7. S5o^i/z + 70xfz - I05xyz^. 9. {a-h)x-{a-h)y. 11. Zax-ay~Zhx + by. 13. a?+ax-\-bx-^ah. 15. a?-ax-{-hx-ah. 17. aac-bx-^-ah-a?. 19. a^cx-ahdx-ahcy-VbHy. 21. adx-\-ady-bcx-bcy. 23. «*+«♦- jcs-aj' + aj+l. 25. 6a26a-2a»c-96'c + 3a6c>. 6. 54a»6«+ 108a«6«- 243a«6». 8. (a + 6)a; + (a + %. 10. 2ac + 2a-3300. 31. a'b+lSa^bx-lda^x^ 33. 29-28a;-a:>. 35. ab-ac+bo-~b\ COMPLETE SQUARES. 95. Since (a + 5)' = a'+2aA + i' and (a-6)» = a'-2«A -. W see that a Trinomial is the exact sc.u- ro of a Rir.nl i' r""^ two of i^ tern, are exact squares, a^^^L^^^Tr etl t plus or minus twice the square root of their product. ^ first term is unity, the coefficient of the ln.st t^rm must eoual the square of half the coefficient of the middle term This is a special ca.e of the preceding Art. which deserves carefuUtl! ^x. 1. i^+l2xy + 9y'=.{2x + 3y)\ Ex. « — *= 1 O- 2 i.^xr fdO (ar-6)» COMPLETE SQUARES. gj 97. A careful study of Art. 68 will enable the rtudenfto ^oogn.0 au expre^ion which is the exact s,uare oLt^l^ Sx. «' + 44> + 9c>-4ad + 8a«-126c.(a-24 + 3<,).. of ^1*"™' °f "*' 'T""'"' *'•'' ^"""^ •'y '^ting the square root, of the square terms of the expresaioa. To determine the.> Z^ the terms containing a and c must have the same si»n • the other EXBROISB XXIV. Express as complete squares ' 1. a:»+10a; + 25. 3. a:'+20£cy+100y. 5. aV+8a»ay+16y. 7. a:«-38ic»+361. 9. a^-Ha"/* + 496'. 11. 4aj'+12a^ + 9y». 13. 16a*-24a«6 + 96». 15. 4a»+256»-20a6. 17. 9aV+496V-42a6i^. 19. 36a»+166'-48a6. 21. Sa^a^-18ax^i/ + 27xi/. 23. (a + 6)'+2/« + 6)c+c'. 25. (a;-y)'+2(a;-y)y + y>. 27. ix + yy—2{x+y)y + i/. 2. a;*+18a:»+81. 4. aV + 4aa;+4. 6. w*-16mV+64w<. 8. Pa^-2lmx7/ + m^9/'. 10. 81a;*- 18a;' -f\ 12. 9.x«-30a;y + 25y>. 14. a» + 96' + 6a6. 16. 25aV-70a EXHROISB XXV. Resolve into factors 1. ic'-?/'. 4. 4ar»-9y'. 7. \Zx'-%h/. 10. bx^-2Qif. 13. (a + 6)'-c'. 16. a'-(6-c)'. 19. (a + 6)' -(c + (/)'. 21. {a + by-{a-by. 23. 26c + 6'+c'-a«. 25. a=a;' + 6»y'+2a6a^-l. 27. l-16a'-256' + 40a6. 29. a' + 6'-c'-rf'-2a6-2c(i. Q1 >,2_ AJ4. J^.n^ o — ou 2. x'-lQ. 5. 16a;* -49?/'. 8. 2563.-8-1. 11. 3a* -276*. 14. {a-bf-c\ 17. a'- (a -6)'. 3. 25-7/«. 6. («*-16. 9. a;'2/*-100a«. 12. 162cc'y'-242c'. 15. a'-(6 + c)'. 18. (rt + 26)'-6». 20. {a-by-{c-dy. 22. a'-2a;y+y'-«'. 24. 26c-6'-c'+a'. 26. a' - 46' - 9c' + 126c. 28. a' + 6»-c'-c^+2a6 + 2c-6aj + 5)'-l. 35. (a:»-25)'-(2a;+10)«. 37. a'b^ + c'(P-a?c'^-b''dK 69 34. (a:»-10)'-36. 36. (2a:» + 3a:-5)'-(a;»-9a,_40)». 38. (a:'-ny)»-(y»_^2^)v '=(•'«' + a' + aa;)(a3» + a» - oa;) = («' + ««+a')(a;2-aa; + a»). _ , . ^ EXBROISB XXVL iCesolve into factors 1. a;< + ar»+l. 3. a;*+9a:2 + 81. 5. a:* + 5a^y2^9y. 7. a^-5a:»y»+4y. 9. 4a;* + 335*2/'+%*. 11. 9a;*-10aV+y*. 13. 4a;*-37a;«2/»+9y. 2. a:* + 4£c'+16. 4. «* + 4y. 6. a!*-3a;»y»+y«. 8. 4a;* +1. 10. a;* + a;' + 26. 12. a;« + a:*+l. 14. 4aj*-13a;V' + 92/*. SUM AND DIFFERENCE OF CUBES. a + 6 ' we can always resolve the sum or the difference of ff,« . i, * any two quantities into two W... "'" °^ *^^ ""^«« °^ 100. Since and ■a'-ofi + i' any two quantities into two factors. -fi'aj. i. 8a» + 276« = (2a)« + ^Si'^s (2a + 36«)(4a'-6ai» + 96*). ; 1 1 1 60 SUM AND DIFFERENCE OF CUBEa Ex,S. a« + 6«-(o»)» + (6»)« Ex. 8. a^-U'^ (a» + 6«)(a« - 6») -(a + 6)(a»-a6 + 6»)(a_6)(a« + a5+6»). 101. Since a; + y is a factor of a:"+y when n is any odd whole number, and a;- y is a factor of x- - y« when n is any whole num- ber, we ce^ always resolve the sum of two equal odd powers, or the difference of any equal powers of any two quantities, into factors. Ex. a»o+6»«-(a»)»+(6«)» - (a« + 6')(a» - a«6» + a«6* - a'i« + h*). Similarly the sum of any even powers can be resolved if both exponents contain the same odd factor. BXBROISB XXVIL Resolve into factors 1. «' + 6». 2. a» + 8. 4. 8aJ + 273/». 5. 27a^+64y». 7. ic'-lOOO. 8. 729iB»-512y». 10. 64a»6»- 1000c'. 11. a:»+y«. 13. 243+y^. 14. ^-^, 16. l-a:»y»o«io 17^ ^e_j.^ 19. a» + 6». 22. a»»-6». 26. a»-6". 28. (a-6)»+a». 20. a»-6» 23. rt»+6»o 26. (a + 6)»-a». 29. {a-bf + m. 31. (a + 6)' + (a-6)». 33. (a-26)» + (2a-6)» 35. a'+6'+3rt6(o + 6). 37. a» _ 3a'A + .3aA« - 6' - c*. 3. 27 + 6». 6. a;»-3/». 9. 125«»y»-343s» 12. a;«+243. 15. a!»-32/«'"» 18. a« + 6«. 21. a»-6» 24. a" +6". 27. (a + 6)' + a». 30. (a + 6)»-86». 32. {a + by -{a -by. 34. (2a-6)»-(a-26)» 36. a»-6»-3a6(a-6). oo. «'— oarroor-ur' — 8. TB1N0MIAL& «1 TRINOMIALS. FIRST OOBFPICIRNT NOT UNITY. understood f™„ the following el^,^,!l ^''"' """"^ ""^ *» JF^-Eesolve 6«»-13a^ + 6/ into facto™. 1. Kaoh factor mu,t evidently contain both :« and v «^!-Jd:f''ii:ttt: 5' *«'- »' "•« -» '-» - ^ .actr^it XTdict' -r~«" - Of the must also be negativa *^'"°' ** ^^«*<^i^« thej 6. It is useless to trj such a factor as 2^ 9 • « . factor of the given expression. ^ ^^ '^"'' ^ ^ '^^^ » The above considerations lead us *n ro,-^«* n t factors, viz.: (x-6y)(6^ ,7 T/o *^/®J®°* *" ^"t two sets of .. tH^ that le a^™ Jr ti^.^!f !: -t^^' - «-. ^ EXBROISE XXVm. Kesolve into factors 1. 2x* + 6xi/ + 2y\ 7. 34a:* + 21a;» + 2. 9. 8aJ+22a;+9. 1 1 . 6aa:' — BSasr^- = *. 13. 2. 23l?+5xi/ + 3y», 4. 15a:»-26a:y + 8v/2. 6. 14a^+83a^-6y» 8. 6a;*-13iB»y«+2y«. 10. 12a:«-2a!y-30y». ««*+(««+ l)ay + a3/». U, iiJ. 10aV-7aW- ^-I7«v+v. icy-SSay. 62 TRINOMIALS. 1 03. To find the factors of ea^+l\xi/ + Zi/ + Sxz-2yz-8s^. Eeject the terms containing z and factor the remaining ones OS a trinomial, thus : 6x» + 1 1 xy + 3y» - (2a; + 3y)(3a; + y). Take two terms whose product is -8a», attach one to each factor, and then find by trial whether the factors thus formed give the terms Sxz - 2yz in the product. Take 4a and - 2a, thus : (2a; + 3y + 4s)(3a: + 7/-22). In the product of these factors the term containing xz will be (2a;)( - 2a) + (3a:)(4a) =. 8«a; the term containing yz will be (3y)(-2a)+y(4a)» -2ya, which proves the factors correct. 104. By rejecting the terms containing each letter in succes- sion, and factoring the remaining trinomials, we can determine the complete factors without further trial, thus:— 6a;> + 8a» - 8a2 = (2a: w iz){Xx - 2a) ; 3y>-2ya-8a2 = ( y - 2a)(3y + 4a). In these factors 4a occurs with 2x and also with Zy; place it, then, with the factor 2a: + 3v/ first obtained; similarly with regard to - 2a. We thus get the same factors as before; the first method, however, is generally the better. 105. To find the factors of a» + i»-|.c»-3a&c. = (a + 6)3 + c«-3a6(a + 6 + c) = (a-l-6 + c){(a + 6)'-(a + 6)c + c'-3a6} = (a -I- 6 + c)(a* + 6* + c« - oi - 6c - ca). From this result the factors of Exs. 10-13 may be written down by inspection. EXERCISE XZIX. Resolve into factors 1. 2«» + 5xy + 2y»-7a;a-8ya + 6a'. 2. 2«»-6a:y + 2y' + 7a;a-5ya + 3a'. 3. 6a'-37ay + 6:i/*-5aw-5y«-^. TRINOMIALa go 4. 6a^-13an/ + 6y»+12«-132/ + 6. 5. a,^ + 2x7/ + y' + 5x + 5y + G. 6. ^x*-l2xi/ + 9y^-Qx + 9i/ + 2. 7. 3^-l2i/+l5z'-ixy + 8yz-8xz. 8. 72a:»-8y»+55a;y+12y-169a; + 20. 9. 15x«-16y*-22aJy'+15««+Uy»«> + 60aj«««. 10. a»-^b'-c» + 3abc. U. a»-6«-c«>3a4(j. 12. a' + b'+8c»-eabc. 13. 8a» + 6»-27o»+18aio. 106. The following exercise consists of a miscellaneous set of examples arranged by combining the elementary forms explained in this chapter. The student should not try to work them all consecutively, but one or two each day will prove a valuable exercise while proceeding with the following chapters. BXBRCISB XXX. Resolve into factors 1. 46V-(a2-62-c')». 3. (rt»-5a)»+10(a'-5a) + 24. 5. 13a;V-9a:*-4y«. 7. a'-b<' + a* + b* + a^b\ 9. a^ + b^-a^b^-a^b\ 11. («+2/)«-7(a;+2/)V + «*. 1 3. a^i V - (a* + b*)xY + a^by. 15. {sc' + xy-yj~{a?-xy-ff. 17. a? + a^~ax{oi? + a?). 19. 8\a^-\-90a?-l0x-l, 21. s(?-2as?-a'x-\-2a\ 23. aJ*-(p»+l)aJ+;,J. 2. 4(aJ + cctf - {a? + b''-c*- (Py. 4. (a''-9a)2 + 4(a«-9a)-140. 6. (a'-b'Xx'^-y^).iabxy. 8. (i^ + b^ + a* + b*~a^b\ 10. a^-b'^-a'^b^ + a^b'. 12. (a + 6)*+2(a=-6y-3(a-6)*. 14. x*-2xy{x'-f)-yi, 16. (a + 6)» + a'+6»+2(a»-68). 18. (a'-b')(x'-y') + iabxy. 20. aa:'-(a2 + 6)a;»+6«. 22. (ao + bdy + (ad-bcy. 24. (a>+6»)«-(a«-6»)'-(o'' + 6«-c»)« I ^^ ^ THINOMIAIA 25. a«+6' + c»^-3«6' + 3a«6. 26. a' + i» + a«i + ai'4a + 6. 27. aB'-y»-aJ3_a^^«_y.^. 28. a:* + a:y + y« + ar« + ary + y». 28. «»-y« + a:«+y«+a:"y«. 30. a« + 6» + a« + 6«-a6. 31. (a«-6'-c«+rf')'-4H-6r)». 32. a:'+«'y+a:»y'4-«:V+ay^.y.. 33. c'(a-6) + «6(a-6)-c(a«-6«). 34. a»-6>-«(a'-6«) + i(a_6). 36. («-y)(y'-«')-(«'-y')(y-«). 36. («:-y)(y'-«^-(a:»-y.)(y-,). 37. (aaj+iy)« + (ay-.6a:)« + oV+cy. 38. 6»+6«o+6c«+). 39. (a-6)(6«-c«)-(a*-6«)(6_c). 40. («'+«y+y»)* + (a:* + ary + yy+(aJ_a:y4.yi)«, 41. a<+16A* + e<-8a»6«-86V-2cV. 42. 8a*6» + 326V + 8cV - a* - 1 66* ~ 1 6c«. 43. (a«-6»)V + y) - {(a + b)* + (a - 6)«}xV. 44. (»»«-n')«{(a:+yy + («-yy}-(a:«-y»)^(„,^.„)4^^^_^^,^ 46. 4{2(«:«+a-)' + 3}' + 28(«'+a:){2(a;'+a:)»+3}-275(a:>+a:)« Jl CHAPTEll VI. SIMPLE EQUATIONS OP ONH UNKNOWN. 107. An Equation is a statement of the equality of two al-e braic expressions. " Thus 3a: - 4 - 2a; - 6 is an - quation. 108. An Identical Equation, or, as it is sometimes called, an t^dentlty, is one m which the two expressions are equal for all values of the letters involved. Thus (a + 6)« + (a-6)'=2(a' + 6") is an identity. 109. A Conditional Equation is one in which the two ex- pressions are not equal for all values of the letters involved but for one or more particular values. * Thus 4a; + 8 = 2x + 2 is true only when a; = - 3. 4^ The term "Equation" is generally used for "Conditional Equation. * 110. A letter to which a particular value or particular values must be given to make the two expressions equal is called an Unknown Quantity. 111. That value of tlie unknown quantity which will make the two sides of an equation identical is called a Root of the equa- tion, and IS said to satisfy the equation. 112. To solve an equation is to find the value or values which will satisfy the equation. 113. A Simple Equation is one in which the Jlrst power only of the unknown a u an f.i t.v nnnm^! anA ;« «i n-j -- - -X J ---'; «*«« io ooau v;aucu iUl euuatlOU of thejirai degree. 66 iSi SIMPLE EQUATIONS OP ONE UNKNOWN. 114. The following axioms are used in solving equations:^ 1. If equals be added to equals the wholes are equal fuld 3 to each side «-3 + 3 = 4.+.3 that is ^,7 ' 2. If equals be taken from equals the remainders are equal ^* 2* + 5-3a; + 7. Take 2« + 5 from each side, then 2a: + 6-(2«:f6) = 3;r-f7-(2a: + 5). ®' C = a: + 2. 3. If equals be multiplied by equals the products are equal. ^* 2aj+l=3. Multiply both sides by 3, then 6a:+3 = 9. 4. If equals be divided by equals the quotients are equaj J^«f 4a: + 8 = 6»+12. Divide both sides by 2, *^^" 2a:^4-3a: + 6. 6. If the product of two or more factors be equal to zero then one or more of the factors must be equal to zero. ' Thus, if (a:-2)(a:-4) = 0, then either a:-2 = or x-4»0 11D. f'rom axioms 1 and 2 we derive the following tton to the other, provided its sign be changed. T^"«'»f 5ar-4 = 2T + 5. take 2x from and add 4 to both sides, *^«" 5a;-2a: = 4 + 6, »^<* 3a? = 9. 116. Transferring a positive quantity from one siMe of an equa- boThl^' ^ "^^ '"^^ *" *u60. 37. x^(a^b) + x\b~c) + x^{c-a) + .r(ab-bc)^x(bc-ca) + ab-ac = 38. a{b - c)x^ + b{c - a)^» + cx'ia -b) + a^b -c) + b^c - a) + c'x{a -b) + ab{a - 6) + bc{b -c) + ca(c - a) = 0. 119. We propose introducing here a few examples of easy equations of a higher degree than the first, but limited to one unknown quantity. 120. All equations simply express conditions or a condition to be satisfied by the letters involved. If a:»-2.r+l = then x must have a numerical value such that when it is substituted for X the two sides of the equation will be identical So also if {x-b){x- the whole quantity must = 0. -E'aj. i.— Solve a?-m? = 0. Factoring, («_ m){x + m) = 0. Therefore, if x-m = or ar + m = the equation will be satisfied- therefore x = morx=~m. ' Ex. ^.— Solve {x -l){x- 2){x - 3) = 0. ^- . , . a: - 1 = the equation is satisfied ; therefore 1 is a root of the equation. SimiKrly 2 and 3 are roots of the equation. 2a:»-4ar-6 = 0. 2(x-3)(ar+l) = 0. Ex. S. — Solve Factoring, Therefore thei-efore «'»3 or art-^ - 1. t1 70 SIMPLE EQUATIONS OP ONE UNKNOWN. 5 ; Hi i t EXERCISE XXXII. What values of x will satisfy 1. {x-a)(x-b) = 0. 3. {x-a){x + b)^0. 6. 3?-{b~c)x-hc = 0. 7. x»-8x+12 = 0. 9. ^~x\a^.b + c)-^x{ah + hc-\-ca)-abc = Q. 2. (a? + «)(«- 6) = 0. 4. (a; + a)(ar + 6) = 0. 6. ar' + (6 + c)a: + 6c=.0. 8. 2^-8aj + 8 = 0. 10. a:"- 6.r»+ 11^.-6 = 0. 12. (a + 6);c«-(a»-62)(a-J) = 0. 14. x'-2bx^-Wx=:^Q. 16. 9.r»-9j;-28 = 0. 18. 25i:»-49 = 0. 20. a:'-9a:» + 23ar-jjfi = 0. 11. 3.T»-10aj+3 = 0. 13. x^-ax-2a? = 0, 15. 6a:'+ar-12 = 0. 17. 4a;'-36 = 0. 19. x»-3x» + 3ar-l=0. 21. a:» + ar»-a:-l=0. PROBLEMS PRODUCINa SIMPLE EQUATIONS OP ONE UNKNOWN. ^ 122. Algebra is extensively used in the solution of problems of practical valua In proceeding to solve a problem the'tirst th^^ dude I H ?r" ''f ""^ "^^ '""''^ ^^^^"' - ^«'«' -d to ex elude aJl that have no bearing on the solution. The next thing n order zs to si^te in algebraic language the facts given, which If correctly done will result in an equation whose Solution wH give the required result. It must be noticed that before we caoi have an equation we must have t^o different algebraic expresli^" of the .a,n. valvs Thus, if we say that John ha^ 5 apples more ^an James we have but one expression; for let x represent the number of apples James has, then John will have x + 6 To obtain an equation we must be able to say that . ; 6 is 'equal to a d^p^rent expresnon. If we add to the fact already gLn th.it John has 10 apples we have the statement: x-f 5 = 10 from which X can be found. 8TMPLE EQUATIONS OP ONE UNKNOWN. 71 The art of expressing in algebraic language the/acts of a prob- lem must be acquired by practice. The following examples will illustrate some of the simpler methods of solving problems: Ex. l.—Whut number is that to winch, if 7 be added, twice the sum will be equal to 32 1 Let then and also therefore .r = the number; x + 7 = the number when 7 is added, 2(.r + 7) = twice the number when 7 is added; 32 = twice the number when 7 is added ; 2(a: + 7) = 32, or a:-|-7 = 16, or x = 9. Ex. ^.— What two numbers are those whose sum is 48 and difference 22 1 Let a; = the number; then, since the sum of the two numbers is 48, the second number must be 48-3*. But the difference of the numbers is 22; therefore, if x is the greater number, a; -(48 -a:) = 22, a:-48 + a- = 22, 2.r--48 = 22. 2j; = 22 + 48 = 70 a; = 35 = first number; 48-a; = 48-35=13 = second number. or or Transposing, therefore We might have assumed in this problem that the second num- ber is a; + 22; then the statement would be* a; + x + 22 = 48 2ar+22 = 48 2jr = 26 a:=13 /. x + 22 = 35 Here the second number is the greater. 72 SIMPLE EQUATIONS OF ONE UNKNOWN. aJ%cr'^\'1^ ^^"^" °^ '^°*^ *^^ ^^ y^''^^« of «"k together h cos V" r f ' " ^"'^^ ^^ ^'^^"^^^^ - ^^« <^'otk. 'Find the cost of a yard of each. Let then therefore and ar = cost in £'a of a yard of cloth; 2.r=.co3t in £'b of a yard of silk; 30^ = cost of 30 yards of cloth, 80ir=. cost of 40 yards of silk. But the whole cost is £66, therefore 30^ + 80a: =66, Z c 110.. = 66,' Hence value of cloth is £^ or 1 2s!*, and value of silk is £f or 24s!' EXBRcisa xxxm. ence 1^"^ '"° ''""'^''' "^''' '""^ ''''''' '^ ^^^ -^^ -hose differ- (oL^nl::^"" "! " ^"^ °^^« *he di/ro,-ence between (2x + 4)(3.. + 6) and (3a: - 2)(2.: - 8) equal to 96 1 above 80 as the sum is below 80. What is the sum ? 4^ The sum of the ages of two brothers is 49, and one of them is 13 years older than the other. Find their a^es 5. I bought 20 yards of cloth for 10 guinea!, part at 1 Is 6d a yard and the rest at 7s. 6d a vard VTr^ ' ^y la. Od. did T buy 1 ^ "^^""y ^^'■^'* "^ eacl» NoTK.-Be careful to express the various sums of monev in fh« denoimimtion. money m the same 6. At an election where 979 votes wero <»,'va« *i. ogether Find differ- bween ruuch them i. Gd. each same ssful ach? ceed ring SIMPIE EQUATIONS OP ONE UNKNOWN. 73 9. A person left $700 to be divided amon. fl,™ .«ch a way that the first was to reoei. ,oTbt:of t" 2^* "S double that of B;. What did eal! Jo"' """^ " """ ■f J.'V^'l^* ""' °"™'^'''' "■* " '» ''hose difference is 10 .„^ .f 15 be added to their sum the whole will be 4a7 ' 12. A father's age is twice as great as that of his son. h„t in .ea«ago,twas«.reeti«.,. as g.at. Find the a^roTXh " 16. The ages of two men differ by 10 years anri i f? t^e^elder was twice as old as the .ou^ K'nd^lii?, Z ^, ^rtfl"^! '™°'." "^"^ *PP'^ "»■* ^ "-^ times as many 15. Find that number of which it can be said th«.f f.. .• an hour, the latter at 22 mi es W^^^^^^^ ^ m Z" '*/' f " what time from starting? " ^^^ "^^^^ ^^ ^ 17. A man bought a number of cows for iJtin « • more horses for $120 apiece ^Sl 7u I ^P'^*'^' ^""^ ^ wi ^i^v apiece, and paid altogether Skoonn vr many did he buy of each kind ? S^^ner ;p.JOO. How TOcentsaponndl P™"" *'>'"">« ""'t"™ -ay be worth 19. A cistern is filled in 20 minutes by 3 ninea o„. „f i.- u a-ow^ £rou7Zh p:;:;::^^' '"' ^--^ ^-^ -""^" 74 SIMPLE EQUATIONS OF ONE JNKNOWN. 20. The sum of $500 is divided among A, B, C and D. A and B have together $280; A and C, $260; A and D, $220. How much does each receive 1 v 21. Divide $1520 among A, B and C, so that A has $100 less than B, &nd B $270 less than C. 22. Find two numbers differing by 8 such that four times the less may exceed twice the greater by 10. 23. A vessel containing some water was filled up by pouring in 42 gallons, and there was then in the vessel seven times as much as at first. How many gallons did the vessel hold 1 24. In a company of 266 persons, composed of men, women and children, there are twice as many men qa there are women, and twice as many women as there are children. How many are there of each ? 25. Divide 90 into two such parts that four times one part may be equal to five times the other. 26. Divide 60 into two such parts that one part is greater than the other by 24. 27. A bill of $100 was paid in dollar and fifty-cent pieces, and 80 more fifty-cent pieces were used than dollars. How many of each were used 1 28. A bill of £100 was paid with guineas and half-crowns, and 48 more half-crowns were used than guineas. How many of each were used 1 29. Thirty yards of cloth and 20 yards of silk cost $120, and the silk cost twice as much as the cloth. How much did each cost per yard % 30. A man bought 30 pounds of sugar of two different kinds, and paid for the whole $2.94. The better kind cost 10 cents a pound, and the poorer kind 7 cents a pound. How many pounds were there of each kind ? 31. A man has three times as many quarters as half-dollars, four times as many dimes as quarters, and twice as many half- dimes as dimes. The whole sum is $7.30. How many coins has he altogether ^ ■i SIMPLE EQUATIONg OF OUIt ONKNOWM. 74 children were there » «•»» each. How many 33. A wine merohant has two kinds of wine. n„« „ .*i. .a cents a qnart and the other 75 oente I onart C„,7k J wiBhe. to make a mixture of 100 g^oLt^l «^™" "''^ ^' How many gallons must he take of^LorwIdT "^'°''- xig xus wire, two sons and three dauchters A »«« ,•» to nave twioo aa mn^u i , ""•"fe^wrs. a son is than aU Zc^dren X therH*:: ■""" I'' ^"' '''' "- ^^ ^ ren together. How much was the share of 38. A bankrupt owes B twice as much as he owp<, A . An much as he owes J »r.A m. xt X ^' *^<^ ^ «« ^..aed am„.;l::lISd earre-cliT """ ^ *" every da^ he wtallt he ;o°l rfr '' ^'"'' ''"' "" of the time he had 2o7mii ! *" ^ P""™' ^' ^^^ <"^^ day. he worked ^"^ *" '""'"^ ^"l "■« °»-^' of ww!h l'';o7^f„i'^'' rr" "^ '"^'^"■= »»*>'- fi^xi. „..._ : ^'"" ""S*"^ and 10 yards broader .„„*»._. co^-J .4"are yaro. more than the former." Knd the si^^f ■;;:i """" I CHAPTER VII. COMMON FACTORS AND COMMON MULTIPLES. HIGHEST COMMON FACTOR. 123. The methods of resolving a single algebraical expression mto factors have already been given. We proceed now to ex- plain how to determine the factors, if any, which are contained in each of two or more given expressions. 124. A Common Factor of two or more algebraical expres- sions is any expression which is a factor of each of them. Two expressions having no common factor except unity are said to be prime to each other. 125. The Highest Common Factor of two or more alge- braical expressions is the factor of highest dimensions and greats est numerical coefficients which is a factor of each of them. The Highest Common Factor is usually denoted by the letters H. C. F. 126. A Multiple of an algebraical expression is any expres- sion of which the given expression is a factor. The terms "Measure" and "Multiple," used in arithmetic, are not very appropriate for algebraical expressions. The former is, therefore, replaced by the correct v^«rm, "Factor;" the latter is retained because we have no other word to supply its place. 127. The H. C. F. of two or more expressions will evidently be the product of aU the factors common to each, and may there- fore be found by the following 'RxjLE.—Hesolve each expression into elementary factors, and the product of all the factors common to each expression will he *J,^ u n m ^. vm jj., V. ^ TC<£ mrea. HlOHiSST COMMON FACTOR. 77 w?nlVV"!^ '"' '^ '"'" ^^"'^ expressions can be reac ily fac- tored Its factors may be tried in succession a^ diviTr^ of Z ^otr'^^^^"' and thus all tbe common facto. ^Zrt^; Ex. 7._Find the H. 0. F. of 45aW, 60a»6c», 35aW. The factors 5, a^ b, c», are all those which are founH in i. expression; therefore 6a%^ is the H. C. F. requhed . Ex, ^.-Find the H. C. F. of a»-6», ai + 6«. a'+6» »' + *' = (« + 6)(a'-a6+6»). The only factor found in each is a + A whiVh ;- fi, * H. C. F. required. ' '* therefore the Ex. A— Find the H. C. F. of (2a:-3)(.:-6)(3^_4) and 15a:»- l28:r'+309.r-220 « any, .„., .e found a»ong tL"lr; J tZTutX^^ « «»< a factor of the second e:.pression beoau7e 2 l^H f factors of 15 and 220. Tmna 3a: 4 wH ? i *™ °°* 5^-36.+66, and by fnrthe^rifl we fild^/sV'Ar'""'" '' BXBROISB X2XIV. Find the H. 0. F. of 1. aVc and a»5U 9 is^/s -5. 7o»6», 14a«6 and 35a5«. 4 17^,^ o.. . ,^ a.S....JVand20.V.:-;i^n^%r^^^^^^ 7. 6^y and I5^y'^20.y. g. 14„., and 21™'„ - 7^ U. ay +y2 and a6c + 6ct/. 19 «8A2^ ,,« , , ^' 78 HIGHEST COMMON FACTOR. 13. ar«-6a; + 6anda:*-llar + 24. 14. as'-Sar-U and a!»+ar-66. 15. x^-l and ar«+20ar+19. 16. 3^-Ux-5l and a;»-7a-170. 17. a^-y*anda:« + y«. 18. ar* + a:y + y* and ar»-y». 19. a^-y> and a:"-y«. 20. sc^+y^ and a:"+y» 21. 3* + X*i/*+f and ar»-y»» 22. ar»-4ar-77 and a^+343. 23. 8ar» + 27y»and6a:»+5ary-62/». 24. a» + 6» + 2a6(a+6) and a»-6». 26. ar* + 4ic»+16andi:»-8. 26. ar'+l and «» + ww«+«Mr+l, 27. 1 26a:' - 64y» and 7 6a:' + 60^:^ + 48a:y». 28. a:(a: + 1)», a:»(a:« - 1 ) and 2a:(a;» - x - 2). 29. 6(0-6)*, 8(a'-6')»and 10(a*-6*). 30. ac{a - b){a - c) and hc{h - a){b - c). 31. (a + by-{c + df, (a + c)»-(6 + c^'and(a + cf)'-(6 + c)". 32. 63^ - 6z^ + 2.r2y - 23^ and 1 2a:> - 1 6a:y + 3y«. 33. 3a»-3a% + a6'-6»and ia*-5ab + b\ 34. 3a:» - 3.i-»y + .ry» - / and 4aH» - a:»y - Bxy*. 35. 20j:* + a:»-l and 75a:* + 15a:' -3a:- 3. 129. The following principles enable us to find the H. C. P. of any two algebraical expressions. For their demonstration see Arte. 136-139. 1. If one expression is a factor of another it will also be a factor of any multiple of the other. 2. If one expression is a factor of two others it will also be a factor of the sum or the difference of any multiples of the others. 3. If one expression be divided by another the remainder, if any, will be the difference between the dividend and a multiple of the divisor, and will therefore contain all the factors common to the two expressions. 130. To find the H. 0. F. of any two algebraical expressions we have from Arts. 127 and 129 the followin Thr „ %'"* crease the H. C. P.. because 2 is not . factor of ZdiviL^; ^at we n n dc are ust be not a )f the 3x -7 Srst in- we HIGHEST COMMON FACTOR. gj reject the factor 39 from a remain,?-., tw The H. 0. P. of :^+4^+3 and ;^+8:r+7 is :r4-l It will be their greatest common measure If ^-o IT sions become 15 and 27, and ^+1 - q r;, .u r ^''^'^'' common measure; but if .3 the ~ u^ *" *'''" ^''^'''' and T4- 1 1 u- I- expressions become 24 and 40 and^+1 =4 which IS noi their greatest common measure Th: explanation lies in the fact that the other factors H 3 and J 7 have no common measure for all values of . but ttev h /' parucular values of .. viz., when . is an, o^', '^L' ''' '" suitable for beginners and 1 ,f ^J^T^^^^' ^«-ever, is not CO -siderable Vrf^irt^^lX:^::^^ ^ to the corresponding Arte, in L. C. M. ^^ "^^^^"^ 136. If F is a factor of A then it is a factor of ml • for «l Z' 18 a factor of A, we may suppose A-aP Z I ' ' thus P is a factor of mA. " ' ^^''^ ^^ == '^^' 137. If P is a factor of ^ and JS then it is a factor of mA ±nli • tor, suppose ^ = a/» and j5 - A P f »,« ^ »^«^r or m^i dmi^; JSaXrt:^:.r^ - - - - -e... „, 82 HIGHEST COMMON FACTOR 5 I isl '.■: ll i 138. Let A and -B denote any two expres- BJ A (p pB 'C)B(q D)G(r rD sions whose H. 0. F. is required. Let them be arranged and the divisions performed ac- cording to Art. 130, and let the various quo- tients and remainders be denoted by the let- ters in the margin; then we have the follow- ing results: — A^pB+C, B'^qC+D, G^rD; :. A^pB+C B'^qC+D '-2){qC'hD)+rD -rD+D ^p{qrD+D)+rD -i)(r+l) "Dipqr+p+r) Therefore 2> is a factor of both A and B. 139. Again, since every expression which divides A and B divides A -pB, that is, C, therefore C contains all the factors common to A and B. Similarly I) contains all the factors com- mon to B and C, that is, all the factors common to A and B; therefore /) is the H. 0. F. required. Also, overy factor of A and B iaa, factor of D. 140. If we are required to find the H. C. F. of three expres- sions, A, B and C, find the H. 0. F. of two of them, A and B; let it be 2). Then the H. C. F, of 2> and C will be the H. 0. F. required; for every factor of D and (7 is a factor of A, B and C, and every factor of i4, 2? and (7 is a factor of D and C; there- fore D is the H. 0. F. of A, B and C. BXBROISB XXXV. Find the H. 0. F. of 2. a» + a:"-Sa;-8anda^+2a:« + a:-4. S. *•+ ar" + a? - 3 and a:" + 3a^ + 53- -»- 3. 4. a«+x»-13a;-4and3a:»+10a:«- 13;r-20. 6. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. HIGHEST COMMON PAOTOB. 3aJ'-17a!»-5ar+10 and 3a:"-23a:>+23ar-6. 2«»-a:> + ar + 4 and ia^-a^-x' + Tx-i. 2«>-19a^ + 38a;-21 and 3a:»-20a:>- 12x + 35. 3a!«-13a:>+23ar-21 and 6a:»+a;«-44a: + 21. a^-3a;»+2a^+ar-l and a:»-a:>-2a:+2. a^+5ai^-!i^-5x and x^+Sa^-x-S. 2a:« - 12a.^+ 19a;»- 6aj + 9 and 4;c3- 18x»+ 19a.- 3. 2«»-lla:>-9 and 4aH»+llaf*+81. «»+3a^-8a:»-9a;-3 and a:"- 2ar*-6a:»+4a:»+ 13.7 + 6. 2«»-5aj* + aJ»-9a«-lla; + 6 and 3aJ^-.8ar*-4ar» + 6^-5a:-3. 9aH»+ll«»-.2 and 81a:»+llar+4. «»-209«+66 and 56a:»-209:c*+ 1. «'+l and a^+x^-a^-a^+x+l, 2a* + Za'x - 9a»a:> and 6a*x - 1 Ta'^x' + 1 4a V - 3aX*. aV-a'6a:»y fa6«ay-6y and 2a'bx'i/-ab'xy^-by, a:»-9^« + 26a:-24, :r»-10;r« + 31^-30 and x»-lla:« + 38;r-.40. «*-10x2+9, a;*+10a:» + 20^-10x-21 . . . . and ar*+4a:'-22a:*-4x+21 ax^-bx*+ax-b and cuifi+(a~by+(a-b)x-b. a^ + 2m.r»+mV-n' an > r^.(m-iy+{n-m)x-r^ «» + (5w»-3)«»+(6m»-15m)ar-18m« and «»+(m-3)a:«-(2m»+3m)a: + 6m». p«'-(^-y)ar«+/. i):r + y and;,««-(p4-g)a:> + (p + y)^_y +(6 f 2c)«+a a' + 6»+c»-3ado and a(a + 2b) + b(b + 2c} + c{o->2a). ay(aH» + 6»)+*a:(V+a««) and aa:rt/« + 6»UA«/A^4.«VA 88 84 LOWEST COMMON MULTIPLE. i i LOWEST COMMON MULTIPLE. 141. A Common Multiple of two or more algebraical expres rZtor^''^ ^^P'-^^^^ion of which each of the given expressions is 142. The Lowest Common Multiple of two or more alge- braical expressions is the expression of lowest dimensions and smallest numerical coefficients of which each of the given expres- sions IS a factor. ^ T n^'^T..^'"'^^'* ^''°''"°'' ^""^^^^^ ^ ^"^"y demoted by the letters Li, \J. M. 143. The L. 0. M. must evidently contain all the elementary factors winch any one of the given expressions contains, but no other factor; therefore for finding the L. C. M. of two or more algebraical expressions we have the followin*^ RvLK.~^esolve each of the given expressions into elementary, factors, and ths product of the highest pouters of all ths different factors which occur will be the L. C. M. required. I I BXBROISB XXXVI. Knd the L. C. M. of 1. 1a% Zab\ 6a6c. 3. 18/.y, 40aV, 75ry. 5. a^-h\ ab-¥h\ a^-^^ab, 7. «»-l, r*-!, r» + a:+l. 2. 6a»6% 106W Ibahcd". 4. 2UW, 35m«n», TOmVp. 6. a'-6', a-6, a»+a/» + 6«. 9. i{^-y%^x-y)\lQ(^x.^y)\ 10. M^-\\h{x+\),^-^^l 11. o»-6», a^-b\ a« + a»6« + 6«, a«-6«. 12. a»+6», a*-a«6» + 6*, a'-b\ a'-b», a*-b*. 13. a:»~6j; + 6, ar«-4, a:»-9, a:«-7«+12, «>-16. more LOWEST COMMON MULTIPLE. 35 16. (x»^iay, {x+2a)\ {x-2a)\ 16. 8(a»-6«)(a-6)» 10(a*-50(a.-6)., l5(a'+5«)^ 17. 12(a»-6«)(a + 6)3, 9(a«-6*)(a-J)2, 24(a»-6»)> 18. a:*-10«»+9, a^-7a: + 6, ar»-7;E-6. 19. a«-6»+c»+2ac, a« + 6«-c»4-2aJ, 6»+c»-a»+25a 20. 4(^-.,.), 20(^^.V-^-^), 12(.^-.^), 8(^-^y). 21. «»-l, a:»+l, ^_^^ ^7_^^ :i^+ar» + ar. Let A and £ denote the two expressions, 2) their H F • let a ^d^i denote the other &<*>™ of A Jd i,, so lul:!^, or^Ltrf °" " 'r*^ "''""^'°™ -*>■"* <"">'»!'« either ^ or ^ as a factor is evidently obD. Then ahD. aD.hD AB D In practice it is easier to divide one of the expressions by the H. C. F., and multiply the quotient by the other expression mdt^riwr .<^«^^«^^^«ltiple of two algebraical expressions is a f th^rt Ir^^^^^ ""^^^^'^^ '-^ using the notation oi tne last Art., if M be any common multiple of A and /^ ;«• «, * contain the facte. „,V, if it contains no othll. th™ ^I'^L 80 LOWEST COMMON MULTIPLE. 146. To find the L. 0. M. of any number of algebraical expres- sions, A, B, C, etc., we have the following RULB.— i'lW thA L. C. M. of A and B, denote it by M;find the L. C. M. of M and C. and proceed as be/ore until all the ex- presstone have been used. Tha last multiple thus found will be the L. C. M. required; For the L. C. M. of M and G contains, without excess or defect the /actors of M and C7, thit is, the factors of A, B and C. iSiml larlyfoT any number of expressions. i t'- ->.$■ Ill If 8 ifti I HX3SRCISB XX2VII. Find the L. C. M. of 1. Jr'_6^+lla;-6andar»-9a!» + 26a:-24. 2. r» _ 9.<;» + 1 9ar + 4 and aH» - 6x'» + 4a; + 1 . 3. 2a:' + 9a:» + 7ar- 3 and 3a:3+ 5a^- 15a: + 4. 4. a:*-a;*-ar+l anda:*-2a:»-a:*-2a;+l. 5. 6i-» + arV-lla^-6y»and6a:»+lla:V-ary«-62^. 6. x'^-a^-da^x'+lWx-ia' and a:*-aa:»-3aV+5a»;r-2a*. 7. ^-7ar-6 anda:»-4a:*+4a;-3. 8. a:«-10a:»+9, a:*+10a:» + 20ar»-10a;-21 and a:* + 4a:»_22a:>-4ar + 21. 9. a;* + 3a:»-a:-3, 2a:* + Sar* -5a:«- 6a; + 3 and 2a:* + 3a:>+2a:»-l. 10. a^ + 4a;»+16, a:^+2a:* + 4a:»+8a:»+16a; + 32 and a;*-2a;* + 4a;«-8a;2+16a;-32. 11. a:» + 6a;»+lla: + 6, «»+7ar'+14a: + 8 and a;» + 8a:2+ ll,\^ + 12. 12. o»a: + aV-2aa;», a'a:-a»a:»-6a.r» and a»-2a»a;-5aa;»+6«». 13. a;>"3a;«+3a;-l, x^-a^^xtl, a:*-2a:»+2a;-l and a;*-2a:» + 2«'-2a;-j-l , i CHAPTER VIII. FRACTIONS. 3^' ~a-b *^ fractions in form but 147. A Fraction is the quotient of one expression by another v^hen ^the^chvision is indicated but cannot reaUy be perforn^e^' Thus -, J, are fractions; b"- ^ ""'"*' not in reality. 148. The expression to be divided is called the Numerator and that by .hich itis to be divided is called the B.^^^, of ffLft"^^ "' ^'^ ^^^^"^^^^^ - -"^^ *^e Tenns 149. Any expression may be written in the form of a fraction by considenng it the quotient of itself by unity. 150. As division is the reverse of multiplication any expres- firs^ ::Lr ^T. \^ '^^^^ ^^^ ^ ^^-^ denominatrX first multiplyxng it by that denominator and writing the product as numerator over the given denominator. ^ 151. Since the numerator is a dividend and the denominator Its divisor we learn, from Arts. 78 and 80. the following facte- 88 REDUOINQ rRACTIONS TO THEIR LOWEST TERMS. ; ! 4. If the signs of all the terms in both numerator and denomi- nator be changed the value of the fraction will remain the same. This is equivalent to multiplying both by - 1. \ 5. If the sign of either numerator or denominator be changed the sign of the fraction will be changed. These examples should be carefully observed. _a_ x-y y-x c-b a-0 b-a REDUCING FRACTIONS TO THEIR LOWEST TERMS. 152. When the numerator and denominator contain no com- mon factor except unity the fraction is said to be in its lowest terms. A fraction is reduced to its lowest terms by dividing both numer- ator and denominator by their H. C. F. 153. In the following examples the factors of the various numerators and denominators are evident by inspection, or may be determined by methods already given: — 2a* - 2a6 2ci(a - b) 2a Ex. 1, Ex. 2. Zab-W db{a-b) 36* a* - a^b - a^b* + a¥ a\a -b)- ab\a -b) a» - a6» a\a-b)-b\a-b) ~ a*~b* a(a^ - h*) a a' 6_a*b-ab* + b' {a^ + b^){a^ - b^) al' + b'"' EXERCISE XXXVIII. Reduce to their lowest terms the following fractions: — md^ . UPmn^ ^* 6a6»' 18aW ** 27a"6V 2. 126»cd'* 125aary 150a6o* 6. 2Umw' 13a»«6» 65ai«6'*«" a tt'-' + 6>' REDUCING FRACTIONS TO THKIB LOWEST TKBMS, - 360g»yV 89 10. 13. 16. 19. 22. 375aV 126a:>* ay 33 V -5ay* 6aV-66'ary» 12a«a;»y-1262a:»y* 12a' +186' 8a*- 186* • g 240j?Vr' ' 180;>yr»* «i/+a6w' 27 Sar'y - 3a?y« 6a:»-6y» ' 16. ^~'^_ ^ 18. 25. 28. g' - g6 - 2&» a' - 3a6 + 26»' o' + 6»-c» + 2a6 20. 23. 26. -- a'6ca? - a'6+4ai* 4ar~4v 8ar»-I6a^ + 8y>' g* - g V + a* aJ»-l «'-6«4-c'-K2ac- '"• iir?Z^72^. a?»+2a:>+2;r+r 32, (?fli)(2^)-ar«(6:r»-7) 34. (iLitfUfeiitf*} 31^ gcg'+(a(4a' -~^* ar* + ar» - ar - 1 * 3a».r« - 2aar« - 1 4a»a;''- 2a'»jr* -• 3aa;» + 1 * 27. oa?'+(6(;-a6-ffc) j- + a^,c - 6»c 18. (^-^4a-)'-2(a r' + 4ar^-lft X* - 10:^>T9 20. ^*(^+y') + .r.v(aN ^ ' ab{x' - y») + ary(aa _ jay o' + 6» + c'-Sa6c a' + 2n6 + 6' _ ai"^ • 26. .^^^ - 3^V - 2y««> + 3y 3a.V+2y* - 2ar'y» - 3/««* -. 22. 6V-2a6c» + aV-a»a;» 28. (y» - «)' - (eg - v')(a5 - ^»^ iRirs. b: — til ■1. • + 6)c 91 CHANGES IN THE FORM OF FRAOTION& 29 («' - l)a?' + (2a' + a)a;»+2 a^ x i ' (a - 1) V - 2(a - l^PCl^JTT * 30. (L±fOXl±:^^+ <')'(1 + a«) «(l-c)(l+c)-c(ir^)(l+-^- (m + n)» - m» - nj - mn(r^TT^* 33. (g^ ^)' + ^aHa - 6)* 4- 8a'6Va - />)« (a + 6)" - 6ab{a + b)* +l^«6»(;r+6p- 33. o.'b + c'd + a^d + bc' ' a6» + cflP + abJTMTacd + abTTb^dTbd^' 34. igi+^y)M?:ift)^^ + (a - 6)»^^ (ax - byf - (a + b){x + zXax - by)T(^Tb)^^' {(6 + c){b + a) + 26(c + a)}-^ -■(^^^^(^Tjyi- CHANGES IN THE FORM OF FRACTIONS. 155. If the degree of the numerator of a fraofmn . i exceed. a,at of the denominator the frao^on I^t" hT ^ " the form ot a mixed or integral exnression hv Ai Z- T'"^^^ '» ator by the denominator. TheJoZt wm tt' T" T"- of the result; the remainder, iflny " the „ " ^r" ""^ the divisor the denominator, 'of theLTl , p^r^"*"'' ^^ 2y* ory^-y^x + y3^-.x» + 2x* X +y' according as we consider .r or y the leading letter In arifH . .. i, written hetween two .Z^^Z^^lLZl;^,:^ is understood. Thus 3* mean, 3 + 1. but a* mean, a. * -f? ■.%. ■M. *'-^"% > o^, \^ IMAGE EVALUATION TEST TARGET (MT-S) i^O h A f/. Photographic Sdences Corporation 1.0 |50 ™'^'* lllll^^ 1^ Ih III 2.2 ! "^ IIIM ■UUU I.I !.25 — ^ 1 .1.4 1.6 ^ g ,c ^ ■^c** 33 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 873-4SC3 o f/ie ^°\/'^ i/.A ^ I I 92 CHANGES IN THE f6rm OF FRACTIONS. fore a fraction the dgn of every term in either numerator or de- nominator must be changed. 156. A mixed expi-ession may be reduced to a complete frac- tion as follows : — Multiply the integral expression by the denominator, to the pro- duct annex the numerator, and under the result write the dmomir nator. fl^Hj'_(g + 6)'-(a« + y ) 2ah a + h Ex. a + &- + 6 a + b "a+b' 157. A fraction whose numerator consists of several terms may be split up into an equivalent number of fractions connected with the proper signs. Ex «* + 3a'6 + 2a'6' _ a* 3^ 2a«6» a 3 2 The truth of the above is evident from the division of poly, nomials, Art 82. BXEBOISB ZIi. Reduce the following fractions to mixed quantities:— 1. 4. 7. 5^+7«-3 a: + 2 • x-y ' 3«»-7««+l 2. 6. 8. a^-ax + x* a + x 2si^-x^+2x~3 x*-x+l '• 3 33;»~3y» + 5 x-y 9a* 3a:"-4«+3 * "* 5x^ + ix-l' Reduce to com>jlete fractions 6. 9. 3a + r x'+x-^V 10. 1 + X 13. 1- 16. a-b+ 1-z g- y x+ y' (a + by a-b 11. 1+X + 14. 2a- A - . 17. a? + 6- l-x' 2ab a + b' 22; -15 12. a'b + 6« a;-3 * a +b' a + b I jrator or de- mplete frao- ', to the pro- the denomi- ',b ■b' '^eral terms } connected 3 2 ' a >n of poly- V + 5 r f-1 6» (g - by a + b («+y). LOWEST COMMON DENOMINATOK. 19.^ + ^^^+^. 20.^-^^^,^^ ^ * + y 21. l+ir + ar»---^±fL. 22 . «. «'-ay + v» 98 a:'-4ar» + 8 a? + a 23 ar4.1 fl-*^ + o ' (ar - 2)« "• 24. 1 +2y + 2y«+2y» + ?y!±V Separate into fractions with a single tenn in the numerator. • 3^~^^- 28. «^ + «y'g + a gVj^ OQ ^^+bcd+cda + dab ik^ An^ p,^ ^ lOOz. ' LOWEST COMMON DENOMINATOR. 158. Since the value of a fraction is unchanged by multinl^' both numerator and denominator hv + 1, . ^ multiplying that two or more fractiZrarbe^e^rbV !' '^"°"- tio^ having a common denomin'ater bf rfl^T^^" '- u.^a by d^ a. z. r/;t r r^:.:^-'^'^ ^ J:,t^^Z "" ^^''^^^^^ ^^^*- -^^^ ^^« lowest com- 3Ac* 5ca' lOa*' L. 0. M. of denominators, SOaic Quotients of 30a6c by denominators. 10a, 66, S^ Fractions required, ^t. J^ I2<^ SOodc' 30ado' 30^* \ M ' il ADDITION AND SUBTRACTION OF FRACTIONS. Ex ^.-Reduce to equivalent fractions with the lowest com- mon denominator 2 3 ar-3 x + V x-V a^^y \ L. C. M. of denominators, ar*- 1. Quotients of a:»-l by denominators, ar-1, x+1, 1. 2ir-2 3a: + 3 a;-3 Fractions required. «"-!' a»-l' i^^' BXEROISB XLI. Reduce to equivalent fractions with the lowest common de- nominator , 2« 4« 7.T • 3 ' 6 • fO* 4. 7. 9, 11. 2. 5. 1-ar' r^' a + 6 a — b ab a - b' a + 6' ^^r^J* a; + 4 ic + 2 8a;- 3 2-3« 4a:» ' ~67"' a b o «i 6 c ic' ca' a6* 0-6' + 6' 6. a; ar«- 5ar + 6' «'-7ar+12* 8. 10. :. 12. 7 8 2a:-4' 3a: + 6' 6a;»-20' a? + 2 ar + 3 a; + 4 1 -ar' a:-«»' JT^* c(a + 6) __ a{b-\-c) (a-b){a-cy {b-a)ib-cy ''' ^bi^Tc)^)* ^p^y ADDITION AND SUBTRACTION Oi^ FRACTIONS. 159. The rule for the addition and subtraction of fractions follows at once from the principle of Division. To divide the sum or difference of quantities we divide the quantities in succes- sion and connect the quotients with the proper signs. Conversely the sum or the difference of two quotients is equal to the quotient of the sum or the difference of the dividen.ii by the common divwori therefore L rs. >west oom- nmon de- e ab' y a? - xy f* 'ractions »^de the 1 suoces- iversely, quotient K>miiion ADDITIOti AND StTBTRACTlON OP FRACTIONS. 05 160. To add two or more fractions we have the following RuLK.-i?.c/«c. the fractions to equivalent ones having tU lowest ^rru^n^n^,^ Add theirn^erators, a^ LLll fum as numerator ^orite the common denominator. lo^ To subtract one fiction from another ^e have the fol- RXJLIL-Jieduos the fractions to equivalent ones mth the lowest ~^^^ fu^^act the numerator of t;.sul>tr!lZ £x. 7.— Simplify —L- + ^ . ^ -3 ^ 1 +x^l -ar'^rr^- Reducing the fractions to equivalent oner ^ aving the lowest common denominator we get 1 fjlj l-« l+a? «-3 _^-x+l +X+X-3 x-l l-«> l-x" I Ex, j9.— Simplify a {a+b)b'{a^b)a (a+i)6 (a-6)a ab{a^ - 6')'"^a» - i'J Simplify 1 » . 2a; 3aj 6a; nXBROISB XLII. • a+ar 2a-a: a; —a n 3a 6a 7a 66 ^- 8- + T+16 + 24- i 96 ADDITION AND SUBTBACTION OP PRACnONS. U 3ir~8 2x+i 6 U 7 ?^.*«-3 33-Uar 21 9. U ^ _1^ 42 1 ^y-1 13. 15. __JL_ ^ 2a& _! 2 2^ «:» 17. %-iL + y « + y «" + ay* 19 ^ " °^ g' •<■ a6 * ia - a' "*" ia + 6«' 21. ^+ ^ -^ y* , gl.gMj^ ^ 4a:- 6 2a; 4a:+2 8.1+ '+1^1 « 2a:^3a: 6a;' 10. ~+L:if +iJL? «* Ac cT"' 12. ^Lt^+^lif ^ + 2a 14. ^^^.i:ii . _±:2« * «-2 a;-3 a;»_e«+6" aJ' + ay ay + yS^.c + y* «-y «+y 18. 20. a^-ay+y* a;>+xy + y«* a + 6 a«-6«+(^T6)-f 22. 4if+2 + ftf 23. — i- + ^y__.^lZ^ «-y a?'+ay + y»'^"^3Zpr- 24. -^±L. + _^zL_j. 2 a;»+a;+l a;»-a;+ 1 "^^+aJ+r 25. -^ilA- + -fLlL+!(^±^ ax + b^ aa-by a^a^-^hy' 26. 1 2a; a? + 3a; + 2 "^a;* + 4a; + 3 ''"PTB^Te* 27 ™lll^- + _JLtf_ 1 • S(a;».a;+l)-'2(a;»+l)+6(^- 2a ^^ + ^^ + ?ilJ 3a;-4 «+ 1 «- 1"*" a;+3 ■*"i32' ; 1 ADDITION AND SUBTRACTION OF FKACTIONa 97 Simplify , 3» 6a; i. 7. 10. 12. 14. 16. 18. a a - b' a + b' a X 2. 6. BXBROISB XLHI. 4ar-7 2a;-9 6 3 • « + y x-y x-y X + y' ♦»+n m + n s. 6. 1 1 ar-l'aj+1* a - b + a - b~a + b' ^+y' y x{a-x)''a{a-x)' ' mn-tt?" m^^mn ^' l^'J^' a+x ^-xy + y» x* + xy + y»' 1 1 «"-19«-i-84""«>-12aj+:36" a;'+8a;+16 ar - 1 !r'+.7a;+10'"«T2* 3g+& a-t-76 a'+3a6 + 26» o' + 606 + 66*' a 6 2a^-.lla:+12 2a:» + 6a:-12- '^' (^Tjx^^^-^^rilJTr:^ i (c-aXx-e-a) x^-{a + b)x + ab (x - a){x - b){x - c)' ^a-Sb Sa-b 7ab{a-b)-.2(a'-b') 3a6(a + 6) - 2(a» + 6»)* 3a^-2a;-i-6 2a!* - a:» + 6a:« - 2 a; + 8 (2«»-3a! + 4)» (2a;»-3a; + 4)* * 26. From -^+-^ + -^ take ^.,±^J^ 98 ADDITION AND SUBTRACTION OF FRACTIONS. 162. In rimplifying algebraical expressions two formula are frequently of service, viz.:— ""nuiaa are ■• (« + *)' + («-*)»= 2(a« + A«). 2. (a + 6)«-(a-i)»-4ai. ^They should be remembered in words as well as in symbols. combilL^I frf '''' ^- ^- ""• °' '^^ *^^'^^ denominators and combined all the fractions at one operation the labor would have been considerably increased. -£•?.;&.— Simplify ^L_ + __y!___. 2a!y x y{^-y? K» - vT T^TTyf- -y : -2. Grouping the first three together, and aJso the last three and then combining the results with the proper sign, we get y(* y) xix^yf i^.y). ^^^(^r;p=^(^r^,; i^^l251_(£iji*_(^jJ^2M^^ 4:r«y» 4xv ADDITION AND SUBTRACTION OF FRACnONi 99 Ex, 5.— Simplify S Observe that *-3 and 3-ar, x-^ and 6 -ar are not different factors except in dgn; they stand to each other in the same rela- tion aa a and -a, and the quotient of one by the other is - 1 The L. C. M. of the denominators is {x-2){x-Z){x-b)- the quo^ tient of the L. C. M. by the first denominator is ar-3- by the second, -(x-2); by tJie third, *-6. In the second cj one factor m the denominator, 6-ar, is the negative of the corres- pondmg factor in the L. 0. M, which gives the negative siim before the quotient; in the third case there are two such factora. and consequently the quotient is positive. We get^ therefore, the following result: — 2{x-^)-Z{x-2)+ix-S) -5 (a:-2)(ar-3)(ar- 6) "(r-2)(ar-3)(;r-5)' The result may be written with a positive sign in the numer- ator by reversing the terms of one factor in the denominator, thus : 5 (a?-2)(a;-3)(6-a:)- In connection with this example read Arts. 56 and 161 (6). Simplify 1. 4 BXBBOISB ZLIV. 2y x-y ' X + y^ 3^-y^' 2 __1 3. l-x l-a^-'i^.^' 4. y 1 2ar 2xy X X ~ y x-¥ y a?-y^' -y X + y «>+y« ^^.y^' "• x-y~^y—;f^' 6. J \ gjL V X 7. 1 x-2 . 8. 4(a:+l) i(x-l) 2(^+1)- *'• 2(^31) -2(i:^)+(^frTy, 9. 5-i. ' • «" * + 1 (« + !)'■ 10. ij-l +»._»_ I 100 EDITION AND SUBTRACTION OF FRACTIONa 11. 13. 1 2z+l '''-'>"' '^^^^y ''-^m-i^w-^T^. . L._'*-20« l-2ar l+2a? 4a:»-l* 14. -?- + J^_Jf^ 25 «(16-ar) 2a?+3 2-3a. ''• -i^rr-^ar^-^ 16. 1:l_%^*_^^z2L_ 17. i + y._J^^_^^:V_ ,^ *-y 2y v« + a./« i' VJr ar; a: « - y g*-xy»' 20. ,,-f!±L_^.__4+c___ c+a 21. 7-^:±--+_^±«f_ c»+o6 22. |::^+izlf l-2« + «> 2-6a:« 6-4^ 6+4* QTB^jTi^+gg^^ygp. 23. i 4-^. ^ ^ ^"(«'+i)""*'?TT-p(?^,. f -i- + fez2!±(ilf)Lt(c - a)« 3a;+5 4a; + 12 1 27. 1 4(^ + ir4(rTI?-i6(;Ti)-^i6(^ri5+^^ 23 1 1 1 I 1 ~4(««+l)- MULTIPLICATION OF FRACTIONa 101 !)• MULTIPLICATION OF FRACTIONS 163. To find the product of ? and i . b d* Since ^ means that a is to be divided by A, *• . c •'• r x6-a/ similarly - xc^-e. 6 "*' 5 '■^' *y ^ ***® product required. Now, a=.6a;, o^cfy, ae ^ bd To find the product of two or more fractions we have therefore tne following RvLiL-MuHtply together all the numerators/or the numerator 0/ the product, and all the denomtnatore /or t/te denominator of the product. ^ 164. If a numerator and a denominator contain the same factor this factor will be found in both numerator and denomi- nator of the product, and must be removed to reduce it to its lowest terma The process may be shortened by cancelling such factors before multiplying, which will evidently give the same result. JLX. I. -—- X -T= — X 3xlOx9xa»6>c« 1 SB ■ 66c 27ca IQab "o x 27xl6xa»W~8 * x(x-2) " x'^2x ' I t: P t 102 MULTIPLICATION OF FRACTIOWai Bx. S. o»+«» .X-i-x-: "-^ a»-aJ (a+xX^ax+x^ 1 • a«- /« ,n/ j BXBROISB XLV. I 3a 2a • WW 2 5a 36 5. 8a»6»x 1 4a»6' g 6a^ 35&*a; ' 7aA»^5i^.* g TarVa- 9a6c » * 18a'd»c^28^- wy» 4ZX (Zxy iQxy 8ai-12^»' 10. ^''^ ~ ^'^y ac^-bcd abe-m^adx-bdy' 12. M^-y*)' 11. ^ZLy'Hz^ ^ ~xy a*+ab ' ex X; _ i« a^- 11a; + 30 a;« _ 3« «* V «/ V a + ary a»+y^a-6^ a'+a6 4. -y- 16. fa'-SV^^!^!^^,^^ /^^_ aVx ^-Sxy^2^y>^ ^^^ '(^^>^2-irr^^. 18. (a + -^}x/a--flU^^x (« + *)'+(«-a^)' 19. -= ■X~, r:;X-^ -X MULTIPLICATION OF FRACTIONa 103 20. 21. «" + 2ary +y>^a;*-y»^x* + ary + y«' «*+«-2 ar»-8ar + 15 a:>+12« + 35 X --: :: ::r- X x'+ix-lb V+6a:-7 ToTHJrp-- {x-^yf-j? («-y)»-a:»^x»-(y + «)i^^Z]^,- 23. w' - win + n' • ^ """s : X - '— ' 24. m» - 3win(m - n) - n» m» - n» m* + w* + mn{m? + n») * a*-3a'a;« + a:* d}-ax-V^x' a* - 4ar* + aa:(2a« + oa;) a*-aj* + ax(2a' + aa;) " 165. The product of two expressions consisting of several frac- tional terms may be found by multiplying each term of the one by each term of the other and connecting the partial products by the proper sign, as in ordinary multiplication, or the two ex- pressions may be combined into single fractions and their product found as in the last exercise. U 3^2A2 3;-4^6"*-4"6~9~6 or -' ff^i4)(f-^)= ^ bx 1 4'^36~6^ 3a:' + 2x+3 3ar-2 9a:» + 5ar-6 a? 5x 1 36 4'*"36 6* h 'I *W* MULTIPLICATION OF FRACTIONa All the TOrions artifices used in the previous chapters to abbre- ™to ae work h, mumplictiou and division, and al'so the v^^'^ t^nsof^tormg, may be appUed in the «une way to fractional ISXBBOISB XLVI 17. (a' + 2 4,y„.-2+2\ 18, /'^xi^l\/y. 1 i\ 12 H. (1 + DIVISION OP FRACnONa 106 toabbre- e varioua ractional )• -i) DIVISION OF FRACTION& 166. Todivide%yf. d Let? Now '*. ^ =y, then - is the quotient required. a bx c "dy* d b a .*. -X bx d X ^^6 "y" lo^n ^^^^^ ''''^ ^'^^'''' ^^ *^''*^^' ""* ^""^ therefore the fol. Rule.— /nwr< ) ^-ab (*«-a^X*'-*') (*'-a^(*>-^)"?Tar- loe DIVISION OF FRACnONS. Simplify - S5ac^ 5x1/' 5. fjLf^^^y + y! * xy-f a?-xy' 7 ^*-^ ^^«*-2aa; a'+4aa; ' ooTi^* BXEROISB XLVII. 2 5^^_Ji * 15a6» • 30aA"* 4. 6. 6x 10 3aj— 3 * X— 1* 8 — ^^^5Mll3 15. - - 1 a b b» b &> li+r + -«+«-«. ^ X a d 16. ^-f + ?_«^f _? ,. «' + 6»-c«+2aft o + J+<, 18. f^'-f!z3^^/*+y ^-y\ (a + c)« - (6 + (£)» • (a - 6)»r(rfr^r mmm COMPLEX FRJLCnONa. 2x-3y • Va^-y"""~^^:y-j• \a-bx b-axj ' \a-bx~r^y 23. f^+ _f^^ ^ A, ^ * (2a + ft)*>\ 107 .»• COMPLEX FRACTIONa Bx.l. ^«. j&. 1- \ ^ J a a + 6 a ' I'X- 1 + X 1+x 1 +« + ««• the ™*. ,..,„^ „ Cea^Tu^^^^',-- -« .. 1+ar 1-4:- 1__ " l-a>-.i *»y multiplying numerator an J 1 + a? denominator by l+«. 1 ^ 1 -i_ ^ " T~r~~. — : by multinlvino k« -» I - t- ■■^- i. T X + X" ~ r-j — ^ '^j «• . I! ! 1 il 108 Simplify 1 ^-1^ COMPLEX FRAOTIONa BXSBOISB SLVni. 2z 2. l-|(*-2)- 3 % - 3^ 5 2^-K^-2) 6. -» i(3:r+l)-| 1(^+1) + a!-l 7. '^^^ 1- 6» + a«* 8. I + X 1 -jg 1 - a; 1 + a; 1 + a? 1 -~~i' 9. 2a6 a*f6' 2ft» 10. 2a' a'+6« a»+6* 2a«"' 11. 26» a»+d« 13. a;-y x'-y' U. 1 - a; 1 +ar •i + i + i abbcea a'-jb + c)* ' abc a+ h a - h e + d c - d a + b a - b' + ■ 12. a + 2b a a + b "^ b a + )ib a • 6 a + b \x a)\x'*'a) \x ~aj\x'^a) 16. ay + y»«« + a!y c-<^'*'c + rf «-l + 1 + x+a X- a 1 . 1 4-a! 16. «-« «+ -..L^ULLl 17 « ft+c f, y + c'-an 18. 3a6o &(? + ca-ai X— a x+ a «-l ft-1 c+1 » & c 111 " - + _ __ a 6 c a b +e h- a + ' 19. 1 « 1 a a a + b \x a) \x at 4-* I ->■ ^> o 1 '-^ g 1 a COMPLEX PKACTION& 109 (a-y)(a-ar)«~(o-y)»(a-a:) 1+g 4g 8aj 1-ar 22. l-^'^l+a;''^r+^~fT^ (l-2m)»+(l + 2m)« 21 (^-'t^')-a-2^! (l-2m)'-(l4-2m)« (»jf|_2y)»_(2«+y)s- y(« - y)» x{x - y)» MISCBLLANK0D8 EXAMPLES IV FRACTIONS. 3. Find the value of — + _JL »hen r ^^ x-a x-b "^"^73* Simplify {a + b a-b a'-b* j^Tl' (aa? + 6y)« + (6ar-ay)' • l ; I 110 8. 9. 10. 11. 12. 13. U. 15. 16. 17. COMPLEX FRACTIONa {«V + y^ + y(a^-y^M - U'(x'-y') + y(g' + y«)}« {a{x + y) + b{x - y)}» - {a{x - y) + 6(x + y)}« " {(g + 6) V + (g - h)yy - {(a , 6) V + (a + ^)y | , {(a: + y)V + (a:-y)»6»)»- {(a:-y)V + (« + y)«6»}>- {ax-ay + bx-byy+2xy{a + by ■2ab{x-yf-iabxi ' \n m / \n m / l(m + n)'-(m-n)»A^?T^« + ^*« »* / «'+l a^+s^'-i-l {a?~xf^1F-x^ + x-\ ■• /^a\' (a;-a)(y +6) « + 6 «_ft when a;> a!+2a x-2a 4a6 a + 6* 2i-x 26+aj"46»-a:» {ay^hxy- {ax-by)* (ay-bxy+ (ax-by)* {a + b){y-x) (a-b)(y + x) * l+ol+61+c- X -V V - when a » 2, br ^ 1 -a I - b' I -c « + y y + z z z-x e — « + as 18. )9. 20. 21. [t^_^Pq±_pr^_^ fx p\ /px q\ \q8 f»8 ^ r^^ qr «/'^iT"^j' Za{a? + gg; + a«) 2a;» + Zaa? - 5a» ja_ 3aar-3a« «*-«' sB' + gS X- a"'"iB»-aaj + a»' (a + 6)(l-a6) a(l-&«) + &(l--a«) (1 - ai>« - (a 4 by ■ (l-a»)(l-i>)-4a6- (c - d)a* + 6(&c - AcQg + ^b'^e - b*d) (be - bd-i-r? — /v/W j. a/A*/, j. A-J 1-J\' — w a — ucu) COMPLEX rRACTIONS. m oo g'- fy'-f g»-3ayg ,3 (26-o-a)«-(2c-a-ft)» (c-a)»-(a-6)» • 24. If y+«+tt-aB, «+M+a5 = Jy, u+a^+y = c«, x^-y^z^du, 23. If a;»-y2! = a», v'-zx^b^^ i^-xy«<^, find the value of «V+6'y + c*« . x+y+z in terms of a, 6, c. 27. Find the value of (g: + l)(g«+l) «.+ 1 when «=-, . y 1 + a 1 + 6 (ajy+l)(o»+l) y+1 28. Find the value of (x + a){x + mb)-{x-ma)(x- b) {mx + a){x + b)-{x-a){mx-b) ^^^"^ "'^'^Td* 29. If «aa+6+e+(; prove l-b' 2ab »- a § -b 8- e 8-d + — ^r— + + __, a c 30. If 2«c=a+d+c prove ade »-a 8-b 8 - e " 8 " 8{8 - a){8 ^b){8- e)' 31. If a:-?^±iz5 prove l±^_(< ^-b + cy + iab a + b + c" b + ax' (b - a + 0)* + iab' ^2.Ux + y + z~0^roye^^^^l^ + y(^Z^.^^tl^ y-» «-« «-y " 33. Iff^ ?_.« L ._ 2 2 !i 112 THEOREMfl IN FKACTIONS. 34. If 6 2ae 1 = Drove , 1 , 4 1 c — a e 1 a + c ^^^^° a - b^b-c^ a 35. Ifar. prove z -+- a* MM 1 . .V(l -z) ( a h r + -UC a i p 1= THEOREMS IN FRACTIONS. 169. The follo^g theorems are of great practical value:- a L Let ^ and - be any two fractions equal to each other, then a +b c +d a -b c -d' a«or na±.nc±.pe. mb±nddbpf.... when m, n, p,... are any multipliers whatever. then Proof ^Aa before, let ^ - - = 1 ^^ b d y"='*» Dividing, ~ a^hx, c^dx, e^fx..,. :. ma^mbx, nc = ndx, pe^pfx.,,, »*a=i=nc±pe....^mbx±ndx±pfx..„ x(mb±ndd=pf...). ina± ne±pe.... ace .... a c nZZ^'^ '■ ""*" "" "^ '""' " «" ■"- ™7« Theory, 11., by letting j = j =,, .„d then .uUtitutlng f„, a. te, .nd for .. i, to ^? and It^. I, • ^^* TflBOBEMS IN FRACTIONS 171. Another proof is sometimes given for Theorem 11. ;— Sineo .-»(|). ..,(J). ..^^), b + d+f b + d+/ Bat ace 6+d+/ 6+rf+/ b~d~f' i?ar.7._If i±f-*fl±5±^ prove ^ = ili:^ 1-x If then or (1 + g)(l - g + a;*) 6 (l-«)(l+a; + «»)"'o Adding numerator and denominator of each fraction, and dividing by their difference: \+a?+\-a* 6 + a l+a»-(l-a:»)"6"37 2 & + 1 6 + Ci; or that is, a? b-a' and inverting each fnvction : «" = b + a ^«. S. — If T ■» -T prove TH£0££MS IN FBACmOHS. 116 ab-ib* ab cd-i<£^ a* a o«-o6+ft« 6»"6«'*'^ Z'"*"''^ a6-46« ab ^* But since a' a a :=£I±i. a e (P 5- __^_^_^ c'-cd + (P cd-id^ '^ cd-id^ ' iino^Acr ;>roo/.— Let ^ - 1 -«, /. a-6a: and c^rfar. also^ . a*-ab + b^ 6V-3»g + 6« ar'-as+l a6-46« " 6»«-46« '^ aj-4 ' c*- crf + cP ^a^d'x + d' a^-x+l cd-4d^ " d»a:-4d' " aj-4 » -, since each fraction ^ a6-46» cd-4cP x-i Ex. 3. If °~^ b-e c-a a + b + e ay + bx bz + cx cy + az^ax + by +cz then each fraction « 1 •, when 0+6+C is po*»0. x+y+z For each fraction equals the sum of the numerators of aU the fractions divided hy the sum of their denominators; a-b+b-c+c-a+a+b+e .*. each fraction ' ay+bx + bz+cx + cy+az+ax+by+oz a+b+o 1 (a+b+c)(x+V'^z\'^ x+v+is when a+i+c is notaO. il 116 THEOREMS IN FBAOnON& If a + 6 + c- the sum of the numerators divided by the Bum of the denominators--, the value of which we do not know Hence the necessity of the condition that a + 6 + c is not-0. ^x.^If^* *+« e+a then 8a + 95 + 5c = 0. a-b 2(6 -c) 3(c-a) (o-i)+(6-c) + (c-o) = 0, .-. m(a-6)+m(6-c)+m(c-a)-0. Hence, if the quantities (a -6), (6-c), (c-a), in the denomi- nators of these fractions can be made to have the same coefficient, the sum of the new denominators will be = 0. Multiplying both numerator and denominator of each fraction in turn by that quantity which will make the common coefficient of (a -6), (6-c), (c-a) the L. C. M. iof the given coefficients, we obtain a+b_ Q{a + b) 3(6 +c) 2(e+a) a-d"6(a-6)"6(6-c)~6(c-a)"*' .-. 6(a + 6)-6A(a-6), 3(6+c) = 6>fe(6-c), 2(c+a) = 6A(c-o). (6a+9b+5o)^U(a-b + b-e+e-a) = 6A(0) = 0; .•. 8a+96+5c-0. Adding Bx, 5.— -If X y (a-%+(6-c)y+(c-a)«=0. Since we require {a-b)x, let us multiply both numerator and denominator of the first fraction by {a-b). For an analogous reason multiply both numerator and denominator of second frac- tion by (6-c), and numerator and denominator of third fraction by {e~a). lit I THEOSEMd IN FRAOnomk 117 .-. -^ ^''"^^^ <^-*')y (o-a) » a+6-o a«-6»-.c(a-6) 6'-e»-a(6-c)"^-a'-6(c-a)"*' /. (a-fi)«-*{a«~i»-c.(o-i)}, (6-c)y-*{A«-c«-a.(6-c)}, (o-a)«-A{o»-a»-6.(o-a)}. Bat o'-ft'-c(a-6) + 6»-«>-a(6-o)+c«-a«-ft(o-o) -a»-6>-ao+6c+fi»-o»-a6+a«+c"-a»-io+a6-0j .'. (a-*)«+(&-o)y+(o-a)«-ife(0)-0. BZBBOISB U 1. If- a y h -, prove -^ — — 1-,^ 2. If ^ = 1 = 5, prove - - ^^^y+/>g a e a ma + nb+pa' 3. Iff -|, find the value of ^±5. 4. If? c ax + b , prove ^—^ has always the same value, what- a cx+d ever be the va^ie of x. ^- ^^-J'P^°^ g'+y (a - 6)« "• ■" A ■* 3» prove == — =«-l _-L 7. If - - ? prove "'+^^^-*-^ 2«& + 35» "--^--, prove I ^^^ THEOREMS IN FBACnONa 10. If * = f ^ 5, prove ^+V+<«^ ^+y»+^ 11. If^«f=,f,prove^^±y!±flz:?^ *• 12 If ''^"^ ae-bd a-6-o+rf"a-A+c^*'^®"sliaJla+4n.o+d; and each ratio i a + i+o+cj 13 It^^-^ll-^Jll «-* . ~~^ == — -— . then shall a + 6 + ■ 0. 14. If a o c show that each fractiou is equal to 2±i+5, and that - « * „ £ 16. If ^i±i!„.5i+ff hx + ay b-o c-a "a-b ' *^®^ « 6 c y 16. If b + 'TTH'TTl* ^^""^ a-6-c 17. If ^!±^zfL'„^±f!::^' a»+ft»-c» be " ca ^ ^6 — »prove each fraction -1. 18. If ^Lt^„f!±fsf^f3j^ each of these fractions -i-tf 8„nno««„ ^ 1 + y ' ^^PPosmg a, + a, + a, not - 0. 19. If^i:Jf,ffjlff..?«-^,, T y a — 6 —-.then --y ?. .fjtli. y±i g+ar 20. If -=l-lil-JlZll« *:?^ a? + V+« THEOREMS IN FBACnONS. 119 > and each y z ition— 1. QOt>-0. 21. If z (a + 6 + c)(ya + ««j + ary) = (x + y + «)(aa; + 6y + <»). JJ. If "5 — — -S — — — , prove d^-yz %^-zx s?-xy ^ o»ar + 6'y + c** = (a* + 4' + c^(a; + y + «). no T* ** ~ ^ ^ - «* ay-bx m. XI ™ — :: — = 1 prove oa?+6y+c« — 0. 24. If 25. If ado a+5 ft+e o-\-a ^rrb)~Mb:r^)'-6{^y P~^« 32a + 35* + 27c -O. ir y ^ , y .« %::7S- J53^=;(^qr)' P'0^%- + 1 + - -0. 26. If y z a{y + z) b{z + x) e{x + y) y , prove -(s'-«)+f(«-»)+-(^-»)-o 27.If^.i:!^_f:^-l,d.owthat a' 6« a'ar+ftV+c** m n r 28. If ^'« - = - and -+^+-=1 prove X y z a* b^ e^ ' ^ m' n' r* w' + n' + r" nr^ ^.hx ky Iz , «" v" «^ /« y «\' a« ft« <5» so. If ^l|^-4^ii±i, each fraction -^^ CHAPTEB IX. ! i M '< SECTIONAL SIMPX.B BQUATIONS. P^e e^unpjes of equation, involving™ h te^"' ^"^ *" proceeding Xlf^it,'^* t^"»' ^^^"^ tenns. .„d then nmltiplied b/ZI 0. D "'"'«*' ""^ '^^ ^«»" ^ ^a?. i. — Solve 5x 5x Y~T" 9 4 3-a- Multiply both sid€« by 4, the L. O. D., ®^ 10ar-.5x»9-(6-2aj) " 10«~6ar-9-6 + 2ar/ f^ s. juationa of proceed to 1 be solved ' Common , and then equations h by the of every 'action ia fng such FRACTIONAL SIMPLE EQUATIONS. 121 Ex. f.~Solve x + i X- i '2-f 3a;- 1 3 6 ""^n^' Multiply by 16, Then 6a?+20-3a?+12 = 30 + 3«-l. Oollecting and transposing, a? =» 3. NoTE.-The student can always test the correctness of his result by snbstituting in the given equation the value found for x, when, if oorreci the two sides will become identical Solve EXEBOISB U. 1. X-i 3. x-3 17 3 " 3* 5a;+3 3-4a; 31 9-5a? x 8 3 2' - 5aj-7 2a;+7 „ ~2 3-=-3aj-14. . 3ar+6 2ar+7 ,^ 3a! 7. -y ^ + 10 = -. 9.? (27-2.) -1-^(7.-54). 6*5' . 10a; + 3 6a;-7 ,^/ 4. -3 2 — 10(a;-l>. 7a;+5 6a;-6 8-5a; 6 "1 12"* J, 3a;-4 6a;+3 ,„ 6. 10. 2ar+7 9a;-8 aJ-11 11. 8a! -15 lla;-l 7a:+2 8 ~ 7 "IT' 7 11 "T"* 12.2.-3U|(3_2.)+1,. 15a; 10 X 13. ^2(2-3)-i(3.-2)»i(4.-3).14. i|^4%|-i-H+. 18. ,-8735, 6 3aj 4^2a; • .- 3-a; 3-1 6+a? x 4 « ^ 3 12' 19. ^(ll-a;)-|(13 + 2*)-](16-3a;). 20. |(*) + |(ll-*)-l4('-2). 9 3 "6 2 16. 7a;-|a;=.^a: + J^-6J. s^-io-^Is g+l 3 a; 5-« 2 «"3 6~* 12S FRACTIONAL SIMPLE KQUATIONa 175. If iho doMomiimtors contain both simple and compound oxproRiiona it i» frccpicntly hoHt to first oomhino the gimple ex pniasions and then clear of fiaotiong. Kx. l.~.~So\vo 9« + 20 4(a!-3) aj S6 » Tmnspoaing ■ , and oondiining, wo got 0f^+2()--_9x 20 4(.r-3) . ft 4/«-3\ Oloaring of fractions, 26.r - 100 - 3G.r - 108, 8 from which we obtain «» TT* Ex, «.— Solve 95+15 8.C- 7 36^ + 16 41 14 64; + 2" 66" ■^66* Transposing, 9j?+15 36.r + 16 41 14 66 "66 Combining fractions on the loft side, i- J 66"l4' ex-7 '6x+2* 8jr-7 6jr+2* Clearing of f motions, 6^+2--112.r + 98; /. 118.r-96 or X--??.-!?. 118 69 Solve 1 ^"^''^ 7 »'-3 4J- + 6 * 14 ■*^6x+2"~r"* EXEROISB lAt, 9(2^-3) ll:r~l 9a; + ll 14 3^+1 7 3. l?i±iZ_i?fJLi.5f:if 4 6J-+13 3^ + 5 2x 18 13j;-iG 16 64: -26 6 PRAOTIONAL SIMPLE EQUATIONS. 12S 5. 7. 9. 6a?-f7 2jr-2 2a;+l 15 ~7>^""T"* 7aj - 6 « - 5 a; 36 6«-101 6* 4ar + 5 9.r-6 2x-Z 10 "77+4""T"* 6. 8. S.r+1 2j;-4 2aj-l 16 7*Tl6"~T"* 4« + 3 8x+19 7ar-29 9 18 6«-12' 10 ^ i'^- Ba? 2 3 " 3x-7 "3" 176. Complex fractions generally should be simplified before proceeding tc find the lowest common multiple of denominators Ex. 7.— Solve o 4* 7x ^"9 1 -9-^ 4 4 10 • Simplifying first and third fractions, 27~4x 1 7jr-27 36 "4 90 * Multiplying by L. 0. D., 136-20a:-46-14»+64. Transposing and collecting like terms, 6a; -36, .-. a: -6. Ex. «.— Solve 25- -« 3 Ux + i} J3_ x+i ■*" 3a: + 2 '^x+l'^ Simplifying complex fractions. 76-ar 80ar+21 23 3(x+l) 5(3x+2) x+l Transposing, -^ _. il + ?1^.±11 . g . 7R-« — fiO lift* j^ oi + 6. •v?i- -rax 3(,T+1) ^6(3x4-2) ft: J 124 FRACTIONAL SIMPLE EQUATIONa Multiplying both aidea by 16(«+ 1)(3«+2), 5(3ir+2)(6-«) + 3(«+l)(80;r + 21)»75(ar+l)(3:r+2). Multiplying out and collecting terms, 8«-27, .-. ar-3f. Solve BXBRoisB un. 1 5? ^|a?4-n 7g+5 ' 9 " 16 ~12"' 2. 10^(*-f) + ^(3f:r + 7)-21i. 3. 2.r-f5 ar-15 ar(5 - 2§) 21 3i . aj 66 - 2aj U 7ar-2 6i 5Jx + 5i *-~8j"- SJ 7^ lOJ "^^46" 6. 3a;- 2 2 -3a; 8fa; - 6 4J.+ l"l^2a:-4^T2-- 4 ^ 7 -lii_l^ 3^ 1 '3 3f 11 II' 8. '-|<^"^) 31 ^-|(^-2) 3 ^36° T""* 3^-^l+^) 1-L 2f + ^ } m. FRAOnOMAL SIMPLE EQUATIONa 2ar-3 3a?- 1 4 in 1 3 ~ 2 x-l 2 * 3«-2 • -. 2 -3a ; Car 2a:-3 a?-2 12. .6ar + i5^±:!?.i:?._:tj^ .6 .2 .9 1, . 3.5a? 24 -3a? „^ 13. .6-— 2 8 •37^^- 14. .15a? + i^55^Z::?2?„i^_:0?^-:18 .6 .2 .9 1] 126 15 4^Jt-22x 6/ :..x 9 33 "■* «r"54;- 177. Other artifices are often employed to lessen the labor of aolvrng equations. In a certain cla^ of examples the actual division of each numerator by its denominator assists greatly in reducmg the equation to a simple form. In others a judicL combmation of frrctions according to their denominators is of great value in facilitating the work. ^a?. i.—Solve ?^+^.16fjf93 18a? + 86 6a? + 26 2a? + 5 2a?+ll %HT"*"2^rpf Divide each numerator by its denominator, 94 Alii -r 11 5 6 f 2a?+5^ 2a?+ll 5 o B + 3 + 2a;+9 2a?+7' '* 2a: + 6 2a?+ll "2a?+9'*"27f7 Dividing by 5, -^+—1— - '^ J. ^ 2«+6 2a?-j.ll 2ir+9 2*+7* N 126 FRACTIONAL SIMPLE EQUATIONS. Combining in pairs, ix+U 4ar+16 (2ar + 6){2x +11) {2x + 9)(2ic + 7) or 4ar+16 4a: +16 4a^+32ar + 56 ix* + 32x + Q3' In these fractions the numerators are equal, the fractions equal, but the denominators unequal. How are we to reconcile the apparent inconsistency? Only by putting the numerator of each fraction = 0; the fractions will then be equal for aU values of the denominators except zero. /. 4a; + 1 6 = or ar =. - 4. This result may be obtained otherwise. Bring both fractiona to the same side of the equation, then I ^a;+16 4a; +16 4«» + 32a: + 35"4a'+32a; + 63"^ or ^ *■'' H4«" + 32a; + 35"4ar» + 32a; + 63j"^* But the second factor is not = 0, /. 4a; + 16 » and a? - -4. Bx. £.— Solve Transposing^ Combining in pairs, Simplifying, Clearing of fractions, 3a;+l a;+2 1 4 2a;+3~6a; + 9'^3x"3* 3a;+l a;+2 4 1 2a; + 3 3(2a; + 3)"" 3 ~3«* 9a; + 3-a;-2 4a;-l 3(2a; + 3) 8a; + 1 3a; 4a;- 1 3(2a; + 3) 3a- * 8a;»+a;-8x»+10a;-S. 1 M /, 9a; = 3 and ^ a - . 3' nUCnONAIi 8IMPLK BQIUATION& 127 e fractions k) reconcile imerator of r all values = — 4. li fractions 5. »--4. Ex, A— Solye — . + ^ 3(«+6) Transposing, Cbmbining^ Transposing^ Combining, Simplifying, Combining, « + l ar+2 («+!)(« +2) 1 ^ 1 6af + 17 ^ 6 ff + l « + 2 ir + 2 6x4.17 ~ « + 2 • % + 5) («+!)(« + 2)* L g^+18 3(a;->-6) + 1 x + a " '^(»+l)(«+2)* « + l X4-2 («+!)(« + 2)* 1 ^ 6a?+18-6a;-12 3(a;+5) « + l a: + 2 "(«+!)(« + 2)* 6 8(a; + 5) « + l ar + 2 («+l)(« + 2)* 7*4- 8 3a;+15 («+l)(a: + 2) (ar+l)(a! + 2)* /. 7«+8-3«+15, or 4«-7andar--. it Solve , bx + Z 2x-Z ^ BXBBOISB lilV. 3. 5. x-l 2x-2 5a;» + a;-3 Tar'-Saj-O 5a; -4 7a:- iO "*• «•+««*- 6a; + c a^ + ax a^-ax -6 a:" -03:* + 6a: + + 6 ir+4a + 6 4ar-f ■a +26 1 ■ 5. 2 ^^"^^ 2a;+38 * 2a:+l~ a:+12 "^* 4 -^+iz^=?J:i ^^ • x-2 x-l x-l'^J^' l-i-a; + a;' 62 \+x 6. l-a: + a:» 63 ' 1 -«* ^ 3ar-l 4a:-9 1 ' 2a:-l"3«^"6' ^11! 128 FftAOnONiX SIMPLE EQUATIONa V. - + 6 11. 2af-3 ar-2 3z + 2' ,« 3+a? 2 + ar 1 +« , 3-« 2-x 1-ar *• «-l «+l ■2*. 12. 29 2 18. JL, T 37 1§. « + 3 « + 3 ?+6«T6* 3 « + l a* «-i"«-i"irp* 17. -+- .+ 14. 16. x-8 2a;-16"24"3«-24* x~i x-b x-7 «-8 «-8 it-5 ir-7 x-i «-10 jt-7 «-9~aj-.6* 9 36 « + l 2a:-l"^3a?-l"6j^' 18. -L_1Z2,^4_2(^) «->-6 x-3 x-6 x~4 • 19 *^ILi!+l?izi? 8a; -30 6ar-4 «-4 2ar-3 "2ar-7 '^IT^' 1 *! m.'^"'® equations known numbers are represented by ?"• .^"^ "" "^^"^ ^'""'''^ "^"**^^^ The si^me artifici and methods are employed in solving these equations as when "gttres are used. Nom-U«uaUy the Jirst letters of the alphabet are employed to repre- ^ kncum numbers, the last letters unknoum numbers But thisTle do- not always hold good, as any letter may be used in either w^ Bx. 1. — Solve Transposing, Factoring, Dividing by (a -6), ax+bcobx-^ao. ax-bx^ao-be, x(a-b)^c{a-b). c(a — b) » — « X X -1. 9 2 i" 'Zx -24* 7 X- -8 B X- •9- 7 X. -4 9 a?-6' FRACTIONAL SIMPLE EQUATIONS. Ex. «.— Solve (a + « + 6)(a + 6 - «) - (a + x){h -x)- ah. Multiplying, (a + 6)» - «> - o6 + ar(6 - a) - «« - aft. 129 JTiB. 5. — Solve Transposing, Ck>rabining, or Clearing of fractiona, f 'V *•/ ""'■ h-a '• Zax- 26 ax-a ax 2 36 26 ~ TT 3gu;-26 CLX- a ax 2 36 26 --_, "3- 6ax -46-3ax + 3« -6aa: 2 66 3 -3aar - 46 + 3a 2 66 a "3* -3aa;-46 + 3a» -46, .*. ax^a and d;—!. lented by ) artifices as when t to repre- thia rale tray. Solve BXBBOISB LV. 1. a(«-6) = 6(a-a?)-(a + 6)a;. 2. {a+x)(b + x)=.{e + x)(d+x). 3. Qx-a 3a;+6 4a:- 6 2x+a' _ a?-a a;-6 ay-c „ 6. ^: +-^-+ 7 = 3. 7. 6+0 o + a a + b a a-o x+a x-o x + a-c' Q aj^^hbx + c px*+qz+r ax+0 px + a ,, (x—a)* x-2a^h * (9 + by ar + a + 26* a{a*+ g») 4. ^ =»aa;+6*. 1 1 6. a + x 1 a6 — aa? be — bx ao — as* o. ar + a + 6 + c»» j — -. a + 6 - + « 10. a+b x — c x—a X -h' 0+0 O+a a+6 12. l-:+:_i+iJ4--8. ISO FRACTIONAL SIMPLE EQUATIONa 16. a + e b + e x + c (a-.b)(x~a) (a-A)(z-6)"(ar-a)(:r-6)+*-''- 17. x-o ar-6 x-e x-a-b-i o a abc ' 18. (''<^y+{x-by+(x-cy^3(x^aXx-bXx-c), 1 2 1 19. 1 2 BXBBOISB LVL Solve MISOELLANKOUS KXAMPLE& L (3a.-l)«+(4^-2).-(5^_3)«. 2. (^ + 2a)(^-a)«-(:.H.26)(^j)a V2:r+i; ix + a *' (^jj -^^Ti' \ * T-* AX/ x+^ x-1 x+l x^-l ' 7 -L+-i- - «-l x-2 x-S 1+a? 3+2a^ g 2aJ-3a;+l 2ar-3 a;'-2ar+2 "^ ar-2' 10. ^=,_^+ * Q 1 _^ 11. (^■*-l){x+2){x+3)=.{x-3Xx + i)fr ; 5). 12. (z+l)(a: + 2)(a: + 3)-(ar-l)(ar-2)(ar-3) + 3(4:r-l)(:r+l). 13. (a?+6)(«+c) + (a?+c)(a?+a) = (2x + a)(ar+6). 14. (a?-a)(«-2o)-r«-3aV«-4/,\ -^m+n. J *)(*+c)' 26)(»-6)» 6)« 1). 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 23. FRACTIONAL SIMPLE EQUATIONS. 3 6 ISl (3ar+l)(2j:-3) (6x-4)(2« + 5)* 1 a (3«-l)(a:-2)"(2ar-l)(«-4)' (a?-2a)»+(x-26)«- 2(«- a-6)». Sabc a*l^ (2a -I- h)l^x hx a + 6"^(a + 6)»'^ a{a^hy -3ca; + — . 2a!-3 3a;-2 5a;»-29a;-4 x-4 "^ a:-8 "a:"-12ar + 32' (a? + a)(a; + 6) + (« + c)(a; + o) - (a; + *)(« + (f) + (a? + d)(a? + c). + :+ ^ n» n x—a x — b T-e x-e x-a x-h x-2a x-2b x-2c + — :+— ; »3. b+e-a c+a-b a+b-c x-2a x-2b x-2c Zx b + e-a c + a-b a + b-e a + b + c* a-x b-x e-x + - + a*— be b^—ac c^-ab a + b + o' a — b b — c e — a a+b b + e a+2b+e x—e x—a x—b x—a x—b x—e ab-bc be - ca ac - ab a^-b* 6'-c' c^-a* x + o x + a x + b x+a x+b x+e cf—abe b^—dbe t^-abc c?-he V-ea (?-ab x + a x + b x + e x+a a x+b b x+e a + b+e X - md X — me x + mb x + ma V u a + o + c + dm^^ ONTARIO C nr o OF EDUCATION I ill I j 'III 182 PB0BU6MS PBODUCINO FRACTIONAL EQUATIONa PROBLEMS PRODUOINQ PRAOTIONAL B.'QUATION3. i« I!?: ^' ri'^f "" ""^ P'°^^"°" '^"**^S ^' fr*°«onal equations IS accomplished by employing the same methods and X^Z wT ,rr'^^f r'^^"^^ ^^^^ ^^^^--^ fruotion7;'i: t^ons. It will be well, however, to give specimen solutio^ of different types of problems frequently occurLg in practic^ ifar. L--A can do a piece of work in 6 days, and 5 can do the Let X -number of days it takes .1 and J5 to do it Then - -amount 4 and ^ can together do in a day. But^ can do it in 6 days, /. J can do 1 in 1 day, and ^ can do it in 8 days, .-. 5 can do 1 in 1 day; 1 1 6^8 therefore ^ and 5 can together do | + 1 in one day. But ^ and 5 can together do 1 in one day. • 1 + ^,^ Simplifying, ^^^l. .-. .-^^^Sfdays. Solltr"^ "^"^J^^ ^" ^^'^ ^^ *^" P^P^« ^^ 24 minutes and 30 mmutos respectively, and emptied by a third in 20 minutes In what tune will it be filled if aU three are working togeCi Let X -numberofminutesitwillbefilledbyaU three together. Then --amount filled in 1 minute by aU three. PROBLEMS PEODUCINa FEACTIONAL EQUATIONS. 183 But ^-amount fiUed by first tap in 1 minute^ . 1 and ~ = amount fiUed by second tap in 1 minute, also - - amount emptied by third tap in one minute; •• 24"^30~20"*°^**^^* ^^^ ^y ^^^ *ap3 ill 1 minute. But - -amount filled in 1 minute by 3 taps, ** X 24 30 20* Simplifying, - - j|q-1, .'. aj- 40 minutes. M^e S.—A person walks to the top of a mountain at the rate of 2J miles an hour, and down the same way at the rate of 3i miles an hour, and is out 6 hours. How far is it to the top of the mountain 1 Let Then and X =. distance to the top of the mountam. X ^ =» time to walk up X 57 ■■ time to walk down. X X '*' 2l * 31 ~ ^l^^l® time. But whole time of walking up and down is 6 hours, • * J. ** K ^ 3aj 2» , 5a; " h H' y*"y"*» •'• T-^» /.«-6mile& JUt £ fK^A *l.^ xi i--i « . - . - _ __ ^^ ^^.^ ,^, .^^„ «u6ween o ana 4 o'oiocfc when the hands of a dock are at right angles. hi 134 PROBLEMS PRODUCINQ FRACTIONAL EQUATIONa l^e minute-hand of a clock moves twelve times faster than he hour-hand. When the hands are at right angles one hand is 15 mmute-spaces ahead of the other. At 3 o'clock the minute- hand IS at 12 and therefore the hands are at right angles at that tune. There is, however, another solution; for the minute- hand may get 15 minute-spaces ahead of the hour-hand. To find the time when that occurs: The minute-hand at 3 o'clock is 15 mmute-spaces behind the hour-hand, and it has to gain this space and get 15 minute-spaces ahead of the hour-hand, so that it has to gam altogether 30 minute-spaces. Let X = number of units of space moved by hour-hand from 3 o'clock to time required. Then 12ar = number of units of space moved by minut©. hand; .'. \2x~x=\\x = spaces gained by minute-hand. But spaces gained is 30 minute-spaces, ,*. 1 la? « 30 minute-spaces, .'. ar=2T\ minute-spaces. But time is shown by minute-hand, which has passed over 12* spaces, /. 12a: = 12x2T^ = W = 32A minutes. Therefore the time is 32,^ minutes past 3 o'clock. EXBBOISB LVIL 1. Find a number whose third part exceeds its fourth part by 14. *^ 2. The half, fourth and fifth of a certain number are together equal to 76. Find the number. 3. Divide 60 into two such parts that a seventh of one part may be equal to an eighth of the other. 4. Divide 45 into two such parts that the first part divided by 2 shall be equal to the second part multiplied by 2. 5. In a mixture of wine and water the wine was 25 gallons more than half the mixture, and the water 5 gallons less than D«e-ihird of ui« mixture. How many galloM were there of each 1 aster than >ne hand is be minute- angles at be minute- l. To find ;lock is 16 this space hat it has hour-hand 7 minute* over I2x rth part together one part divided gallons ass than of each 1 PROBLKMS PRODUCING FRACTIONAL EQUATIONS. 136 6. Divide 46 into two such parts that if one part be divided by 7 and the other by 3 the sum of the quotients shall be 10. 7. A can do a piece of work in 5 days and B can do it in 4 tiays. How long will it take A and £ together to do it ! 8. il can do a piece of work in 5 days, i? in 6 days, and C in 7^ days. In what time will they do it, all working together 1 9. A can do a piece of work in 2| days, B in 3J days, and C in 3f days. In what time wiU they do it, all working together? 10. Two men who can separately do a piece of work in 15 days and 16 days can, with the help of another, do it in 6 days. How long would it take the third man to do it alone 1 11. ^ does ^ of a piece of work in 10 days, when B comes to help him, and they finish the work in 3 days more. How long would it have taken B alone to do the whole work 1 12. A and B together can reap a field in 12 hours, A and C together in 16 hours, and A by himself in 20 hours. In what time can B and C together reap it ] In what time can A, B and C together reap it 1 13. ^ and B together can do a piece of work in 12 days, A and C in 15 days, B and C in 20 days. In what time can they do it, all working together 1 14. A tank can be filled in 15 minutes by two pipes, A and B, running together. After A has been running by itself for 6 minutes B is also turned on, and the tank is filled in 13 minutes more. In what time may it be filled by each pipe separately ? 15 1 cistern could be filled by two pipes ii 6 hours and 8 hours respectively, and could be emptied by a third in 12 hours. In what time would the cistern be filled if the pipes were all running together ? 16. A tank can be filled by three pipes in 1 hour and 20 minutes, 3 hours and 20 minutes and 5 hours respectively. In ,,5 „.„ voiiia. wv Miicu waoa mi inreo pipea are run- ning together 1 186 PROBLEMS PBODUCINQ FRACTIONAL EQUATIONS. 17. A fish was caught whose tail weighed 9 pounds, his head weighed as much as his taH and half his body, and his body weighed as much as his head and taH together. Find the weight of the fish. 18. A hare is 50 leaps before a greyhound, and takes 4 leaps to the greyhound's 3 leaps; but 2 of the greyhound's = 3 of the hare'a How many leaps must the greyhound take to catch the harel 19. Find the time between 2 and 3 o'clock when the hour and minute hands of a watch are, Ist, coincident; 2nd, in exactly opposite directions; 3rd, at right angles to each other. 20. Find the respective times between 7 and 8 o'clock when the hour and minute hands of a watch are, Ist, exactly opposite to each other; 2nd, at right angles to each other; 3rd, coincident 21. It is between 2 and 3 o'clock, but a person looking at his watch and mistaking the hour-hand for the minute-hand fancies that the time of day is 55 minutes earlier than it really is. What is the true time t 22. A horse was sold at a loss for $200, but if it had been sold for $250 the gain would have been f of the loss when sold for |200. Find the value of the horse. 23. A merchant adds yearly to his capital J of it, but takes from it, at the end of each year, $5000 for expenses. At the end of the third year, after deducting the last $5000, he has twice his original capital. How much had he at first! 24. A trader maintained himself for three years at an expense of $260 a year, and each year increased that part of his stock which was not so expended by J of it. At the end of the third year his original stock was doubled. What was his original stock! 26. A cask contains 12 gallons of wine and 18 gallons of water; another contains 9 gallons of wine and 3 gallons of water. How many gallons must be drawn from each cask to produce a mixture containing 7 gallons of wine and 7 gallons of water f PROBLEMS PRODUCINQ FRACTIONAL EQUATIONS. 137 26. A man rowed down the river a distance of 11 nules in U hours with the stream, and on his return rowed back again in 3| hours. Find the rate of the stream per hour. 27. A* boatman moves 5 miles in f of an hour, rowing with the tide; and in returning it takes him 1| houra, rowing against a tide one-half as strong. What is the velocity of the stronger tidel * 28. A boatman rowing with the tide moves n miles in t hours. Returning he uses ti hours to accomplish the same distance, row- ing against a tide m times as strong as the first What is the velocity of the stronger tide ? 29. A train which travels 32 miles an hour is | of an hour m advance of a second train which travels 42 miles an hour. In how long a time will the last overtake the first ? 30. A train travelling b miles per hour is m hours in advance of a second train which travels a miles per hour. In how long a time will the last overtake the first? Discuss the result when a>b; a^b; o < b. 31. An express train which travels 42 miles per hour starta 50 minutes after a freight train, which it overtakes in 2 hour« 6 minutes. What is the velocity of the freight train t 32. If J, who is travelling, makes | of a mile more per hour he will employ only J of the time, but if he makes ^ of a mile less per hour he will be on the route 2^ hours more. Find the length of the route and the speed. 33. At 12 o'clock the hands of a watch are together. At what hour will they be opposite to each other 1 34. il and 5 accomplish a piece of work in m days; il and (7 can do it in n days, and 5 and C in p days. How many days will it take each to do the work alone» and how many if they work together? 36. If a men or 6 boys can dig m acres ira n days, required the number of boys whose assistance will be required t© aimAIa /« = , men to dig {m-krp) acres in {n-p) days. ^ ' 10 . - •f" 138 PROBLEMS PRODUOINQ FRACTIONAL EQUATIONa 36. If ^ can do a piece of work in 2m days, and B and A in n days, and ^ and C in m+ ^ days, find the number of days in which A^BaadC together would do the work. 37. Two friends at a distance of 78 miles agree to meet in an intermediate locality, and set out at the same moment, one from A, travelling 6^ miles per hour; the other from B, travelling 74 miles per hour. When and where do they meet? 38. A person, after paying a poor rate and also an income tax 11 ; ^ ^^ ^' ^"^ ^^^^ remaining. The poor rate amounts to ^22 10s. more than the income tax. Find the original income and the number of pence in the £ in the poor rate. 39. What must be the value of n in order that ^" "^ ^-_ may be equal to ^ when a is ^ 1 ^ « + o 40. A person, after paying an income tax of 6d. in the £, gave away t^ of his remaining income, and had £540 left. What was his original income 1 41. I bought a certain number of eggs at 2 a penny and the same number at 3 a penny. I sold them at 5 for twopence and lost a penny. How many eggs did I buy 1 42 The sum of £330 is laid out in two investments, by one of which 15% IS gained and by the other 8% is lost, and the amount of the returns is £345. Find each investment. 43. Find the weight of a mass of copper and tin, 40 pounds more copper than tin, to which if a quantity of copper f heavier than the tin be added there wOl be 11 pounds of copper for every 3 pounds of tin. ^ 44. The first digit of a certain number exceeds the second digit by 4, and when the number is divided by the sum of the diirits the quotient is 7. Find it. 46. One-half of a population can read; of the remainder 42% can read and write; of the remainder again 16% can read, write and cipher; whUe 243,600 caa neither read, write nor cipher What is the population \ PROBLEMS PRODUCING FRACTIONAL EQUATIONa 139 46. Divide ^607 Is. 8d. into two sums such that the simple interest of the greater sum for 2 years at 3^% shall ciceed that of the less for 2^ years at 3^% by £18 I6s. 47. A person possessed of £5222 invested a part in 5?& stock at 105, and the remainder in 3% stock at 96. How much did he invest in each kind of stock if his whole income amounts to £191 16s. 8d.f 48. The hour is between 2 and 3 o'clock, and the minute-hand is in advance of the hour-hand by 14^ minute-spaces of the dial What o'clock is iti 49. 112 pounds of bronze contains by weight 70% of copper and 30% of tin. With how much copper must it be melted in order that it may contain 84 % of copper 1 50. Find the time between A and A -I- 1 o'clock when the minute hand ia m minute-divisions before the hour-hand. I i CHAPTEE X, SIMULTANEOUS EQUATIONS OP THE PIEST DBaRBB. 180. If one equation contain two or more unknown quantities an xn^Jimte number of values may be found that will satisfy the equation. "^ alius, when :r + y = 6 any value may be given to y, and a corres- ponding value will be found for x. \ If x = 2 then y = 4, " « = 3 " y = 3, " ar=l «« y„6, and so on. Any connected pair of these values substituted for ar and y will satisfy the equation, and are called its roots. 181. But i^ in connection with the preceding examples, another equation be given, expressing a diff^ent relation between a: and v then only om pair of values for ar and y can be found which wS satisfy both the equations. Thus, if not only a; + y ■■ 6 but ar-y=s2, then the only values of x and y that will satisfy both equations are a?- 4 and y- 2. -i v«i Such equations are called simultaneoUB because they are both true at the same time, i.e., they are satisfied by the same values of a? and y. 182. If there are two unknown numbers to be found then two -_ j^„„.,.^^ aiuss, Ko given, £Hju«uou9 are said to be L._ SIMULTANEOUS EQUATIOAS OP THE FIRST DEGREE. 141 independent when they express different relations between the tmknown quantitiea Thus x + yTm\ and 2x-}-iy^% are not independent^ since they express the same relation b^ tween x and y. J^^L " '*«« «»kno™ are given rtr« independent equations mil be required. GeneraUy. if „ unknown numbers a« to be T:^:':^'' *° "-^ - '^'^'^' ^-«- *» ••"^^ 184. Different methods are adopted in solving dmple simnl- Ex. i.— Solve (1) (2) (3) (2) FIBST MBTHOa ar + y-8, 2ar-3y-3. Multiply (1) by 3, then 3a: + 3y = 24, »>«* 2ar-3y- 2. ^^^^S' 5^»26, ... *-5|. To find y, multiply (1) by 2 and subtract from the result (2) ^'^ 6y-U. .•.y-2f The object of this method is to make in turn the coefficiente of .and y the same in both equationa Thus (1) wa. muTt ^^ by 3 to make the coefficients of y the same in both equatW ZI^tZ ZT^'^ ' "^ "^'^'^ *^« coefficientsTfTthJ same. Then, addmg m the first instance and subtracting in *h. second, one of the unknowns disappear or is ./.^^^S T. ^:^ZX:J-^ -^Pnx-is called .^^^" i t 142 SIMULTANEOUS EQUATIONS OF THE FIBST DEGREl. 185. Having found the value of either x or y the remaining value can be found by substitution. Thus, in the preceding example, * 26 « + y-8, 2ar-3y-2, (1) (2) we found x^-. Then substituting this value of x in (1) we get JRr. ^.—Solve 26 ..y.8-~--. 8K00ND MBTHOD. 2« + 3y-7, 3«-y-5. From (1) 2af-7-3y or x^'^~% Substitute this value of a: in (2), thoa 21-9y-2y-10 -lly--ll y-1. .. X ^ 2 ^• This method is called elimination by substitution. Simplifying^ or or Since y=»l, (1) (2) 186. Ex. A— Solve THIBD MBTHOD. 5a; + 4y=58, 3« + 7y = 67. (1) (2) iRKE, SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. 143 (1) (2) remaining WM preceding U From (1) 5:r. .58-4y. :.zJ^-^\ (1) 1 From (2) 3ar = = 67-7y, .....«^:^ (1) we get U o since the two values of x must be the same; . 68-4y 67-7y -5 3 • 1 Simplifying, or or and 174-12y-335-35y, 35y-12y-336-174, 23y=161 y-7. « 1 If y-7 then ^_58-4y 68-4x7 ^ «- E -0. This method is called elimination by comparison, 187. The following examples should be carefully noted:— Ex, 4.— Solve 2 3 - + - = * y 6 4 = 12, - + - = 16. X y To find ar, multiply (1) by 4 and (2) by 3. Then 8 12 - +— = X y >48 and 15 12 « y • 45. Subtracting (4) from (3), - - - .3; :. 3ar« .-7 and JT* 7 -3' Similarly y can be found. (1) (2) (3) (4) ' 1 - i 1 1 1** SHCULTANEOUS EQUATIONS OF THE FIRST DEGREE. ^*.-f.-SolT. 2« + 3y-16^, ^, Divide both (1) and (2) by ly, Th«a 2^3 y *."-"*• (») 3 4 y^--21. ^rom (3) and (4) the values of . and y can be obtained iRr.A-Solve ' *(y + 7) = y(;r+l), . 2a; + 20-3y+l. aearmg(l) of brackets, xy+7x^xy+y; .*. 7«-y. Substituting this value of y in (2) ;r can be readily found. (4) as in 0) (2) (3) Solve axERoisa Lvm. 1. 4;r4-3y-i31, 3«+2y-22. 2. 3ar-2y«.7, 8ar + 2y = 48. 3. 7«+3y=17. 6«+3y-13. 4. 5ar + 7y=-43, llar+9y»69. 5. 3y-2ar=ll, 13ir-5y-l. 6. 8ar-9y-l, 7. 2^+3^-6. S. 10:r-|=69, 3*+^y-6i. lOy-f »49. f SIMULTANEOUS EQUATIONS OF THE FIRST DEQKKE. 146 *v * y « 9. - + - « 2. 3 4 ^ 3x+4y-26. 10. 3ar + 7y-79, 2y-|.9. 11. 2a:-^-3aj-l, 2« 3y + — -y + 2. 12. lJa?-% = 5, 2 5 13. 4^x-Sy = iQ, 4Ja; + 3iy-215. 1 1 K g^+gy-S. 14. 2Jx + 3Jty=.74, 4Jar-6iy-l. 16. ga?+gy=-43, g*+gy-42. 17 ?f±^4.* ft 17. _^_+-=.8, 7y-3a; -y-11. 18.-^+1-—., 3-4a: 5y-7 ~6~ + ^ 2"- jg 2(15x-H3y) 7 -7(dr-2y+l) 12{y-3+-(y4.7J)}. 20. «-i^-5. . «+10 „ 4y — 0--3. 21.2«-y±.^7 + ?2^. . 8-a? „., 2y+l 4y— ^-24J-J^. 22. 5x-&t2»32, 3y+ -X- -9. 23. (M;+y-6, JT+Ay-o. 24. a;+y«ao, 1 4 I ! 146 SIMULTANEOUS EQUATIONS OP THE FIRST DEGEEE. 25, ax^by^ 27 * y €UV bv a a b 31. (a+6)ar-(a-5)y«^ •o 4a;- 2 ^ --3-+4y-3-19, 8y-6 32. «+y 6 x-y "z* « + 5y=a36. 81. _^^~yy 2ar+2/ 1 3 e -. 35. a?-f6 y-3 -3, 11 4 """g"""*"^^- 4ar-lly„0. 8-if^-6. 86. (« + 7)(y-3) + 6 (y+7)(a;+3)-64, 37. ? + ?-?? 15 13 • » y"y "a "l6* «• "-^-«. t» iCiU 3a f-l^.ir. e y > 2* *2 ,„ 40. ?-?.l, « y 4* 7 4 6 «^y"3* SIMULTANEOUS EQUATIONS OP THE FIRST DEGREE. 147 41. f«*l-". r«^|->«- 43. x—m m — n y—m wi + n* as m* - n* 45. y m*+ n*' «+ 1 m+n+p y+l m-n+p' x-l m+n-p a . y-l tn-n-p 5 X ^ — 5--1. 44. x + y+ 1 ^ ;> + 1 x-y+ l^p-V x + y + 1 1 +m «-y-l°"l-w»' 46. x- a + e y - a-\-b* X +e y +b a +b a +0* 47. ^±^^fn, x-y+l y-x + l x-y+l 49. x+y-'mxy. nm. 61. y-ximruey. a ^ ft 6 + y a — x* e d d-x o + y' 48. 6-a-l « + j^ , hx + ay a+h + 2 ~ ' h bx-ammay-b. 60. my+ar— pary, ny+X'-qxy, 62. «+y a «-y 6-c' « + a + b y+b a+c 53. («»+n)a?-(»n-n)y«»4»nn, 64. X V ^ m+n m-n 'P'-^, + i -l a; p-+ i y 148 SIMULTANEOUS EQUATIONS OP THREE UNKNOWNa SIMPLE. SIMULTANBOCTS EQUATIONS OF THBBB UNKNOWNS. «»t there will i"^' t T" ""^"^ """ ^ elimi-ated, «, ^*. i. — Solve 3a-+6y-7«=12, To eliminate z between (1) and (2). ^'^^^P^^ 0) bj 7 and (2) by 4, then 14«~21y+ 28^ = 28, 12ar + 20y-28a=.4a Adding together (4) and (6), 26a;-y-7g. Multiply (1) by 2 and add (3), then But ,f-^y-^3- 26ar- y.76. .-. multiplying (6) by 7 and subtracting (7^ 173ar-519, .-. ;r-3. Substituting values of a: and y in (1), «, i 189. It is not always necessary to go thro«ah fK- Various methods of shortening the worlflin * ' P""*^ in a«^;„i m, * ^® '^^^^ ^I present th«m«oi»™ (1) (2) (3) (4) SIMULTANEOUS EQUATIONS OF THREE UNKNOWNS. 149 Solve Add together x + y-1, a:+«-5. 2a:+2y+2a-15, . ^ ^ 15 .-. x + y+zr^-^. But Subtracting (1) from (4), ar + y. Znm ■1. 13 2* (1) (2) (3) (4) (1) Similarly, by subtracting (2) and (3) in turn from (4) we get x> 3 A 5 2 and y- 2* In this example the student might add (1) and (2) together, and from the result subtract (3), in which case y would be found! Similarly, by adding (1) and (3) and subtracting (2) z would be found. Solve 1. 5x+3y-6«a4, 3*- y+2«-.8, a?-2y + 2«-2. 3. x+ y+ aia«6, 6x+ 4y + 3«-22, 15aj+10y+6a-63. 5. y-x+Mmm -5, z-y~xmm -25, x+y+«-35. 7. 2aJ-3y-3, 3y-4«-i7, 4«-5a;-2. 9. ax-k-byk-ez'ma^ 6, 2a: + 3y- ««20, 7ar-4y+3«-35. 4. 4ar-3y+ «»9, 9«+ y-5«=»16, ar-4y + 3« = 2. 6. 15y = 24a-10;r+41, 15ar-12y-16z + 10, 18a?- 7«+14y-13. 8. 7«-3y«30, 9y-6«-=34, a; + y + «-33. 10. bz+oy tmOf as TCx m-bf ayk-bxmmg. ! i] '■a fl 1 m SIMDWANBODS WJCAWOM 11. y+2+3--i. IS. ?-i + l .38 X by z 5 • 3aj 2y « ■ 6 ' 1-1. i 161 5« 2y « "To"* 15. i + l_l « y «, -»a, 1 1 1 « y « -^ -a >7. "^'-2. y + l ' y+2 ^ .fl-*. « + 3 1 «+3''2' 19.1.?.,. « y ' 2 3 ^ y~.-^- OF THREE UNKNOWNa 19 ^ . 3 3 2*+3*+5y-17. 1 3 2 3*+4^+3y-19, 1 5 1 6 i*+gX+-y,18. 11 ^ 2 14. -+--5, 3 4 y""a""^' 3 4 --- »5. « a; '8. ?+i-? * y « 3 2 0. 2-0. iT « 3 0. 18. ?il^ « + l a;+l 3«+ar y+r 2, 2. -2. 20. -^„1 «+y 5* y 1 y + « 6* : « It SIMtJLTANEOUS EQUATIONS OF THREE UNKNOWNa 161 21. ^-^-20, -15, 4y - 5a 23. 9jy-20(a?+y), 10aa-24(x+«), lly« = 30(y+a). 4y- 3x xz 2«- 3« y« 32. 26.53!, o Sx» "To" '« + y, '« + «, 2y2 27. - + - + - « 3, « y « o 6 , + - - 1, X y z z -0. 24. 26. 28. a? y 10 6 —o, « aj 21 45 ,„ y 2a *y- 3(«+y), a;a- 8(ar+ a), 7ya-24(y+a). * y « , - + I+--1, 6 a ay a+y a» .6, 2a b X y 29. y +«-«■■ a, a + a;-y — 6, x + y-Z'ae. 31. (x+2)(2y+l)-.(2*+9)y, (x-2)(3a + l)-( a; + 3)(3a-l), (y+l)(a + 2)-(y + 3)( a+1). 32. 6x(y + a)-4y(a + ar)-3a(aj + y), - + - + --9. X y z S3. ar+y+a—a + ft-c, &r - «y 4- «Mr - ogr -«- 6a - or » o^ - (a + 6)0. >a yg y+« 80. 2np»(^ + m)ar-(j9-m)y, 2mp = (m + n)y - (w - n)z, 2wn«(n+;?)a -(n-|>)«. id 162 PROBLEMS PRODUCING SIMULTANEOUS EQUATIONS. , " k i. ir. in m ii i PROBLEMS PRODUOINQ SIMPLE SIMUL- TANEOUS EQUATIONS. 190. It is often convenient and sometimes necessary to use more than one unknown quantity in solving problems. If we use two unknowns, x and y, we must have two independent equa. tions resulting from the statement of the problem. If there are three unknowns employed then the problem must admit of three independent equations; and generally the problem must furnish as many independent equations as there are unknowns. If there be more equations than unknown numbers some of the equations are superfluous or contradictory, in other words, too much has been ^ven; if there be less equations than unknown numbers then tJie problem is indeterminate, that is, more than one solu- tion can be obtained. ^^. i.-mie Hum of two numbers divided bv 2 gives as a quo^ tient 24, and the difference between them diviued by 2 gives as a quotients. What are the numbera I Let X «■ one number and 2/ - the other number. Then also and x+y -2- -2*. -2--17J «-y-34. From ^3^ and (d\ ^^ h^a -._ h _ _. _ (1) (2) (8) 3 SIMUL- (1) (2) PROBLEMS PRODUCING SIMULTANEOUS EQUATIONS. 168 Ex. S.—A certain fracuon equals 1 when 7 is added to its denominator, and equals 2 when 13 is added to its numerator. Find the fraction. Let X «= numerator of fraction and y = denominator of fractic \ Therefore _f_ =, 1 y+7 2* also 1±H„2 Simplifying (1) and (2), 2:r-y==7 ^gj '"<^ ar-2y--13. (4j Solving (3) and (4) by usual methods we get ar=9, y= 11. i?x 5^_The sum of the two digits of a number is 8, and if 36 be added to the number the digits wiU be interchanged. What la the number 1 Let ar = right-hand digit of the number and y = left-hand " « «« nu^b": '"" ' "^''''"*' *^' ^^^ ^ *'^' P'*^ lOy+.-the If the digits are interchanged lOar+y-the new number. .-. 36 + 10y+«-10a:+y, .j. «+y=8. ^2) 9a;-9y = 36 *-y-4, ^3^ «+y-8. yjv Therefore from /.« will it take each alone 43. A cistern has three pipes /I R „r,j n . U in a minu^s, ^ and (7 in'*^;l^ ^d Ld ^^ ^ ^'" How long will it take each alone te fill tt J "'""'""' """"*«» tH^/intCit brLtT^ rrrrv* t;*--^ 116 square feet more; but if it were 4 tX 7°"" """***" longer it would contain 113 l2~ t T "*"" "^ ^ *««' length and breadth. ""*""■»"'««' "ore. Required it. _ 45. If the sides of a rectangular field w.™ ...,. ,. . S^ the a«a would be increased by"22o"s,u:;";;::dra t^^ PROBLEMS PRODUCING SIMULTANEOUS EQUATIONS. 161 length were increased and the breadth were diminished each by 6 yards the area would be diminished by 185 square yards. What is its area 1 46. If a given rectangular floor had been 3 feet longer and 2 feet broader it would have contained 64 square feet more; but if it had been 2 feet longer and 3 feet broader it would have con- tained 68 square feet more. Find the kjigth and breadth of the floor. 47. A cask, 5, contains 12 gallons of wine and 4 gallons of water; another cask, C, contains 8 gall. -^f wine and 12 gallons of water. How many gallons must be . d,wn from each cask so as to produce by their mixture 7 gallons of wine end 7 gallons of water 1 48. A cask, A, contains 12 gallons of wine and 18 gallons of water, and another cask, B, contains 9 gallons of wine and 3 gal- lons of water. How many gallons must be drawn from each cask so as to produce by their mixture 7 gallons of wine and 7 gallons of water 1 E it I CHAPTER XI. SQtTABB AND OUBB BOOT. SQUARE ROOT. V^^'^^^rs^^^LZT"''''^ "P'-- « one o, From «, ; •''"',<-*«) X (-*.«)- 1 6a.. "" "' * ^ *«: In thi. chapter *o "ol^:^ ~« "r'"-- Since a..„,.„.,f;:j;^;«;^^t^^^^ P'y-ng together the square rootaTtf!>° ^ '<"""■ ">? "'■W- Again, aW- afe ^ .«, "°^,f ^^f *«>'-* facto™ „( aV. 0>e product of the square oota „f th ^."^ """^ "' "'^'"^ ^ "*« »' Generally, then, to' find Zl:Z''^''T '**"" < * ' "■• the «,uare root of each factor and ITw!,." '^T'^^ '^'^"< bes so obtained; the result SbeZ^^ °«*'""'' ""« l""""- flOM'tity. ' '"TU be the square root of the given •Extract square root of SlaW. .4\ ^x «._Find vsm^. V64.8. V?.,.^ yji5_4._ ^_^ — 1- «• w- SQUARE BOOT. 168 193. To find the square root of a fraction. a a a^ ^ \a} a Hence, to find the square root of a fraction: — Find the square root of the numerator for a new numerator, and square root of the denominator for a new denominator; the new fraction thus obtained will be the result required. Ex.1 Uo* __2a f36(a + 6)« 6(0 + 6)' 194. We now proceed to explain the method of extracting tne square root of a multinomial. The following mode of arranging the square of any expression should be carefully noticed: — (a + 6)»-a» + (2a + 6)6, (1) (a+6 + c)' = a' + (2a + 6)5 + (2a + 26 + c)c, (2) (a + 6+c+rf)» = a'+(2a + 6)6 + (2a + 26 + c)c + (2a + 26 + 2c+rf)(i, (3) and 80 on for the square of any number of terms. From (1) it is seen that a + 6 is the square root of a' + (2a + 6)6 ora» + 2a6 + 6'. To find the Jirst term, a, of the root, it is necessary to extract the square root of a', the^r«< term of a' + (2a + 6)6. To find the second term, 6, we subtract the square of a from a' + 2d6 + 6'», and into the first term of the remainder, 2a6 + 6' or (2a + 6)6, divide 2a or double the Jirst term of the root; the quo- tient will be 6 or second term required. We now add 6 to 2a, and multiply the sum by 6; this product subtracted from the re- mainder 2a6 + 6^ will leave no remainder, hence the root has been found. M^J I I 164 SQUARE ROOT. •*»0« The process of fin«i' ^'T '^y "o^ 1« stated in 71Z7. "^ ~'^"*'"8 »' '»» Arraago the terms of a,t^ "°'~ ■"agnitudo of indices of one of tT^ l^?"^.'"" '" "'° '"^'' "t the squ.™ ^t of the fl^W 1. ?? "'™''"^' "'«'' ^^ke fi™t tenn of the „„t. S«btr;l ''°™ ""^ "»''" « «>« pression and bring down fh '"''"'™ *""■ ""e given ei of the ™ot and sft dZ te r::^,, ^^"^ *"" '^^^^ ■l;v.sor; divide the fln,t term o^hi " "!" f "' *«"" »' « trial- of this divisor, and ^i1^^Ti^2"T '^ ""> «"* '«™ "-^d also to the first te™ of Zdi .""* '""" <" ">« -^t f visor by the second Z^otlt'^": ^""'P'^ ""' «•«■?•»*« from the a«t remainder; tZ H th? l ""'' "'""'•'"' **" ™»"lt has been found. ' ** """ I"* "o remainder the root *" '■~''^"*'=' *» '^»-" -t of 4„.+ ,2„4.94. 4a' + 12a6 + 9i« rru "XPI.ANATIOW. ■lUe square root of 4a> is 2« a the result from ia^^Uab + nUh.TT^^ ^"^ *"^ subtracting I>oubIing 2a for a trial-divisot tt find thTf'" " ''''*^^*' and give the quotient 34. • secL f . ^" ^" ^'^^^« l^aA 3* to 4a and multiplying 'Cl^^Z "' ^' " '*• ^^^-g there is no .mainderV, a.isTe ^ reTsfuir t^ *'' ^ » ^ ^. , . 4"irea square root. ^-^d the square root of «'*«+ 162a* + 6561 a|*|+162a4 + 866irafi+8l 2a*+8r; +16"2^aT6561 + 162ai + 6561 Remainder is n . -- SQUARE BOOT. 165 196. Again, from (2) it Lb seen that a + b+o ia the square root of a* + (2a + 6)6 + (2a + 26 + c)c or a' + 6» + c* + 2a6 + 26c ■»- 2ca. To find the root a + 6+c from a' + (2a + 6)6 + (2a + 26 +c)c, we find first (a + 6) as in preceding case, and then treat (a + 6) as one term, and proceed as before. For after finding a + 6 there will be a remainder, (2a + 26 + c)o or 2ac+26c + c*. Doubling a + 6 and dividing the first term of the product into 2ao we obtain the quotient, c, the third term required. Adding c to 2(a + 6) for a complete divisor, and multiplying the sum by e, we find there is no remainder, and .*. a + 6 + o is the root required. Similarly the square root a + b+e+d oi a' + (2a + 6)6 + (2a + 26+c)o+(2a + 26+2c+d)d can be found. 197. The method of the extraction of the square root of an expression of more than three terms can best be made dear to the beginner by a few exa.nples. Ex. 1. — Extract the square root of a*+4a' + 2a*-4a + l. a* + 4a'+2a"-4a+l('a» + 2a-l Ans. a* 2a» + 2a; + 4a» + 4a» + 2a»- + 4a« -4a+l 2o' + 4a -i; -2a'- -2a»- -4a + l -4a + l Ex. 2. — Extract the square root of 9a;« - 1 2ar'2/» + 1 Gar's/* - 24a:*3/« + 4y« + 1 6ay. Re-arrange as follows: — 9a:« - 24«*y' - 12aY+ 16aY + 16ay + 43/« ^^ 6«*-4iry'; -24a^'- 12«»2/*+16«'y* (Zs?-ixy^-2y^ Ana. -24a;*2/' + 16a;'3/* 6a^-8a;|^-2.v"; - 12a:»i/"+16a:3/» ^-W -12aj»y»+16«y»+4y« 1 . V' iV I f II : pi ! ( ., 'Ji 160 8QUABE BOOT. i, 1 1 Extract the square root of 3.^+12.^36, ^-8.+ i6,4aV+4a6.+6. 6. 49^ + 66ar»y + 30a^y«+8ary»+y«/ 7. a^-2a:»y + 3a:»y«-2ary»+y«. 8. 4a«-12a»ar + 6aV+6aV + aV. 10. 4ar*+9-30ar-20;r»+37a:>. 11. ^•+26^+10:.*-.4ar»-20a:»+16-24:., 12. 4a»+16c«+16aV-32aV. 13. 4-12a-lla*+5a«-4a»+4a«+14a« 14. ^+8^-4^y-4^+8:.y-lo^y.^^ 1. 25^ 31^,34^^-30^,.^-8%.:,^ 17. «'+4a4 + 46»+9c'+6ac+12dc 18. «'+2a»i + 3a*i.^4a»6.+3a.AV2a*»^i. 19. 9-24a.+58.^-ll6a3^.i29^_i4o^^ 20. 9a«-12a5.24ac-16*c.4..^167 21. 25;.y-30;r»/ + 29:r'y*-12ay+4t/« 22 4^^.12^^^17,V-12y:^^4^; 23. 25^-20ary + 4y3 + 9*>-12y« + 30:««. 24. 4.:«(;r»-y) + y»(y^2)+y»(4^+i). SQUABS BOOT. 167 198. When fractional terms occur in an expression its square root can be obtained in the same manner as whe) the terms are not fractional, but the beginner will require to exercise more care. ifz-Find the square root of ^-^ + *|aV- ?<««+|:. d^ (At ii , , 3 . af /■a' 3 if — — — — 4. —arse* oar 4 — r ax 4- 9 2 ^48 4 ^4(3 4 ^2 9 2o« 3 c^z 43 zo- o \ arx *o , . ^_ ax I — —— -^ — a Or 3 4 / 2 48 "2" J a*3^ ^16 -r-2'^-*"2>'+3'*^-4'^+4 1 «-j 3 ^ aJ« + — a 3r aar 4- — ^3 4 ^4 BXBBOISB TiXTT. Find the square root of 7 4 2. a!«-4«»y + 6«y-6V + 5y- — +C X V . , 4 10 20 25 24 16 a; X* ar ar ir IT A ji • 63^ X 1 - o« 2a „ 26 fi« If if I: ,1 fi ^^^ SQUARE ROOT. 6. 4a'-12o6 + a6»+9i»- 36» h* 2 "16* 7. «* + 8:c«+24 + l|+5? it" y* «» ly « y,- 9. $^+^+-^+4-?!'-!^ V X^ i? jg ^ • a' 6> ~ + ^_^4.2«<^ ««? *c W ed 10. ^+~+-l + ^_l%?^ «^ 4c bd ed 9 16^25^4 6+TF-y-Io + T-6- m. The following are examples of a more difficult character No fixed rule can be given for facilitating the extraction ^^h: root«; much must be left to the teacher and thetg'nmTy o the pup.h Sometimes it will be necessa^ to removeT^r^ket be obtamed by inspection and sometimes by factoring, ^a;. i.— Extract the square root of 16a3(a + 6 + c) + 4aic(6 + c) + 4a«(i«+c») + 16a>ic+6V. Remove brackets and arrange according to powers of a. 16a|+16a»(6 + c) + 4a»(A' + o» + 4i.) + 4a6.(6 + c) + 5V 8aH2a(6T;)7J6^p^^ r^a' + 2a(5 -, c) + ^ 16a»(6 + c) + 4a»(&' + c» + 2M 8a> + 4a(6 + c) + 60^ + 4a»(26c) + 4a6cOT^T^ + 4a«(2&c) + 4a&c(6 + c) + 6V iKr. ^. — Extract the square root of Re-arranging we get the equivalent expression, rninh ia Mvi«1anflir 4'ltA <....._ .« ..«.«•. _ _ SQUARE ROOT, 169 Ex. 3. — Extract the square root of This expression may be arranged as follows: i^-yzy-{x^-yz){y'-zx){z^-xy) + {y'-zxy-{x'-yz)(y'~za:)(z'-xy) •^{9?-xyy-{3^-yz){y^-zx){a?-xy) = (^-2^){(^-y«)'-(y'-»a?)(«"-ary)} +anal. +anal. = (a:»-y»){ar*- 2a:V + 2r'2'-y'2'+ay+««»-a;»y«} +anal. + anaJ = {!^~yz){o^ + xf + xs?-^a?yz}+faxaX.+&naL = x{a?-yz){3^ + y»+!i?- Zxyz) + anal. + anal. = {!^-^'iy-^^-Zxyz){x{7?-yz) + y{y'-zx)-^z{7?-xy)} = {a^ + y' + s^-3xyz)(a^+y'+:^-3xyz) = (a^+2/«+»»-3ary»)». .*. Root required is a^ + iy + ^-Sxyz, EXBBOISB LXm. Extract the square root of 1. 3(3a»-2a6 + 62)(a»+36») + 6«(a + 46)«. 2. a«(a-5A)(a-i) + i'(3a-6)»-3a»6». 3. (a-6)* + 2(a*+6*)-2(a«+t«)(a-6)«. 4. ax{ax + l){ax + 2){ax + 3) f 1. Hence show that the product of any four consecutive numbers plus one is a perfect square. 5. aV + *'-c') + 2(a+6)(6 + c)ac + 2a«(o6 + ac + 4c) + 6>c». 6. (a'+b^+c?f + 2{ab + bc+cay - 3(a« + 6« + c»)(a6 + ac+ic)«. 8. a* + 6* + c* + i an )1g braical expression is one of the three equal factors of which the expression is composed. Thxis the cube root of a« is a\ for o» x a" x a*r=a\ The cube root of -a« is -a«, for (-a»)x(-o») x(-a«)- -a*. Hence we see that the cube root of a positive quan+it; h positive, and the cube root of a negative quantity is negative. 201. We find the cube root of a monomial by extracting the cube root of its difierent factors and then multiplying the different quantities so obtained together. The product is the cube root required. 202. To find the cube root of a fraction, extract the cube root of the numerator, and divide the result by the cube root of the denominator. The resulting fraction is the root required. Hx. i.— Knd the cube root of 27a»6V. '^27a»iV=3a6V. Ex. j^.— Find the cube root of 27c»rf"e>»' 4 _8oW^_2aJV 27cfd^e^~3^d*?' 203. To find the cube root of i\ polynomial is a more tedious pro cess. Let it be required to find the cube root of a' + 3a'6 + 3a6' + b\ ifow, we know that (a + 6)» = a»+3a'6 + 3a6» + 6» or its equivalent] a»+(3a«+3a6+6')6. (1) Therefore a + b is the cube root of (1). To find a, or first term of cube root, we extract the cube root of first term, a». Subtracting a» from the given expression there remains 3a'6+3a6»+i»or 6(3a»+3a6 + 6«). To find second term, b, of the cube root, we must square a, multiply the result by 3, and divide the product into the first term of the remainder, 3a»6. To CUBE ROOT. 171 obtftin the complete divisor, 3a' +306+5*, multiply the product of first and second terms of the root by 3, and add the result to three times the square of the first term of the root; then add the square of the second term of the root to the previous sum. Multi- ply the complete divisor now obtained by the second term of the root, and subtract the product from the first remainder. If the second remednder is zero the expression we have obtained is the cube root. Ex. i.— Find the cube root of 8x*+36a;'y+54ay+27y*. 8x* + 36a:"y + 54ay + 27y» (Ix + 3y Ans. + 36a:'2/ + 54a?y>+27y" + 36a:*y + 54ary» + 27y» 3x(2.r)«-12«« 3x2a?x3y=18a;y (3y)'=9y' 12a!*+18a;y + 9y> Here 2a; is the first term, 12a;' the trial-divisor, and 12a;'+ 18a;^ + 9^* the complete divisor. 204. To find the cube root of a quantity whose root consists of more than two terms. Since {(a + 6)+c}»-(a+6)»+ 3(a + 6)«e + 3(o+6)c"+c« -(o + 6)»+{3(a + 6)» +3(a + 6)c +c«}c, we see that the third term, c, can be fourd by the rule employed in finding the second term, that is, we first find a + 6 and then treat a+6 as one term to find c. The second trial-divisor is now 3(a+6)', and the complete divisor 3(a+6)'+3(a + 6)c+c'. The divisor, 3(a+ 6)' +3(0 + 6)0 + 0", is multiplied by the third term, c, and the product subtracted from the 'second remainder, 3(o + 6)'<; + 3(0+ 6)0" +0". There is now no remainder, hence the cube root has been found. bVWB _»~-.I_i.; ^e £ J. a+6+c+d^ we first find a-\-b+e^ then treat (a+6+0) as one term i tli II 178 CUBE ROOT. to find the fourth term, d. The student must bear in mind that the method employed to find a cube root of three, fo,-, eta, terms IS exactly the same as that employed in finding the first two terms. Ex. «.— Find the cube root of a!«-3aj»+6a:"~3a?-l. 3j^-3a:« + ar>; - 3a:» + 6a:» - 3a; - 1 3(x«-;r)»-3(:r»-T)(-l) + (-l)« -3a:* + 6aH»-3a;-l -3ar* + 6a:»-3a:-l In this example, to find second trial-divisor, we treated a»-x as one term, and then proceeded to find third term in precisely the same way as we found the second term. EXEROISB LXIV. Find the cube root of 1. x» + 6j:«y+12ay+8y». 2. ar»+12aj»+48x + 64. 3. a»-9a«+27a-27. 4. T«-3(Mr»+6aV-3a»ar-a«. 5. ^ + 3«» + 6a:* + 7«»+6«>+3a;+l. 6. a:« - SaH' + 15a;*- 20x8+1 5a;' -6«+l. 7. ^a* - 36a*6>+ 54a»6*-276«. 8. «»-6»+o"-3a»6 + 3a>o + 36>a+36>c + 3c»a-3c«6>-6a6c. 9. 8a;«-36a;»+66a;*-63a;»+33a;«-9a;+l. 10. l-9a;439a;"-99a;»+156a;*-144aJ»+64a*. 11. 64j:» + ig2«6+U4a;*-32a:»-.36aJ+12a;-l. CUBE ROOT. 173 12. a« + 9a»&_ 136a«6«+729a6»-729A«. 13. «•- 126c» + 606V-1606»c« + 'M'jAV-1J26»c+C46«. U. ««-^ + 12-6x». a«_J_ J__l *^' 8 27a«'^ 3a» 2* 16. a;» + i + 3(ar+iy 206. The square root of a* is a\ and the square root of a' is a; therefore the fourth root, a, of the given expression, a*, can be found by extracting the square root of the expression and then the square root of the result. 207. To find the sixth root of an expression we notice that a, the sixth root of a', is the cube root of a', which is the square root of a*. Therefore we can find the sixth root by taking the square root of the expression and then the cube root of the result, or vtc0 vw9^a'*b'*; (1)» (2), (3) and (4) are known as " Index Lawa" 210. To prove the Index Laws:— L a-x«--(axaxa.... to m factors) x(axa.... to n factors) "axaxa.... to (m + n) times factors -a"*** (by definition). IL (a-)» = a*xo-xa"'.... to n factors = «"*^*^- *••'*»• (FromL) -a" riL («*)" =• «6 X aft X o6. . . . to n factors -(axaxa.... ton factors) x(6xft.... ton factors) -a»x6"»a'*6» THEOBT OF INDICEa 176 - a)' ana -Xr^r""^»' factors bob axaxa to n factors ' b xbx b.... to n factors fiitilrtii 'b-' P 211. We proceed now to assign meanings to a', a« and a"*. Assuming that the laws which are proved to be true when the indices are positive integers to hold good when the indices are zeroy negative or fractional^ we arrive at the following results: — (a) a«=l. a*^xa^> From (1), ^m-(-0 __ fgm , Dividing by a"», From (1), But, from (a), a™ a"* a"* X a""* = a"*""* = a*. a^xa""™!. a"*' «""*=»-—. Dividing by a"», (c) o«= ^o^. It has been proved that (a*)" — a'** when m and n are positive integers; and as we sussume the same law to hold good for frac- tional indices, p ?xj /. w .o» Extracting the g^ root of each side. :, o«- ^o». urn n 176 THEORY OP INDICES. numerator o{ the ft™andThen^h ' 'T "'"'^*^' '^ ^^« b7 the denominator. For .nstane; "^'"''^ "'^^^^^ 2*'-l, 3"'=- 33-9, «*» i?'^ a«= iT^, 2*- V2'^ -/32. 212. Again, since a^xo-^a'-H*, Dividing bj a*, a^-^xa* — a"'-'»-H» = a" -^» Of conversely, a" >a m-» 213. Prove that a» x A" = (aM". (a*x4-)'' = (a")»x(6")» I 1 Extracting the n*- root of each side, i i I a«x6»=(a6)». This result is of great use in surds. For exa„.ple, Prove v'2 X a/3~» VJ. ^2-2*, 1/3 » 3*; /. -^Sx v^3=24x3*=,6i-.i/6; Again, prove 4^Jx^i^ ^12. ,\ '^3xl?'T„3*x4*-l2*-^i2. Art. 209 (3) THEORY OF INDICES. 177 Also, from the above result, we can find the value of such a product as '9' S x '^ 2. For i?'3=3*=3* = (3y=9*. and ^ 2 = 2^ = 2* - (2»)* = (8)* ; .'. i?' 3 X i?^¥ = 9* X 8* = (9 X 8)* = (72)* =-^72. Ex. 7.— Find the value of x^^-« x a*+*-* x af>+*-\ Ex. 2. — Find the value of a:*"*xa:*-*xa:*"". Ex. 5.— Simplify ab^c x a~hc^, a6*cxa-i6c*=a*-46*+V+* J^ar. 4.— Simplify \^^j . /256\-|_/625\| V625/ ~ V256J f^625\n» /5\» 125 The student will obf?()rv3 that we have extractec'. the fourth root of ^v^ before cu^injj it. We might have raised j—r to the third power and then extracted the fourth root, but the largeneiis of the numbers involved would have made the pixKsess cumbrous. If* 17 R THBOBY OF INDICEa Bx. 5.— Multiply «i - 2«* + 1 by ar* - 1. From observation we see that ** - 2a;* + 1 „ (ar* . i)i. -a:-3ar* + 3x*-l. Otherwue^ x^ - 2z* + 1 « -2a:*+ a?* - a?^ + 2.r* - 1 « -3a:*+3a:*-l ^x. + ;,-iy. *r • + ^-'y; ar»y-« + 2 + ar-VY^y-i ^. ^-ly a?V'* +l + l+ar-y l+ar-y We might have performed this division hv firo* «i, with negative indices into equivalen ones ^'r^'^^^^ ?""^ and then proceeding as m olnary irlZn^ "^ "' "'""' «V"'+2 + a:-y»^+2 + ?^ Thus, and X y THEOBT OF INDICES. 179 Ex, 7. — Extract the square root of 9«-* - 18ar-»y* + \^x-*y - 6a;-»y* + y". 9ar-« -18a:-«y*+ 9-c"'y + 6j;-'y-6a;-'y*+y* 214. In solving the following examples let the beginner bear in mind the index laws and the meaning of such expressions as a*, a"*, and no serious difficulty will be experienced. BXBROISB LXV. Express with fractional exponents 2. -^ay^a", ^x'y'**, ^a»6V, 6^a»6c»«*. Express with radical signs 3. J, o*6*, 4a!*y"*, 3a;*y"*. Expi^^s with positive exponents 4. a-«, 3«-»y-', 6i:-«y, a^y"', g-.^.yi - Write in the form of integral expressions 3ary « a c* x~^ x'* THEOBr OP INDICES. c'xcA cf*xrfi^ i , a-xo'xa ;i V /».~i 8. aixVa, c-ixVc, yixi^y, ;.SxV^ 9. a«6ic-ixa*J-Md 10. .W.^-ly-Vi .Syi,ix,-J^-i^-j a 12. -- ^ c^ «.A n^ a* a ;* c* ^S ^ 13. 14. 15. /16a-«\-f / 9a* \-# . i if Multiply BXBROISB LXVI. 1. x^ -x^ + l by ar* + 1. 2. a»6-«+2 + a-«i» by a»d-«-2 + a-»6«. 3. 4a:-»+3a;-« + 2ar-»+l by «-«-ar-»+l. 4. ar^+ajPyP+y* by «*-«Py*+y*. 6. s*-xyi+x^y^t/^ by x + ar4y4+y. 6. a- + .r(ai-6i).aM by ^r* - a:(ai . fii) . «M. 7. aK"-i)«»_y(«-i)-. by a!*-y«». !"*1!i THEORY OF INDICES. ISl Divide 8. «' -ay* + s^y - y* by x^ - y^. 9. a*-a*6 + a6*-2a*6» + J* by a* -ai* +a*6-6*. 10. a^-a^ by a?-a\ 11. a; - 2(a;* - «"*) + 2(3^* - ar"*) - «"» by «* - x~K 12. (a;")»-l by x"-!. Simplify 13. (2a;i + 32/l)(2a;* - 3y*)(4a:* + 6a; V + ^V^) (4a;4-6a;*yi+9y*). 14. i^'aj-lx -^aj+lx ^x^-x+lx ^a? + x+l x(x*-l)*. 2-H _ 2 X 2" 7 ' 15. Show that 2»+» X 4 8' 2* X ^2""*^* 16. Simplify -2S+i\^i. Fir.d the square root of 17. x^ - 2a;*y* + 2a;*i&* + y* - 2y*a* + »S. 18. a;* + 4a;*2/* - 23^2* + 4y* - 4yM + »*. 19. 4a;-*+12a?-»+9a;-". 20. a^+4a;+2-4aj-*+«-". 21. « - 2 + 2a!"* + jc-> - 2a;"* + x-\ 22. 4o - 12a* 6* + 96* + 16a*c* - 246* c* + 16c* S;» I £ II 18S THKOBT OP IMMOM. 25. (ar + a:-»)»-4(ar-ar->). Simplify 27. -^+^ ^ «'-*• 28. ^.r!±2fe+ri) /^-1\« 29. 3a-V + 6a-»ar-12 ' «^V:^iia-V-12a-»a: + 63* 30. _lzizid±iyL:fM_ ^~8-2a;i+12yi-3;rV" 31. *^"''+a!y-^ + ya;-^+^-* * ay-V-' + a + yaj-^e"-- 34. /^^/_fir"' 30. -—- ^ 36. {(a*)*"*}^ THEOBT OP INDICES. 87. {(a + by - 4a6}4 . |^*+ 2a6(a + 6) J*. Find the H. 0. M. of 39. !+« + «* + a;* and 2a: + 2a:* + 3*" + 3«*. Find the L. C. M. of 40. oa:»-l, ax*+l, (a*a:-l)« a*«»-l, a*a:»+l. 41. I£a;*+y*+«*-0, prove (x + y + z)*=i27xi/z. Simplify 42 (^'xW.W a^-K '^ a:e+« '^a-H^' 43. (8* + 4*)xl6-* 188 44. 9" X 3» X -— - 27* F79 45.27« + 16«-H-^.i:3^ 8-i 4"* 46. {x+l){x'+x + l)-' + (x-l)(x'^x+l)-^ + 2{x'+x»+\)'\ 47. 1 +a;'*-*+a:*-'^ 1 + af-'^+af*-p l + xp-'*+ x*-»* I CHAPTER XIII. SURDS. 215. When a root is indicated but cannot be exactly deter- mined it is called a SurA Thus ^/\ ^4, v'g ^re ^rdn When the root is indicated but ca^ be exactly determined it is said to have the/orm of a surd, ^ VJ, f 8. Surds are also caUed trrcUional numbers. 216. A Quadratic Surd is one in which the second root is 217. The product of a rational factor (i.e., a factor not con. 218. When there is no rational foctor outside the radical siim the surd is said to be entire, aA V b, \^J, 219. A mixed surd can be expressed as an entire surd Thus ; :; ? ^ "?'• u ^-^-»y «^^oan be expressed as «i entire surd, for in the chapter on Indices it has been proved or Now, - i 1 I SURD& 185 220. Hence to reduce a mixed surd to an entire surd, raise he mtiona factor to the power indicated by the root to be ex- tracted multiply the result by the factor under the radicaJ sign, and write the radical sign over the product. 221. An entire surd can often be expressed as a mixed surd. Thus V'8'»2a/2", v'50-5V'2. For ^8-V4^ = (4x2)*=4^2*==2x2* = 2^2: Similarly, VTO - V^blTi = (25 x 2)* = (25)* x 2* = 5 V2. OeneraUy, v'^ - (a-6)« =, (a«)» xb^^a.b^^aV^J, 222. Hence, to reduce (when possible) an entire surd to a mixed surd, separate the quantity under the radical sign into two factors, of one of wUch the required root can be obtained, and set the root outside the radical sign. 223. The expression under the radical sign is called the Surd* ftctor or Base. When the surd-factor h b^ ^nnall as possible, and integral, the surd is said to be in its simplest/arm, 224. Sunilar Surds are those which have, or may be made to have, the same st^d-factor and surd-index. Thus 2^2 3^/2" V50, are similar surds; so also are \/a, v^Pa SV^ ' The student will notice tJiat before two su^s can be said to be simUar they must have the same quantity, under the radical sign, and the same ^,rd-index_or number which indicates the root to be extracted. Thus ^a and Va are not similar surds, for they have not the same surd-index. Sometimes the term radtcal-tndex is used for that which indicates the root to be extracted. 225. When the surd-factor is a fraction, and it is required to reduce the surd to ite simplest form, it wUl be necessary to ex- press the fraction in the form of an equivalent fraction, with a 13 * t i 1 % i J '4 i IMAGE EVALUATION TEST TARGET (MT-S) fe ■i< 4 ^ mo •^ ^ 1.0 I.I is 12.5 Z2 M 1.8 '•25 1.4 ||i.6 4 6" — ► p 4> ^ /} /2 % ^. r **-^* .« _ w '/ Photographic Sciences Corporation ^ * \ :\ v n.- >»^ 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 873-45C3 Q- mo \ ^ <^ I 186 SURDS. denominator whose root can be taken; for the expression under the radical sign must be made integral. For instance, In this example the cube root of 25 cannot be taken; but as 25 = 5' and 25x5 = 5^, tlierefore if we multiply both numerator and denominator by 5 the new denominatoi' will be a perfect cube, and its cube root can be removed outside the radical sign. Ex. 1. — Express in the form of a cubic surd 2, 3, -, {a-bV - . Ex. 2. — Reduce the following entwe surds to the form of a mixed surd : — \ ^m ^r2. ^w, sS' J£' ^^5- (1) 1?'l08='V^27 X 4=i?'27x i?'T = 3^T. (2) Vf2= -v/Se X 2= VdQx v1==QV'2. (3) ^7^= ^Ixyxf^ \^l¥fx \^^ = y^T^, <'>N|5-vfi-s|»-"-i>'"- (6) if 9« ^ 4/ \/26V \ iOabc^ UOabc* X Rki-ot^^Oaic*. 166V Sj— '^ IQb^c^" 2bc (6) ^m^ ^27 X 15= -^27 X ^l5 = 3 vTs. Ex. 3. — Express as tntire surds the following mixed surds: n^Vb^, 3V2T, 5^/32, 30*^^, o»^^», 2a; ^J"^, SUBDS. (1) a'bVT^^ V^x VT^^ Va^I^ xhc^ Va'l^. (2) 3l/2l-.'/9 X Vn-VTsg. (3) 6V^32= A/2B X ^32= VSOO. (4) 3c»^^= ^2f?x ^^- ^27^i?. (6) a»^^- -C^^ X v^^= -^^J (6) 2ar^^- >^32?x 1?^^= f 32^. f (a?+y)'(ar + y) . 187 4 226. Surds are Mwipia or oompov/nd according aa thej contain one or more terms. BXBBOISB LXVn. Express as mixed surds 1. '/ar'A ^8^, ^54aVy», v^2i, i^T25^. 2. f 1000a, i?'r60^^ ^"I08m»w", '^1372a«6i«. 3. i^'a* - 3a»fe + 3a»^.' - a6», VbOa? - lOOaft + 506». 4 ^ ^- \9' Nile' 8* \/2' V6* ^- s/^rTA' N/^^TF' n/2^' V Express as entire surds 6. 3\/2, 2^7, 3^6, 6^9, 21^6. 7. 3i/a, 4a V3;, 3^, ^^sM» I \ 188 SUBDa n).E3, («+ftv /m /lz_y^ /^'+^ 8. (m + ; Simplify ^ 9. 2v'80Sw, 91^81^;^ 6V^726. 10 '/^ - ♦01 ^»r«^ 11. aari^ 2a»6*f6V. 12. Show that \'2b, VIE, ^ are similar surds. 13. Show that 21?'^'. ^8iS. 1 ^ are similar surds. 14. Show that VbO, V72, V32, ^ are similar surds. 227. Surds having the same radical sign are said to be of the same order. Thus i?' a, 2 i?'^, i ^3, are each of the third order. 228. It is often necessary to bring surds of different orders to the same order. This can be done as follows--— - i * 1 « Let a* 6- be surds of difterent orders. Then o*-o^ and - — — -i- JL _L I i^-6-*; but a-«-(a-)«» and 6«»-(6-)""., therefore a«-T5= and ft«-T6=: It is evident that T'^ and TF are of the same order. Hence, to reduce surds to the oame order, find the L. 0. M. of their radical indices for a common radical index. Raise each expression under the radical sign to the power indicated by the number obtained by di^-iding its radical index into the common radical index, and write over the result the common radical index. I SURDS. 189 Or, express the different sards in the form of quantities having fractional exponents with a common denominator. The common denominator wiU be the common radical index. Thus V2 and ^ 3 may be expressed in this form: 2* and 3* or -^2* and ^3«. JSx. i.— Beduce to a common radical index -/If and ^T. i^3-3*-3*-(3»)*-^35. £x, 2. — Beduce to common index ^"^ and V^. 1^8 - 8* - 8^ « (8»)* - ^8» V'l - 2* - 2* - (2»)»^ - V2». BXBBOISB LXVm. Reduce to a common radical index 1. V5, ^J. 2. iTe", VJ. 3. ^2, ^3. 4. ^1, sn, 6. 1/2, ^6. ^6. 6. ^7. ^6. V'llo. 7. i^'S, l^'e, ^\0. 8. iTa' - a6 +^, ;'>'iT6. 9. //?, ^. 229. Surds can be compared with respect to magnitude by reducing them to th.e same order, and then expressing them as entire surds. •^«. i. — Which is the greater, V'¥ or ^ 3"t i^3 - i?'2»- -^8. ^J- -^35- -^9. A ^3 is greater than V'a! \ 190 SUBDa Ex. ;8.— Which is the greater, ^3" or ^TX /. I?' 6 is greater than ^ J EXBBOISB T.vrg 1. Which is the greater, 3 v'T or 2 vTHl 2. Arrange in cjder of magnitude 9 V\ 6 V^T, 5 \^10. 3. Arrange in order of magnitude 4^4J3^F, 6^F. 4. Arrange in order of magnitude 6 ^ Tj 4 ^ 9^ 3 ^18. 230. Similar surds, in their simplest forms, may be added or subtracted by adding or subtracting their rational factors and then affixing the common radical factor. Ex. 1. — Find the algebraic sum of V 50 + ^72 - 3 V^ 2^ _ '/2 ^50 = 51/2, ^72 = 31/2", -L = - a/J. V2 2 .'. Algebraic sum = (5 + 6 - 3 + ) v'2^ = -— v'2! ^ar. ;».— SirapHfy 2 ^320 - 2 v^iO. 1?'320=i?'64 x6=-4^6; /. 2 f 320-8^6. i^'iO- ^875-2^6, /. 3^40-6^5". /. Result - 8 ^5" - 6 i?' 6 - (8 - 6) f F - 2 ^ 6. 6t7RDa 191 231. Surds having the same radical index can be multiplied together by finding the product of their rational factors or coeffi- cients, and affixing to the result the product of their bases with the common radical index written over it. ■^«« •?• — Find the product of V'o^and V'A'. Va X l/P- V^. ^o' i^a-.a*and V^i»-ft*, Ex. 2. — Find the product of a v'T and 6 v'^ For V =c* and V d^d*, „._ n/— i 1 i - /. Vex Vc? = c»xcP» = (cci)''- v^^, /. cVc xb\^ d^ah\^ cd. Hence «^e rule. 232. Surds having different radical indices can be multiplied together by first reducing them to surds of the same order or common radical index, and then finding their product by 'the preceding rule. Ex. 7.— Find the product of ^2", ^3" and -^4 The L. C. M. of 2, 3 and 5 is 30, therefore the common radical index is 30. . Hence l/"2 = "^2S, ^3«'fF», .'. ^/2 X 1^3" X '^T- "^2« X 3" X i* - Vill'*x3Wx2»- •f2«'x3» \ t;i! 192 SUBDS. Bx. ^.—Pind the product of ^~^ and 71 n, — i Va >a"^. Hence the rule. of the mvidend b, the ™«<»a. Jtof 7/^! ^^^^d ttl «'««i over It, of the one bape bv the nfhoi. t* 4.v , • •ii^ «rfe„ they oaa be ^^u!^ ^^e le X Jl°' preceding rule appUed. ®'' ""^^ *^® -»«. i.— Divide ^9" by ^16. ^9- ^15 = 9*^16* = (^)* = /?\*^. I? Fx. £— Divide ^S by -^I. -^a?. 5.— Divide ^6 by i^'T". .-. i?'5 ^ i?'^- ^^'S^-- ^7»« W? 234. m multiplication and division of compound surds differ in no respect from the multiplication and diviL of pd^om^l 8U&D& 198 rational expressions, except that surd terms ai« used instead of rational terms. An example or two will illustrate this. Bx. i.— Multiply together i^S + v'2" and V^S" - v'2^. Here we have the sum of two expressions multiplied by their difference, therefore the product » the difference of their squares; /. (V'3"+ ^1){VZ- '/2')-3-2-l. Ex, «.— Find the product of 2 ^^ + ^2 and ^T + VJ, 2^27x16 + 2 + 2^15+^4x125 Ex. S. — Square ^ + ^3" + 1. (V'2'+^3+l)« -(V2")» + (i?''3)' , 1 + 2a/2"x ^¥ + 2\/2" + 2^3 -= 2 + i?' 9" + 1 + 2 f 8~x9 + 2 v^¥ + 2 i?'! „3+^9" + 2f72 + 2v'2" + 2^3 Ex. ^—Divide v^ 8 + 3 v'24 + 5 ^56 by Vu. VF + 3 \/24 + 5 i^56 + VH 4 ^ +3\/2+5 NT* 194 SUBOS. Simplify ■XBBOISB LXt 2. »v/?-^3^7 + 2^?; 8^IO + 3^Io + 2Vro. 3. 4^n+3Vn-6v/TT. 2*/3-5^3+9V3: 7. '^2+3*'32+lv'l28.6*/m 8. ■«^76+v/48-v'T47+*/300. 9. 20v'2?6-^6 + v'l26-5v'l80. 10. 3ir3-6ir48+^2i3. 11. ^27?-^8?+V>'125j. 12. fM-'f'^+ff32b. 13 1^5;+ V4^_ ViSi^^ U. ««^-3yn2?SJ+2aJv'3l37i-M^28?J. 16. *'32a'4.+6V724 + 3^l28:V-4V288iW. "•28 . 7 ./2 ft;* SURD& EXSIBOISB UOSJL ■ Perform the following divisions: — 1. V2434-V3; ^m^^I; ^81^-^3. 2. '^2i3-^^3■; ^66^1^7; ^600-^4. ^' N/12'^N/7' N/9"^N/2' \35^\6" 4. 3v^6+46\/2'-^3V3. 5. 42i/5'-30v^3'-^2V'l5. 6. 84A/l5 + 168v'6"-r3l/2T. 7. 66\/30-84Vl0 + 100v^l44-4\^35. 8. 301?'T-36^I0 + 30^90-^3^20. 9. 50^l8 + 18i?'20-48^6"^2^30 197 ii.-4r» a-6 1-a 12. Vb - l' Va -Vb' 1 - -/a' a* + ab + l^ aca^V^-bcy'Vle a + Vab+ b cVxy 13. -S/Hi^-^SG; ^"g-ri/S. 15. ^2i^\^^; i?'8^-^2«y. 198 SXTBDa 4f .1* Z^;:JX' ';^^^f^ aw cannot be portent theorems are^e^ucld :-! ^ '''' *^^ '^"^^ ^■ rational quantities Ld lAa surd Th 1' 7~ ' ^^ ' ^^ c-a is a rational quantity V. \! . "'" ^* =^-«- ^ut national quantitie ;re;eSe vT a f T" '^^^"^ *- a surd equals a rati;nal q^Tit^ 1^1" 'r '^'^' *'^* ^' « + 1/4 does not»a "^possible. Therefore II. A surd and a rational ,uanHe, cannot e,uVlTvf 7. V'^ + ^ Va - a; V^MT '+V^r:^' g Vx + 3a+Vx-3a x + VioTT^ 9. _^ 21/3 2^2"- V3 v2+^3' ^4+^r ~vnjT' 7-2VI 15+6VT 10. 4-V6 2 + V5 11 y3 f 1 V£- 1 v3-i^vr7T 12. U. BUBDa 20 + 30V2 5-2\/2^ s + Vs " 2-V3" 205 a + ar +^(0+3?)* 13. L±}^ g-t-2V5' 1 + V5 '^ 2 + V5 * 2\/a + b + SV^TTl 15. 2\/a + 6 - SVTTTb' 241. It is often necessary to find the value of a surd expres Tvoired^'' * P^'**i<'^lar value is given to the unknown quantity J^ar.--Find the value ot x*+x^ + y^ when x^^^!!l±l and 1/3-1 V3 - 1 y= \/3 + 1* "V3- l"^ V3+I/ " V3 + I Vf - 1 VS-l"" V3 + 1* -(4)»-l = 16. This example might have been^solved by rationalizing the de- nominator, of ^|±i and ^i^l before substituting. Thus, V;3_+_j_ (V3+l)' 4 + 2\/3 \/3-l 3-1 2 and V3- L iJ^lzIL' 4-2V3 V3+1 3-1 2 .-. «'-(2 + l/3)«»7+4V^ andy>=(2-V3)»=7-4V3; .-. «" + rry + 3^-7 + 4^3+1 +7-41/3 •2+1/3 «2-l/3, 16. ■ 206 SURDa fhf t K1 '^"^^^g «^a°^Ple instead of directly substituting the problem is reduced to a simpler form bj means of a weU known theorem in fractions. (Art 169, Theorem T ) ^or.— Find the velue of ^^^^t^^^IZ. Let Va -i- x-Va -X Va +z + Va ~x k y when x> 2ab T+b ,»• Va + x~\/a ~x Adding numemtor and denominator of each fraction, and divid- ing bj their difference, 2Va+x k+ 1 2\/7r^"A;- 1 Va +a: k + 1 or Squaring, Again^ adding numerator and denominator of each fraction, and dmdmg by their difference, ™cuon, 2a^2A«+2 2a:" 4* * or ^_*'+l ._ « 2;fe 2r°''! A»+i- Substituting now for x its value -?^ we obtain 1+0* 2b 2k or 1 + A» >fc»+l 1 + 2& + 6' 1 + 2^ + ;fc« 1 -26-f6>"l-2yfe + /5;«* Extracting square root of each side, 1 4-^ 1 +6 1 -ifc'"*'T^rT- 8UBD& 207 If we take the positive sign we obtain for k the value */ if we take the ruigative sign we find A - I . Therefore the value of the fraction « 6 or ■=- . Had this problem been solved by ciirect substitution the pro- cess would have been somewhat as follows: Va + x + Va ~ X Va + X- Va - \~~2ab' \ ^af fl + 26 + &« | l - 25 + y \ l + 6» "^N/ l+6» ir+26Tp f i - 26 + y 1 + 6 1-6 Vl + 6» Vl + 6« 1+6 1-6 Vl + 6« Vl+6« (l+6)±(l-6) = (l+6)T(l-6) 2 26 1 , . 2^ or .^ = - or 6. N.B.— The double value, it will be noticed, is obtained by using the double sign before the square root of the second terms of numerator and denominator. Had the double sign been used in all the terms the result would not have been changed. 208 8URD& aXEBOISB LXXVL 1. Find the ™l.e of ^^±f ,Hen .-?(*-«) 2. Find the value of ^^^^±^+^'^'^"^^ 3. Find the value of ^!i±J + V^^^TTI g 4. Find the value of ^+i + a:+ I ^ten ar^LLVj * a? 2 * 5.Findth«W«e„fJEI|„he„. = 4V2. «. Given v/3» 1.7320608, dnd the value of -^__ 2 + VJ ■ T. Find the value of —i±f__ . !-» v^ 8. Find the value of ^^^ ^ 1 -a? V3 9. Find the value of V^^ZlY^^^ u 2a 10. K2,n-^+l,2.»y^i,fi^dthevaJueof mn + ^/(^rix^rrr) in tenns-of . and y. Va + 6a;-Va-6«* IMAGINARY EXPRESSIONa (109 IMAGINARY EXPRESSIONS. 243. The square root of a positive quantity may be positive or negative. For ( + a) x ( 4 a) = a»; also (-a) x ( - a) = a»; there- fore V^ - ± a. As both ( + a)» and ( - o)» are positive it is easily seen that the square ro ot of -a» cannot be either +a or -o. Such an expression as V~^ is therefore called an tmpoaaible, or tmagina^T/, expression. We mean by an imaginary expression one which, if raised to an even power, wiU give a negative ex- pression. AU other expressions are said to be reaL 244. All imaginary square roots may be made to take the same algebraic form. Thus V^ = Vxx(-.i) - V'x'v::!. 245. The beginner should note carefully such results as the following : — (1) V^ X V~^« - 1, i.e., (V^y^ - 1. (2) {V~^y = (V-Ty X (\r:i) =(-i)x v^ = - V- L (3) (V^ly =. {V^y X (\/^)»= ( - 1) X ( - 1) - + 1 Generally, 1/ - 1 raised to an even power is rational, to an odd power, imagina/ry. If the power is even and a multiple of 4 the result is positive; if even, and not a multiple of 4, it is negative. Ex. 7.— Express V~^ in the form of aV^^. V-4 = V4x -i iiV - i. \ 210 IMAOINART EXPRESSIONS. Sx. ^.—Simplify y~ll fi}inoe \/-9-3v^-l and V^S- ^3 . v/~. l^Vz, Ex. A-Multiply Vb + V^ by V^ - V"^. « V/30 + 6 V"rT _ ViO . y— + V48 « 1/30 + (6 - ViO) \/"^ + Vis - V30 + (6 - 2\/i0) V^ + 4V/3. ^a?. 4.— Multiply a + V^ by a - V~. BXEROISB LXXVIL Express in the form of aV^ 1. V"=nS25, Vlls, ^381, V3144, V^\ 2. V^16, V^:~64, V^Tg?, V~^8h;?. Find the value of 3. (V^)«. (V^T)-, (VTT)T, (y^ri)!. (vrT)« Add together 4. \/^r26, ^"^49, yri2T, - v"::^. IMAOIMABT EXPRESSIONS. 211 6. 6 + V - 16, 3-V-i, 8 + V-i. 6. 3 + 2V~, 4 - 2 VTi, 7 + 3 V^, 4 + 2bV~^. Multiply 7. V - 16 X V - 9, V - 25 X V - 49, V - 92/'^ x V - 16;?^. 8. (3 + 6Vn")x(4-7\/'3T), (3z-8V~)x(3x + 3V'~). 9. (2a + 36\/^)x(c-c;\/^), (m + bV~^) x {n - bV^). Divido 10. X V-x' V - lOx' V-l V-x V-x V-b^ V-bx 11. Ifar. -1 +V-3 find th'3 value of «* + a; + 1. 12. M«- l-V-3 2 find the value of «* — x + 1. 13. Simplify 7+ V^^ 8 -I- 3V-3 4(2 - V - 3) 2 - V^ 2 + \/T3 " 1 _ -v/Ts ■ 14. Ifar -\+V-S and y» -1 -V-S 2 prove «■ — y ajid v' = «. CHAPTEK XIV. ii t ■ SIMPLB EQUATIONS INVOLVINa 8UEDS tte power i„d,«.ted by the ^Sl^de!! * *""" """ *" ■*■* -'•-Solve Vi?+3Sri6-2ar + 2. Squaring both sides, .'. -Sx— -20 or a:«4. JV. «._Solve V'iTTj + i.s. Transposing, vSTg.g.^ Si'""'"?. ^-9.81-18. + ^. .*. 18;r»i90 or ar-S. »e Je of ^:^ »;«:«„„ "n.f"' "^^ "^-^^ "^ -•o «« and then sq«arinX Lsid™ T. """^ *"""" 'o '^« "ti""^. which can I nZvrf bv it ,^ 7^ ""' """"'" ""» '""l. i^oved by the method of the preceding article SIMPLE EQUATIONS INVOLVING SURDS. 218 Ex. 2.-— Solve Va + 4 + V^+TB - 11. Transposing, V«+15 - 1 1 - y/TTl Squaring both sides, a:4-16»121- 22 VJ+l + a: + 4. Transposing, 22\/ar + 4-110; .*. Va: + 4 = 5. » Squaring again, « + 4 - 26 ; .*. ar-21. Sometimes it is more convenient to keep both surds on one side and take all the other terms to the other side. Ex. ;e.— Solve V^+4 - V J^ - 4. Squaring, a: + 4 + a;-4~2Va:« -- 16-16. Simplifying, Squaring again. li'-i - Va:*- 16-8-a!; ar*- 16-64 -16.r + a^. .*. 16a: = 80 and a: — 5. Ex. 5.— Solve Vl2a?+1+Vl2ar -18. Vl2a;+l-\/l2.t; Vl2aj+1 + VS 18 Vl2a:+1-Vl2a;" 1 ' Therefore, adding numerator and denominator of each fraction, and dividing by difference, 2\/l2aj+l 19 or 2l/l2a; 17 /12ar+l 19 Squaring both sides. 12a; 122; -I- 1 12a; 17* 289' 214 SIMPLE EQUATIONS INVOLVING SURDS. Dividing numerators by denominators, Simplifying, and ^*. ^— Solve Vx + \/J73 ^ 12a: "^ 289 289 *"86i' Multiplying through by the denominator V^3, Transposing, ^^(^ = 2-3.. Squaring both sides. «:« + 3a: = 4 - 4:r + a:>. .*. 7a: = 4 or a: a - 7* 249. Sometimes when the eomtinTi ««^* • ™rd3 it a«,«n,es the form oi a ll" . ™ f'" 'i''^'"*'" surda In such ca^esTdear of S' T "'''' °'^'^« "' «etho. . .. e,.«^. r- -:: troSi;!i- ^«. i.— Solve \/a: - 6 + V^^S = Vi^^6. Squaring both sides, ar-5 + ar + 5 + 2\/i»'^r26 = 4a:-6. Simplifying and transposing, 2\/?:r25 = 2a:-6 or Squaring again, V?":r25«a:-3. «>-25 = a:>-.6a: + 9. /. 6a:«34 or a:-6|. * SIMPLE EQUATIONS INVOLVING SURDa Ex. ^.— Solve \/^+3 4 V J78 - 2l/«. Squaring both sides, « + 3 + ar + 8 + 2\/x>+llar + 24-4«. Simplifying and transposing, 2V'ar'+llaj + 24 = 2a; - 11. Squaring again, 4ar»+44ar+96 = 4a:»-44ar+ 121. I: .*. 88a? as 25 or a; i 25 '88' BXBROISB LXXVIII. Solve the following equations: 1. 2\/;m^ = V28. 3. 4 = 2\/^-3. 6. 9 + 4\/^=ll. 7. ll-4\/5^ = 9. 9. ViM^ = oViTTs. 2. SViJTs^VlSar-S. 4. 5-3\/^=4. 6. 6-3V7=4. 8. 7 + 2\!^3;^=..5. 10. 2.r-8 = \/4.r'-.12ar+32. 11. V'a:-5-V'ar-4-l. 12. V^?T6 + V^«15. 13. Vx-ie + Var-S. 14. V.t:-6 + Va;+7 = 6. 15. \/a;-l+\/ar-4-3-0. 16. V^^l=3-l/^4. 17. Va;+Va:-9 19, y/7-\h^^~i^ 36 Va;-9" 105 Va; - 15* 18. \/a: + 7 + VI = on */ _ . -/ : aU V .c i- V j; _ 4 « 28 V^+7" 8 216 SIMPLE EQUATIONS INVOLVING SURDa 21. \/ar+V^r2l--^. 23. V'^"- 6 \/x + 2* 22. 24. V^-3 ^Va?+l \/^+ 3 V7- 2* V^ + ie V^+32 — B3 -— , Va; + 4 Var +12 25. i±v:k?=3. 26 ^^ -*• "^^^ + ^ n 1-Vl-a; ' 3a;-V'4a: + ar' 27. V«+~l + \/a;+16 = 2\/^+25. 28. V'4a + a:=:2v'6 + ar-Va:. 29. Vi + x+Vx+l=2Vx~^2. 30. V4^+5-\/^ = a/JT3. 31. -_= + _L^ -=. Vx+l Vx-l Var»-1 32. V{x-ay + 2ab + b^^x-a + b. 33. \/(a; + a)«+2a6 + 6« = 6_a-a;. 34. x^+{x-9)^^36{x-9)-\ ?5. (12 + ar)^ ;r* =6. 36. (a:-3)*+a:* —. 38. 40. 41. 42. Vl-.t (x-3)« ViTi 2-Vl+ar 2 + Vl-a: 1 ^ 1 _2x 2 2 5ar - 1 \/5a: - 1 J9. ___ =1+ V5x + 1 x + V2 + x^ X-V2 + X' 6 5 x + \/5 + a:' x-Vb + x* = ar + 5. = « + 8. 40. V . / 4a + d; + Va + ;i; = 2Vx + 2a. CHAPTPB XV. SYMMETRICAL EXPRESSIONS. 250. An expression is said to be symmetrical with regard to any number of letters when its value is not altered by inter- changing any two of the given letters. Thus, if we interchange a and b m ab, a + b, a^ + ab + b^ (.-a)(.-6), we get ba, 6+^ b+ba + a', (X - bXx - a). But these are the same expressions as before; they are therefore said to be symmetrical with regard to a and b. If m the last expression we interchange a and x we get (a - x)(a - 6), which is not the same as before; it is therefore not symmetrical with regard to a and x. Again, a + b + c, a'+b^+c*-ab-bc-ca, (x~a)(x~b)(x-c\ ar« unaltered by interchanging any two of thi l.tZ a.^/ C iZ7aX r "" ^'""^'""^ "''" ^^^^^^ ^ *^^ ^'-e wifh"'tI.H ;T""" "'^ '' ''^^"'^ "^^^ *- symmetrical with regard to four or more letters, but we shall confine our at- tention chiefly to those fomed from three. 251. The simplest form of a symmetrical expression is the sum or the produc of the letter, involved, thus a^b^c or abc If one term mvolvmg one or more of the letters, be given, the re- maimng terms necessary to form a symmetrical group can be 7t once written down by using all the letters in theLme'w:;; tlu a' be the given term, a^^b^.^ is the group required.^ 'si J riv IT.'sf :r"" -"*-*--'• ^-- «'* we write a'(6 J) ^Hc^a)^c\a + b) to complete the symmetry. In the latter J 218 SYMMETRICAL EXPRESSIONa y ^1 of another; therefore to complete the sjmmetiy we must have the square of each into the first power of each of ths others. If two or more terms be given we can complete the symmetry by writing a group as before from each term; thus from a^-bc we write a^+b^^c^^ah^u^ca. But from ab^bc we should only write om group, viz., ab+bc+ca, because the two given terms belong to the same symmetrical group. A symmetrical group of terms is evidently homogeneous. 252. If two or more symmetrical expressions be combined in Miy way by addition, subtraction, multiplication or division or If a root be extracted, the result in any case will be symmetrical ; for If two letters be interchanged in the given expressions the same letters will be interchanged in the result, and since the given expressions are unchanged the result will be unchanged t.«., It will be symmetrical. ' 253. The product of two homogeneous expressions will evi- dently be homogeneous and of a number of dimensions equal to the sum of the dimensions of the factors, since each term of the product js formed by multiplying two terms, one being taken from each factor. Similarly, the quotient of two homogeneous expressions will be homogeneous, and of a number of dimensions equal to the difference between the dimensions of the dividend and the divisor. This Art. and the preceding are of great assistance in testing the accuracy of algebraical work and in remembering the factors of algebraical expressions. 254. Many expressions which have not the perfect symmetry described m Art. 250 are still unaffected by particular inter changes of the letters involved. For example, the expression {az-b^) -{ay^bxy is unaltered by interchanging a and x, pro- viding we at the same time interchange b and y; or we may in- terchange a and b providing we also interchange x and y. Again the expressions a'6+6»c + c»a and {a-b){b-c){c-a) remain the SYMMETRICAL BXPKasiONa 219 same if we change a into i. J into e. and c into a,- but if we inter- d.ange any <^„ letter, tl>e latter expression has the san,e numeri- cal value as before, but of the opposite sign, while the former expression .s entirely different. Expressions like the latterTre sometunes caJlod Alternating Expressions. Expressions Jht" are unaffected by a series of changes similar to that just Riven are of the greatest importance, and the greater portion of Z chapter is devoted to their consideration. 255. We shall now explain the formation and investigate the ZdT-1. •'"'i™'" °'"' "' =^"™^'™^ expressions t lerred to m the preceding Art. Place any three letters, a. b, \ on the circumference of a circle • then in passing around the circle they will be found in the order' o one of the three groups, ahc, boa, cab. No other order is posai- ble so long as we pass around the circle in the same direction, and each group xs derived from the preceding by changing a into 6, ft into c, and c into a. Similarly, if any expression whatever contammg one or more of these letters be written down, two others may be derived from it by the same series of changed. Thus from x + a we get x + b and x + c. a- 6 we get 6-c and c-a. ^^ib-c) we get b\c~a) and c'(a-b). (b-cXx-a) we get (c-a)(ar-6) and (a-bXx-c). 256. Similarly, from x + b we get x-i-c B^d x + a; from c^fa-b^ wegeta^(. o)and 6^(c-a), etc. Thus from any o'ne oTtheLe expressions the other two may be obtained by the same series oi changes, the expressions following each other in the same way as a, b, c follow each other on the circle. If now we form a compound expression by taJcing the sum or the product of the three expressions so obtained, it will remain unchanged when a IS changed into 6. b into o, and c into a, just as a + b'c^blZa orabc^bca Forexample, (a~bXb-c)(c-a)^^b-.c)(c-a)(a~b)- these examples the letters are said to be written ^ circulaToixler! (I i< 1 •li 220 SYMMETRICAL EXPRESSIONS. and the expressions so formed are said to be symmetrical in circu- lar order, to distinguish them from those which are perfectly symmetrical. 257. The orderly manner in which the letters of symmetrical expressions follow each other enables us to transform or simplify such expressions without writing down all the terms which com- pose them. This will be understood from the following examples : Ex. 1 — Arrange in powers of x the expression c{x-a){x-h)->ra{x-h){x-c)^h{x~c){x-a). We observe that the letters a, 6, c in the successive terms follow each other in circular order. Expanding the fix-st term we get c{x-a){3i-h)^o^-c{a-\-h)x-\-ahc. The coefficient of .r* is c; in the next term it will be a, and in the next h-; hence the coefficient of «* in the result is a + 6 + c. The coefficient of z is -{hc+ca)', in the following terms it will be -{ca-\rah) and -(aJ + Jc); hence the coefficient of ar in the result is -2(ai-f-ic+CQ). From ahc we know we shall get hca and cab in the following terms. The whole result is therefore (a -I- 6 -I- c)a:» - 2(a6 + 6c + ca)x + 3a6& Ex. «.— Simplify (a - h){h -c) + (b- c)(c -a) + {e-a){a- b). (a - b\{b -c) = ab + bc-ca-b\ Prom ab + bc-ca we learn that if all the terms were expanded we should have all the products ab, be, ca three times over, twice with the positive sign and once with the negative sign. From -6» we learn that we should have the square of each letter taken with the negative sign. The result is therefore ab + bc + ca-a'- b'*- o". Ex. 5.— Find the value of x^ + i/+i^-xy-yz-zx in terms of a, b, c, when x = a + b, y = ft + c, z = c + a. This expression consists of two parts, each symmetrical in itself; tho value of each must be found separately. SYMMETRICAL EXPRE8SI0N& 221 Now, *>-(a + i)«-a«+A«+2a6, from which we see that we shall have al the squares twice over, and also twice all the pro- ducts, VIZ., 2(a« + 6» + c» + a6 + 6c + ca); ^ .«'^'*i,T°^''';,*^l*^'^"*'^"*^*'''^*'«' ^^«™ ^hi«»» ^e see that we shall have all the squares once and the products three times m., a» + 6»+c»+3(a6 + 6c + ca), ' Subtracting this result from the former we get the final result. <* +-^) + 2(6c + ca + a%. is the result required. The student should write the terms in full a few times and care ully observe the order in which the letters follow ea^h other antil he can write the required result from examining a sinde term, as in the preceding example. 258. The three following combinations of letters occur so fre- quently that they should be carefully remembered :- 1. (a-b) + {b-c) + (c-a)^0. 2. c(a-b) + a{b-c) + b{c-a) = 0. ^'i^-<^){^-h){x-c)^^^ia^b^c)x^^iab^bc + ca)x-abc. W- BXBROISB T.VR- TX. Simplify 1. (a + 6)» + (6 + c)'+(c + a)'. 2. (a-6)»+(i-c)» + (c-a)>. 3. (a + 6)(6 + c) + (6 + c)(c + a) + (e + a){a + h), 4. (a-6)(6-c) + (6-c)(c-a) + (c-a)(a-6). 222 SYMMETRICAL EXPRESSIONS. 6. (a - b)(b + c) + (6 - e){c + a) + {e- a)(a + b). 6. (a + b){b -c) + {b + c){c -a) + (e + a){a - b). 7. (a + 6-c)'+(6 + c-a)»+(c + a-6)». 8. (o-6-c)'+(6-c-a)«+(c-a-6)«. 9. (a-6)(a + 6-c) + (6-c)(6 + 5-a) + (c-a)(c+a-6). 10. (a + 6)(a-6 + c) + (6 + c)(6-c + a) + (c + a)(c-a + 6). 1 1. {a-b){ma + mb-nc) + {b-c){mb + mc~na) + {p-a){mc + ma-nt\ 12 1 , 1 , 1 *(a-6){6-c) {b-c){c-ay{o~a}{a-b)' 13. « 6 {a^b){a-c) {b-c){b-a)'^(c-a){c-by a + b b + e c + a 14 • I - ■ - . ^ ^^ ' {b - c){c -a) (c - o)(a - 6) (a - 6)(6 - c)* j5^ (2a;-y-g)« + (2y-g-a;)» + (2^- a.-y)» 16. 17. (y - «)' + (« - xy +{x- yf x + y-z y + z-x z + x-y {y - z){z - x) {^•-x){x-y)'^{x-y){y-zy x+p y+z z+x («" - y«)(2/' - ««) (y* - «a:)(«» - ary) («» - xyXx* - yz) ' '(« + «)(« + y) (« + 2/)(y + «) (y +«)(« + a;)* 19 {(^'^ + ^)' - (^^ + «)('^^ + <')}+ ^^o similar terms {{ax f 6)' - (ca; + a)(ca; + 6)} + two similar terms ' Arrange in powers of x 20. {x + a){x + b) + {x + b){x +c) + (x + c)(x + a). 21. (x-a){x-b) + (x-b){x-o) + {x-c)(x-a). 22. c{x - a){x - 6) + a{x-b){x - c) + i(a; - c)(a; - a). 23. c(fl5 - a){x + 6) + a(a: - 6)(« + c) + b{x - e){x + a). ) SYMMETRICAL EXPRESSIONS. 223 24. (a-b){x-e) +(b-c)(x-a) +(c-a)(jr-6). 26. (a-b){z-ey + {b-c){x-ay+{e-a)(x-b)\ 26. (a - b){x - e)' + (b- c)(x - a)» + {e- a)(x - 6)» 27. (a - 6)(ar - a){x - b) + (6 - c)(a: - b){x -c) + (c-a){x- e){x - a). 28. (a + b){x + a){x-b)\-{b + e)(x + b){x-c) + {e + a){x + c){x-a). 29. a(6 - c){x -bc)+ b{e - a){x - ca) + c(a - b){x - ab). 30. o(6 - e){x - bey + b{o - o)(« - cay + c(a - 6)(ar - aby. It 2a = a + b + e simplify the following: — 31. 8(8-a) + {8-b){8-c). 32. («-a)» + (»-6)' + (,_c)« + ««. 33. {8-a)(8-b) + {8-b){a-e) + {8-e){a-a)+8'. 34. a(«-o) + 5(«-6) + c(»-c) + 2«'. 35. a{8-b){8-c) + b{8~c){8-a) + c{8-a){8-b) + 2{8-a)(8-b){8-e). 36. (*-a)»+ (*-&)> + (,-c)»+ 2(«-a)(«-6) + 2{8-b)(8-c) + 2(«-c)(«-a). 37. (»-a)» + («-6)» + 3(*-a)(«-4)c. 38. {8-ay + (8-by + {8-cy+Sabc. 39. *{(» - 26)(« - 2c) + (» - 2c)(« - 2a) + (* - 2a.)(s - 2ft) + 2s»} -(s-2a)(«-26)(«-2c). If 2« = o + 6 + cand 2aS" = a« + 6' + c», show that 40. a«(5« - a«) + b%S' - b^ + c%S' - c») = 8.(5 - a)(a - 6)(, - c). ^ 41. (-S'«-a>)(^»-6») + (6'«-62)(^»-c«) + (^2_c2)(^,_^2) = 4«(B-a)(»-6)(«-c). *^ 259. Factor o(6« - c«) + 6(c» - a«) + c(a» - i"). Removing the brackets, adding and subtracting b', and re- arranging the terms, we get a(6»-c») + 6(c»-a») + c(a«-6>) = (a-J)(6«-c«)-(a«-ft')(^-c) «B ^a — 0){^u — c)[if-rc— (a + 0) J -(o-J)(6-c)(c-a). i I I - I 224 87MMETHI0AL EXPRESSIONS. By changing the gigna and re-arranging the terms the given expreswon may be reduced to either of the following forma, a«(6-c)+6»(c-a)+c»(a-6), ab{a-b)^.bc{b-c)^ea{o-a), whose factors are, therefore, known to be -{a.h){b-e)(e-a) The three forms of this example should be carefully noted and used in working the following exercise. By their use a large number of the examples may be solved mentaUy. SimUar re- marks apply to Exercises LXXXI.-LXXXV. ^x.--Simplify -i^^±^ + _J^±S!__4._Ji+f)' M«-A)(a-c) {b~e){b^a)^{c-a){c-Ty We observe that the terms of one factor in ea^h denominator are the reverse of the regular circular order; we therefore take -{a-b){b-o){c-a) for the L. C. M. of the denominators. Re- ducing the fractions to the L. C. D. we get (6-c)(ar + a)» for the first numerator, and we know that the other numerators may be obtained from this by the usual interchange of letters Then (6-c)(ar« + 2a.r + a»)-(6-cK + 2a(6-c)ar-Ka»(6-c), from which we see that when the sum of all the numerators is taken the terms involving x^ and x will vanish. The sum of the given fractions. ^»»«refore, is ^''(^ - ^) + ^'(^ - a) + o»(a - &) ^ . •-(a-6)(6-c)(o-o) ^' *'"® ^®^"^* required. Simplify BXEROISU T.vw J _a(p + c) ^ b{c + a) c{a + b) ' {o~a)(a-b) (a-b){b-c)'^(b-c){c~ay 4. a' ' {a-b){a -c)'^ (b-c){b -a)'^ {o - a){c - b)' 3 ^^ , ^^ 1 '^ ' (a-c)(6-c) (b-a){c-ay{o-b){a-by a^ -bo b*-ca c* - ab 7T 1\7Z TT "'"T— TTTf T + t: :—. . ^c«y^a=-y^ \^a-0)\p-C) (6 - c)(c - o)' 6. 6. SYMMETRICAL EXPRESSIONS. X - a X— b x-'O 225 a(a-b){a-e) 6(6-c)(6-a) c(o-a)(c-6)* {x-ay {x-by (x - c)« (a - b){a -c) (6 - c){b -ay{e- a){c - b)' y (x-a){x-b) {x-b){x-e) {x-c)(x-a) ' (o-c)(6-c) {b-a){c-a) (o-b){a-by 8 <*\'^-^)i^-<') f>*{x-e){x-a) d*{x-a){x-b) (a.b){a-c) "^ {b-c){b-a) "^ {c-a){c~b) ' • (a-b){a-c)'^{b-c){b-a)'^{o-a){o-:y 10 (^+«^)(l+^c) (1+^(1+ eg) (Ucg)(Ug6) (a-6)(6-c) (6-c)(c-g) ■*■ {c-g)(g-6) • 11 <»-^ . ^-co-<» , (g - b)(b - c)(c - g ) ' a + b b + c + a {a + b){b + c){e + g)* J 2 (aa; + 6)(6a; + c){cx + g) - (gg -f c)(fta; + a ){cx + ft) 13. 14. {ax + 6)» + {bx + c)" + (ca: + g)» - (gx + c)» - (Aa? + g)» - {ex + 6)» ' a b e (o--6)(g-c)(a:-g)"*"(6-c)(6l^r6) + (c_o)(c-6)(aj-c)' (g-6)(g-c)(ar-g)"^(6-c)(6-g)(x-6)"*'(c-g)(c-6)(ar-.o)' 260. Ex.l. g(6«-c») + 6(c»-g») + c(g»-6») -(g-6)(i»-c»)-(a»-6»)(ft-c) -(g-6)(6-c){6' + ic + c'-(g»+g6 + J«)} « (g - b){b - c)(c - g)(g + b + c). Similarly, by rearranging the terms as in Art. 259, the factors of the first two examples in the following exercise may be ob tained. Ex. ^.—Simplify (a + b)(a + c) (b+c)(b + aS (e+aVi^4.h\ bcia-bXa-c) ca{b-c){b-a) g6(-c)(a + b){a + c)^a{b-c){a* + ah-hbc + ca) ''a%b~c) + (ab + bc + ca){a(b-c)}. Now, the numerators of the other .ractions may be obtained from this by the usual change of letters; and by separating the expression into two parts as above we see that the second part will vanish, because ab + be + ca remains constant and a{b - c) + b(c - a) + c(a - 6) » 0. The numerator, therefore, reduces to a\b -c) + 6»(c - a) + c»{a - b), and the whole of the given expression reduces to — ; — . abo BXBROISB liXXXI. 1 . Factor a'(6 - c) + 6'(c - a) + c^(a - b). 2. Factor ab{a^ - 6>) + bc{b* - c*) + ca{e* - a'). Simplify 2 a(b^+bc+c^ ^ b{c^ + ca+a^) c{a* + ab + b^ ' (c -a){a -b) (a - b){b ~c)'^{b- c){c - a)' 4 a^(a+.^) ^ bc{b + c) ca{c + a) ' (a-c)(b-c) (6-a)(c-a)"^(c-d)(a-i)' b* ■ bc{a-b){a~c) ca{b - c)(b -a)'^ab{c - a)(o - 6)' • {o-a){a-b) {a-b}{b-a)'*'{b~c){o-ay y a{a + b){a + c) ^ b(b + c)(b + a) c{c + a){c + b) {a-b){a-c) {b-c){b-a) "^ (o-a){c-b) ' g (a + b){x-a)(x-b) ^ (b + c){x-b)(z-c) {c + a){x~c )(x ^a) {a-c){b-c) {b-a){c-a) "^ {o-b){a^^^' 9. (g-a)' ^ ^^~i)A_4._j^-g^' 1« - ^X" - c> (6 - c)(6 - o) (c - a)(c - 6) • SYMMETRICAL EXPRESSIONS. 227 10 (l + «'^)0+«'g) , (l+^'c)(l-<-ya) (l+c*a){l+c'b) ' (a-6)(a-c) {b-c){b-a) "^ {c-a){c-b) ' 11. 12. b + o-a e + a-b a-hb-c (6 + c)(c - o)(a - 6) "'' (o + o)(a - 6)(6 - c) "*■ (a + 6)(6 - c)(c - a) • {a -by {b-cy {b + c-2a)(c + a- 26)'*'(c + a- 26)(a + b-2c) {c-ay (a + 6-2c)(6 + c-2a)* 13 (<^ + ^)* + (fta; + c)* + (ex + g)* - (ax + c)* - {bx + ay - {ex + by {a'x + 6'')(62^ + c''){ - 6») + Ac(ft»- c») + ca(c» - a»). Simplify 3. ~ g* i« ^ (a - b){a - c) ■*■ (6 - c)(6 - a) "^ (c - a)(c - b)' 4. .^(*^<'') _^+ a') <;(a«^. 6») (c»-a»)(a»-6») (a'-6«)(6«-c»)"^(i«-c«)(c»-o»)" •c(c-a)(c-6) a(a-6)(a-c)^6(6-c)(6-a)* g^ a'(a + b)(a + c) ^ b%b + c){b + g) c'fc+aVc + ft^ (a-*)(a-c) (b^c){b-a) '^ (e-a)(o-b)'' 7 o| A» ^ • (a-6)(a-c)(A+c)"^(6-c)(6-a)(o + a)-'(c-a)(c-6)(a + 6)- ' (a-6)(a-c) (i-c)(6-a)'*"(c-a)(c-6)' 9 «(^-«)' ^ b(x-by c{x^cf ' {a-b)(a-c) (6-c)(6-o)'^(c-a)(c-~^' 10. («-*)* _^ ^f*-^) (6 + o-2a)(c + a-26) (o + a-Jb){a + b-27) (c - «)« (a + 6-2c)(6 + c-.2a)" (a'oj + by + (6»ar + c)'+ (o»^ + a)»"q^V?^)nrf6V+"^)r:(^^ Solve 13. (a-6)(«-c«)>+(6-^c)(ar-a«)'+(c-a)(ar-6»)»,0. U. (a-ft)(«-c)(^-c») + (ft-c)(ar-o)(;r-a»)+(c-a)(.r-ftj(ar-6>)-0. SYMMETRICAL EXPRESSIONS. 229 262.— J^x o«(6»_c») + 6»(o»_a») + cV-6') - (a' - 6»)(6> - c») _ (a» « 6»)(6» _ c») -(a-.6)(6-c){(a + 6)(6«+6c + c»)-(a» + aft + 6>)(6 + c)} -(a-6)(6-c)(c-a)(a6+6o + ca). ^ Similarly the first two examples in the following exercise may be factored. BXEROISB LXXXIII. 1 . Factor a»(6' - c>) + 6»(c' - a«) + c»(a' - 6»). 2. Factor a»6»(a - 6) + 6V(6 - c) + c»a«(o - a). Simplify 3 «'(^ + )-.0. I 280 SYMMETRICAL EXPRESSIONS. 11. c{a-b){x + a?){x + b^) + a{b-c){x + h:*){x + <:^ + b{c - a){x + c*){x + o') = 0. 12. (a " b){x + bc){x + ca) + (6 - c){x + ca){x + ab) + (c - a){x + ab){x + he) =» 0. 13. (a^x + 6)(6»a; + c){»(a + 6)(a' + a6 + 6'); and since the expression on the left is what the given expression becomes when for c we write 0, the factors on the right are what the factors of the given expression become when for c we write 0. The given expression is perfectly symmetrical with regard to a, 6 and c, therefore its factors are also symmetrical. = 5(a + b){b + c){c-^ a){d} + 6' + c' + aft + ic + ca). Ex. ^.—Factor {x - yf + (y - »)' + (« - xf. Let x-y'S'a, y-Z'^b^ then « - ar »» - (a + 6) and {x-yY + {y-zf^-{z-xf =.a^ + I^-(a + by » - 5ab{a + 6)(a' + ab + 6') -= 5(a; - y)(y - z){z - x){x^ + y* + i^-xy-yz- zx). To express a^ + ab + 6' in terms of x, y and 2, it is best to write it in the form (a + 6)' - ab^ when the substitutions can be easily effected. 23. 24. SYMMETRICAL EXPBESSIOHa 231 Factor BXBROISB T.XXXIV. 1. (a + 6)»-a»-6» 2. (a + 6)» - a» - 6». 3. (a + by-a'~b\ 4. (a-6)»-a» + i». 5. (a-by-a'+bl 6. a' - (a - 6)» - 6». 7. (ar-y)» + (y-«)' + (»-ar)». 8. (x-y)' + (y-zy + (z-xy. 9. a(6-c)» + 6(c-a)» + c(a-i)'. 10. a'»(6-c)»+6»(c-a)»+c»(a-5)». 11. (a; + 2/)* + a^ + 2/\ 12. (a: - y)* + (y -«)* + («_ ar)*. 13. {a + b + cy-c^-b^-€^. 14. (a + 6-c)5-a»-6» + c». 15. {x + i/ + zy-(x + i/-zy-{x-y + zy-{-x + y + zy. 16. 8(ar + y + «)»-(a: + y)»-(y + «)»_(a + a?)». 17. (a-6)» + (6-c)» + (c-rf)=+((£-a)». 18. {a-by + {b-cy+{c-d)'+{d-ay. Prove 19. {(a:-y)»+(y-«)'+(«-a:)'}« = 2{(«-y)*-r(y-«)* + («_.-,)4}. 20. 2H{x-yy+(y-zy+{z-xy}{(x-yy+(y-zy+{z-xy} -'2i{{x-yy+{y-zy+{z^xy}*. Ifa + i + c = prove a^+ b^ + c^ a' + 6» + c» o' + i' + c« 21. 22. 5 3 2 ' o» + ft» + c^ o» + A» + c» a< + 6« + c" a« + 6» + c» a* + A* + c* 23. If a + 6 + c + } =a»+6»+c»+24aJa 13. ^{{a + b + 7c)(a-.b)'+(b + e + 7a){b-ey + (c + a + 7b){c-a)*} ''(a + b + c)*-27abe. U. (4a+46+c)(a-6)'+(46 + 4c+a)(6-c)«+(4c+4a+6)(c-a)« -9(a« + 6»+c»)_(a+6+c)». 16. a'(6 + c)« + i»(c + a)>+c«(a + 6)>4.2a5c(a + 6 + c)-2(a6 + 6c+ca)>. 16. (6+c)"(c+a)«(a + 6)« + 2a«W-o*(6 Lc)«-6*(o + a)»_c*(a + 6)> = 2(6c+co+a6)» 17. (6+c)"(c+«)'(a + 4)'+2a2iV-a«6«(a+6)«-iV(6+c)«-c«a»(o+a)« = 2a6c(a + i+c)* 18. 2(a + 6 + c)«+(a + i+c)(a6 + Ac + ca) + aJfl = (2a + 6 + c)(a + 26+c)(o + ft + 2c). 19. 4(o + 6 + c)'+2(o + 6 + c)(aA + 6c+ca)-a6o - (a + 26 + 2c)(2a + b + 2c)(2a + 26 + o). 20. (3a-26-c)'(36-2c-a)»+(36-2c-a)'(3c-2a_6)» + (3c - 2a - 6)»(3a - 26 - c)» = 49(a6 + 6c + ca _ a» - 6«- c«)«. 21. (a+6-c)»(6 + c-a)» + (6 + c-o)»(c + a-6)«+(c + a-6)»(a + 6-c)« + 24a6c(a + 6-c)(6 + c-a)(c + a-6) - (2a6 + 26c + 2co - o' - 6' - c^». I ! 1 '(. I !'■ t i I tii if! II itl-Ji. 19 CHAPTEE XVI. THEORY OP DIVISORS AND OOMPLBTB SQUARES. 266. Before proceeding to study the subject to which this chap- ter is devoted it will be necessary for the student to clearly com- prehend the exact meaning of the word " condition " in Algebra. A simple example will make its meaning clear. 267. Is ar - 2 a factor of x*-ax + bf A little consideration will show that the answer depends upon the values of a and b. If a = 5 and 6 = 6 then x-2 is a factor of a^ - ax + b. Here, then, are two conditions which, being fulfilled, render .r - 2 a factor of the other expression. But if a = 1 and 6 = - 2, or if o t= 3 and b = 2. it is also a factor. Hence we conclude that the conditions a — 5 and 6=<6 were sufficient but not necessary. If we give a and b any values such that 6 - 2a -f 4 = 0, we shall find that a? - 2 is a factor of x'-ax + b; but if such values be given that i - 2a + 4 is not = 0, then it is not a factor. Therefore 6 - 2a + 4 = is the necessary and sufi&cient condition that a: - 2 may be a factor of a:* - aa: -I- 6. 268. To find the conditions that ii?'\-px-\-q may be a factor of «• -f aa? -{-bx-k-c for all values of x. Proceed as in ordinary division as follows: — ^•\rpx + q)a?-\-aoi?-\-bx-^c(x + {a -p) a? ■¥ px^ -f- qx {a-p)3^+ {b-q)x + e {a-p)3^+p{a-p)x + q(a-p) Now, if b-q''p{a-p) and C'=q{a-p), the remainder is zero for aU values of x^ and the first ^xpi'ei^ion in a factor of the THEORY OP DIVISORS AND COMPLETE SQUARES. 236 second. These are the necessary and sufficient conditions re- quired. If, however, we divide the former equation by the latter , . h — q p we obtain -^=-- or q(b-q)^pc, a necessary condition that the first expression may be a factor of the second, but which in itself is not sufficient. 269. One expression may be a factor of another for particular values of their leading letter though not for all values. Thus, if ar=. 10 both ar - 2 and a; - 3 are factors of a;* - 6ar+ 16; for ar'-6a;+16 = (.r-2)(x-3)-(ar-10) = (X - 2)(a; - 3) when x = 10. Such values may be found by dividing one expression by the other and equating any remainder which contains x to zero; for if the remainder is zero the division is complete. -^ar.— Find values of x which will render ar* - 2a? + 3 a factor of a:'-a:' + 5a?-21. Dividing in the usual way we obtain x+\ for quotient with ix - 24 remainder which vanishes when a? = 6. If we place - 1 in the quotient instead of + 1, as before, the remainder is 2a:» - 18, which vanishes when .r=. ±3. Again, we may take - 7 for the last term, giving 8a:«-12a; remainder, which is zero when ar = or - . Each of these values of x renders the former expression an exact factor of the latter. This, however, is using the word "factor" in a restricted sense. Properly speaking, one 'expres- sion is a factor of another only when the remainder ia always zero. 270. To find the condition that aa^ + bx^c may be a complete square for all values of x. If the given expression is a complete square it must be the square oi xV^-^Vc, since no other expression when squared could give the terms aa^ and c; therefore oa^ + 6a; + c = ^a:Va" + l/7^» = aa!»4-9i/7;;; ^j... and therefore 6- 2V^ac or b^-miac is the condition required. '■ r-^1 F--. M-- 236 THEORY OP DIVISORS AND COMPLETE SQUARES. It ahould be observed that the preceding condition merely enables us to write the given expression in the form of the square of a binomial; but it does not ensure that the numerical value of the expression will be an exact square when for the various letters we substitute numbers which satisfy the given condition. For example, let a = 2, i = 4, c = 2, and a; =10, then aa^ + bx + c = 242, which is not an exact square. Again, let a= - 1, 6 = 2, c= -1 and a; =10, then ax'+bx + e — - 81, a negative number, whilst all square numbers are posi- tive. Since 6' = iac we have (ui^ + bx + e = a(x*+-x-t . , B"^-'^)* which shows that if a is an exact square and b'^=i^ac then the whole expression is also an exact square for all values of the letters which satisfy these conditions. EXBROISB I.XXXV1. 1. What values of a and b will render a;*- 2a; + 3 a factor of 2a:* - 4^* + 9a:' + aa: + 6 f 2. What values of a and b will render a:" — 3a; + a a factor of a;* + a;"-5a;2 + 7a; + 66/ 3. Find the values of a and b in order that a?.-^ Zxy + 4^ may exactly divide sfi + Tar^y + ^x^y"^ + ^3?y^ + aa?y^ + bxtf + 1 23/*. 4. What value of x will render a:* + 6a;'+lla:' + 3a; + 31 a per- fect square! 6. What values of x will render cfi + Zofi + ^s^ + la^ + Tx^ - 7x + 25 a perfect cube 1 6. What value of m will make 16a;*-47na;'+20a;' + 2wa; + 4 an exact square for all values of a; / 7. What value of x will render m^x'+px+pq + q^ an exact ■quare for all values of the other letters involved 1 THEORY OF CiviSOiiS AtJD COMPLETE SQUARES. 23? 8. What values of z will render x* + 2aa:» + 36V - 4a»ar + 46* an exact square for all values of a and b'/ 9. What value of x will make x* +px + y a factor of x' + ax^ + bx + cf «nn^" f^^f^^^^^^^o^ that ax^^2bx + c may be an exact square for aU values of x. 11. Knd the condition that a:^ + Sbx* + 3cx + d may be a com- plete cube for all values of x. 12. If ax»+2bx + c is a factor of a:^ + 3bx^ + 3cx + d then the former is a complete square and the latter a complete cube. aJlZ ^^ f'-"^''^ ^^^^^ -^ ^ + 2a,r' + 3bx + 4c then ax- + 2bx + 2c IS an exact square and u^ + 3ax' + 66a: + 8c is an exact cube. 14. lix^ + max + a^ is a factor of x* - a^ + a^x' - a'x + a^ then 15. If (ar-l)2 is a factor of ^ + ax' + bx + c then 6 + 2c = l Is the converse true 1 16. Ux^ + qx + r contains a square factor then 4«7' + 27'-'-0 Is the converse true 1 ^ • - v. 17. Find the conditions that :>^ + a:fi + bx^ + cx + d shall be a perfect square for all values of x. 19. Finda value of :r which will render («'+»+ l):r>+(„'+„.w - (" - n + 1) a complete square for all values of n. 20^ Find values for a and * which render the faction ^x + (g - l))x + 2 a? - 36' 3^+ {a-l)x + 3(a2 + 2a6 + 3b'') *^® ^™® ^^^ ^^^ values of it ^ 21. Find values for ar a^d y which render th« f...f,v^ ^s- + (OJ - a)z + 26(a: - 2c) . (y-6)s+3a(y_3^ independent of the value of «. I^k; ♦»|lf!|K»S»«,a 238 TUEOKY OP DIVISORS AND COMPLETE SQUAREa 271. A Function of « is any algebraical expression whose value depends, in whole or in part, upon the value of x. 2 2« - Thus 2a; -3. — and \/ar^+6 are functions of x; but — , nT 3aj ^^ and (a - h){x -c) + {b- c)(x -a) + {c- a)(x - b) are not functions of X, because when reduced to their simplest forms their values are independent of the value of x. 2TL A Rational Integral Function of a- is a function in which X does not appear in the denominator of a fraction, and is not aflfected by any root sign. Thus a:'-4a:'+-a: + V7 is a rational integral function of af, o but a:* - 2x + and x + Vbx + o are not; the former is not in- 2 -a; tegral and the latter is not rational with regard to x. 273. If any rational integral function of x vanishes when a; = m then the function is divisible by x-m. Let a-¥bx + e3^ + d3^ + &o.y be any rational integral function of x which vanishes when x='m, that is, let a + 6r» + cm'' + dm' + &c. = 0, then «-m is a factor of the proposed expression. For a + bx + c3^ + dx^ + &o. = a + bx + ex* -h da^ + &0. - (a + bm + cm^ + dm* + &c.) «= b(x - w) + c{a^ - m') + d{3^ - m') + &c. = (a? - »n){6 + c{x + m) + d{x* + mx + m") + &c.}, which shows that a? - w* is a factor. Cor, If a-[-bx + cx'* + da^ + &c. be divided by ar-m the re- mainder will be a + bm + cm* + dm' + &c.; for this expression does not contain x, and when it is subtracted from the given expres- sion the remainder is divisible by a; - m. 274. The Hymho\f{x) is frequently used to denote any function of X. The value of such a function when x = m ia then conven- iently expressed by/(«i)j thus, iif{x) denotes ax* + bx + e, f{m) ' THEORY OF DIVISORS AND COMPLETE SQUARES. 239 denotes arr? ■¥bm-\-e. This notation enables us to give the result of Art. 273 very neatly as follows : — 275. Iff{x\ omy rational integral function of x^ he divided hy x-m, the remainder will be f{m); and iff{m) be zero then x-m is a factor of fix). Divide /(a;) by ar - m until the remainder no longer contains x. Let Q be the quotient suad R the remainder; then/(a?) = ^(aj-wi) + R is an identity, and therefore true for all values of x. Let ar=»m, then Q{x - m) = and we get /(m) - R; but R is independent of ar, therefore R^f{m) for all valuj^i ;; x; and if /(m) = 0, 72-0, which proves the proposition. Ex. 1. — Factor a^-bx^- 46aj - 40. The expression vanishes when for x we substitute 10, - 1 or - 4; therefore a; - 10, a; + 1 and a; + 4 are factors. There can be no other factor containing x, since the given expression is of only three dimensions. There can be no numerical factor since the first term of the product of these factors, viz., a:", is the first term of the given expression. Two special cases are worthy of note in connection with ex- amples like the preceding: — (1) When the sum of the positive coefficients is equal to the sum of the negative coefficients, a: - 1 is a factor. (2) When the coefficients are all positive, and the sum of the coefficients of the odd powers is equal to the sum of the coeffi- cients of the even powers, a; + 1 is a factor. Ex. ^.—Factor (a + 6 + c)* - (6 + c)* - (c + a)* - (a + b)* +a*+b*+ c\ For a substitute and the expression vanishes; therefore a-O that is, a, is a factor. The expression is symmetrical with regard to a, 6 and c, there- fore b and c are also factors. — ,^ — -,...-« ..J vTi X «iiiii,-ii!3i.vjiia, titici ciU4"»j wuere is anoi«iier factor of one dimension. ril 240 THEORY OP DIVISORS AND COMPLETE SQUARES. ill ill I The factors a, 6, c by themselves form a symmetrical group, therefore the other factor must also be symmetrical in itself. The only symmetrical expression of one dimension is a + b + e, .-. (a + 6+c)*-(6 + c)*-(c + a)*-(a + 6)*+a« + 6* + c«-ira6c(a + 6 + c) where iV is a number independent of a, b, c. To find i\r, give a, b, c any values which will not make the ex- pression vanish; in this case a = 6 = c = 1, then 3*-2*-2*-2*+l + l + l = 3iVor iV=12; therefore the given expression =. I2abc{a + b + c). Ex. A— Factor (a - bf + (6 - cf + (c - a)». For a substitute b and the expression vanishes, therefore a-b is a factor; then (a - b){b - c){e - a) is a factor by symmetry. The expression is symmetrical and of 5 dimensions, therefore there is another symmetrical factor of 2 dimensions. Let this factor be m{a^ + b^ + c') ^ n{ab + be ^ ca), in which m and n are not functions of a, b, and c, and will therefore be the same for all vtlues of these letter- Then (a-6)" + (ft-c)»+(c-a)» = (a - b){b - c)(c - a) {m(a» + 6« + «») + n{ab + be + ea)}. We have now two unknown quantities, «i and n. It will therefore be necessary to have two independent equations to de- termine them. These may be obtained by giving two sets of values to a, 6, c. First, let a = 0, 6-1, +c«-a6-6c-ca). 276. If X- mis a factor of any rational imtegral function of x^ when m is substituted for x the result will vanish. Let a + 6a? + c«» + rfa:» + or {a:c-a^''^{bo'-b'c){ab'-a:b\ ac-ac' be' - o'c ab' - a'b a'c - ad the condition required, A ^'^ ~ °^' bc'-b'c ana x - — - or x--^—-^ is the common factor required. ab'-a'b Again, if in (1) wid (2) we substitute the values of m from (3) and (4), we get a{a'o- - 9a; + 7 when «"— 2a: + 3. Dividing the given expression by a:« - 2ar - 3 we obtain the quotient 3.c» - 2x* + 5^' - 7^ - 2x + 3, with remainder - 9a; + 16. Now, since the dividend is equal to the product of the divisor and quotient, plus the remainder, we have 3a:»-8x«-&c.=(a:»-2a;-3)(3jH*-2x* + 5a:»-7x>-2x + 3)-9j;+16. But since, in this particular example, ar« = 2.r + 3, the first factor is zero, therefore the product is zero, and the given expression re- duces to -9a;+ 16. Again, a:' - 2a: - 3 - (ar - 3)(.r + 1 ) = 0, therefore ar = 3 or - 1, and - 9aj + 16 =. - 11 or 25, which are the required results. The division is most conveniently performed by Homers' Method, Art. 86. ^ar. A— If o + 6 = c, (a' - 6»)> + (i« - c«)« + (c« - «»)« - a* + 6* + c* For (a« - 67 + (6« - c')" + (,/ - a«)« - (a* +b* + c*) - a* + 6* + c* - 2a»6« - 26V - 2c^a^ - (a' + 6» - c> + 2a6)(a' + 6« - c* _ 2a6) - (a + 6 + c){a + 6 - c){a - 6 + c)(a -b-e) »= 0, since a + 6 - c « 0. ... (a« - by + (6« - cy + (c» - a»)» = a^ + 6* + c*. The same result would evidently follow if any of the other factors were zero. ti X (« ai BXBROISB LXXXVIL 1. Show, without actual division, that x-a is a factor of i>^-{a-b)x^ + {d-ab)x-ad. 2. Show that a - 6, 6 - c and c ~ a are each factors of o"(6 - o) + 6»(o - a) + c*'{a - b) for all positive integral values of n. t' THEORY OF DIVISORS AND COMPLETE SQUARES. 243 3. Find the remainder when a:» + a" + 6" - Saix is divided by x-a + b, x + a-b, and hy x-a-b. 4. Show that a, a-x and a-2x are each factors of (a - b){a -b- x){a + 26 - 2x) + b(b - x){^ia - 26 - 2x). 5. Show that {ab - xyf -(a + b-x-y) {ab(x + y)- xy(a + b)} ='{x-a){x-b){y-a){y-b). G. Find the value of a for which the fraction x^-ax^ + \ ^x - a - 4 «»-(«+ l).c2 + 23j,-a-7 admits of reduction, and reduce it to its lowest terms. Find the value of 7. ^+290.i:*+279a;«-2892a:'-586ar-312 when a;=. -289. 8. 3j;«-ll^+19x*-13aH»-a:«4-10 whena;»=2^-3. 9. 4^- 12j;^ + 5ar' + 5a;2-6a; + 3 when a; = ± - V 3". 10. 2a:» + 803^* - 398.r3 + 1605a;2 - 1204a: + 422 when a;'^ + 401a; = 402. Ifa + 6 + c = prove Exs. 1 1-19. 11. c? -bc = b^ - ca^c^ -ab and a' + 6» + c" = 3aSc. 12. (a + 6)(6 + c){c + a) + aic - 0. 1 3. a{b'^ + 6c + c") + 6(c2 + ca + a") + c{a? + a6 + 6^) = Q. 1 4. (a" + by + (6' + c^)' + (c» + ay = 3(a* + 6* + c% 15. (a + 6)(6 + c) + (6 + c)(c + a)f(c + a)(a + 6) = a6 + 6cH-ca. 16. (a - 6)(6 - c) + (6 - c)(c - a) + (c - a)(a - 6) = 3(a6 + 6c + ca). 17. (a6 + 6c + caf = a26'^ + 6V + c V. 18. (a'^ - Wf + (6^ - 2cy + (c' - 2a'')2 = 3(a* + 6* + c*). 19. AS I'. «. ^ 20^^ + 60 26'^ + ca 2c« + a6 7 = 1. 244 THEORY OF DIVISORS AND COMPLETlE SQUARES. Ux + y + Z'mxi/z prove Exg. 20-22. 20. ixyz l-or* l-y» l--«»"(l-ar')(l-y»)(l_«.j- 21. JL±±+y_±±^ z + x 22. , — -r. + .^_Ji+y)(y+«)(«+ar) i-^y i-yz i-zx (1 - xyj(TZ^^)(rr^' {^-y){y -z){z-r) ^+^-Z±+ " - X 1 + ^y 1 + y. 1 + «^'(i + xyxf:rj;^j(iT^' 23. Ifar» + y» + a» + 2ary^=l then ft . ,, . *- 1 + xyz. 24. If .:» + y.: + 1 and :^ +;,^ + ^^ + 1 have a common factor of the form x + a, tlien (p - 1)« ~ q(p - I) +l^o, 25. The expressions aa:' + 6a: + c and ax' + mbx + m^e have a common factor if (m + l)»ac = m6». th!fi ^^,^-'*?'-^'' ^^d ma:» + n.r+j. have a common factor of the first degree m .:, then (pa - mcf » (no -p^Xna - mbf 27. li x^ + mx^n and :^-H;,a: + y have a common factor, then tien!'if t ?'' TV '' ^^^^' hj^^mx + n, then the quo- tient IS a factor of mx^ + qx + r. 29 If or + a and X -a are both facto™ of x»+;,:«:« + ^a: + r, then pq ^r, * ' 30^ If .:» - 1 is a factor of ^ +;,,:> + ^^ + «., the other factor is 31. If x-a and a:-6 are each factors otx'+x+l, then a»-6» = 0. 32 If 0. + ,;, and ;r + n are each factors of x^ + ax^ + b, then <*» — — , m + n J^'a " "^ ^ «f^ + * ^^'i ^ ^/'^ + y have a common factor of the second degree m x, then a^bq^{b-q)\ 34. ««^ + 6a:« + c and ca:» + 6a: + ahaveacommonfactor which to a complete square, then o'6* - 460(0* - a*). \- xyz. tor of ■ve a )r of then quo- Jien >r is -0. hen the ich THEORY OP DIVISOBS AND COMPLETE SQUAREa 246 36. If a + 6 + c-O then abc is the H. 0. F. of a» + 6» + e« and o»+6» + c», and 15a6c(c>-a6) is their L. 0. M. T ^n^^'^J ^ *^^ ^' ^' ^' ^^ ^ + «^ + * and x'+mx + a, their L.C.M.isa:>+(a + m-c)a:« + (am-c»)a: + (a-c)(w-c)o. 277. Prom the principle that the square of any real number 18 positive we are able to deduce various important conclusions with regard to the numerical value of expressions when any num- bers whatsoever are substituted for the various letters involved we give three examples. Bx. i.— Show that the sum of the squares of any two real quantities is greater than twice their product. Let a and b represent the quantities. Then (a - by is positive; .*. a*-2ab + 6» is positive ; .-. a»+ 6' > 2ab, which proves the proposition. JEx. ^.-Prove that 4ar»- 24x + 41 is positive for aU real values of X, and find what value of x will give the expression the least value possibla 4a:»-24a; + 41=.4{a:»-6ar+10J}-4{(a;-3)>+li}. Now, (x-Zy is positive 'or all values of x; 1^ is positive and 4 is positive, there^-e tfie whole expression is always positive. The value + (c-a)» = 0. Now, since the sum of any number of positive quantities can- not be zero, each of thsHA fhrna fssrsr-.s ^. rerms uiUst be zero, and Ihereiore 246 THEORY OP DIVISORS AND COMPLETE SQUARES. BXBROISB LXXXVHL 1. Prove that the sum of any positive number (except unity) and Its reciprocal is greater than 2. ^' 2 Prove that the square of the sum of any two numbers is greater than four times their product. 3. Prove a 2a»6» for all real values of a and b. 4. Prove a^ + b' + c^> ab + bc + ca for all real values of a. b and c, except when a=>b^c. «s oi a, o 6. Prove bc(b + c) + ca(c + a) + ab(a + J) > 6abc when a. 6 and o are real, unequal, and positive. 7. Prove that ^i^ V^anri ^''^ --^- J ^ 2 ' ^ "^ a»d ^-:;:^ are in order of magnituda 8. Show that ^5jt?>?Lti' 9. If X is real prove that a:* - 8x + 22 can never be less thaa 6. 10. Show^that the least vi. ue of a--ar+l is obtained by making ar«=. - . 11. Show that the greatest value of 24a;-ar» is 144. 12. Show that the area of a square is greater than that of a rectangle of the same perimeter. 13. If a straight line be divided irto two equal and also into two unequal parts, the squares on the unequal parts are together greater than four times the rectangle contained by half the line and the line between the points of section. 14. Show that the area of the largest rectangle ^hich can be encbsed on three sides by a line 60 feet in length is 460 square by THEORY OP DIVISORS AND COMPLETE SQUAHEa 247 15. Ifa«+26« + c>-2J(a + c)thepa-6-o. i6. If a* + 6* + 6V + c'a»-2a6c(a + 6) then a^h^e, 17. If a« + 6> + c»-3a6c and a + h + c is not zero then ar^h=^c. 19. li{a^ + b' + d»){x^ + y»+z^^(ax + by + czythen ''- ^t ^l a b c' 20. If x' + y'^ix-2t/ + 5^0 then a:»2 and y=l. 21. Prove a»+i»+c» >. » or < 3aAc, according as a + 6 + cis positive, zero or negative. 22. If a, b, e are real numbers, not aU equal, then (a-b)(b^c) + (6-c)(c-a) + (o-a)(a-i) is negative. ^ ^^ ^ 23. If a, 6, c are real, unequal and positive, then (a + b + cXab + bc + ca) > 9abo and (a + i + c)» > 27a J 2Vabcd, ab + cd> 2V^M, and thence aHc + abH + a<»d + bcd^ > 4aic<^. f ■ CHAPTEK XVII. QUADRATIC EQUATIONS OP ONE UNKNOWN. 278. Equations involving unknown quantities of two dimen- sions, and no higher, are called quadratic equations. Thu, ^.6. + 8-0. ..-9.0. ''^ZZl'^ "" '^"■^^ equations. 279. Quadratics of one unknown may be either pure or ad- fected. A pure quadratic contains the square of the unknown quantity, and no other power; whereas an adfected quadratic contains the first power as well as the second of the unknown quantity. For instance, ar'-lG-O is a pure quadratic, but «« + 8j: + 1 6 - is an adfected quadratic. y 280. A quadratic equation is the statement that the product of two factors, each of one dimension, is equal to zero. These factors may be rational, real and irrational, or imaginary. Thus, in the equation ar» + 8;ir + 1 5 - 0, the factors are {x + 3) and (a? + 6)^ar i therefore are rational. The factors of a:« - 2 = are « - V 2 and ar + ^ 2, and therefore are real and irrational. The equation a;' + 4 - has for factors (ar + 2 \/ - 1) and (ar - 2 i^^), and therefore consists of the product of two imaginary quantities! 281. When the factors of a quadratic expression are obtained we can at once write down the roots of the corresponding equar tion. This has already been explained in Arts. 120-122. ( 11 13 16 17. 19. 21. 23. OK •fir* OTADBATIO IQUATIOKS OF ONE TOKNOWN. 249 ife. 7 — Solve a;" - 9 • 0. Factoring, («-3)(»;3).0; ,.*_3or-3. „^* *-*>'" **-(m-»):.-m„.0. BXBROieai LXXXDL Solve by factoring 1. «>+9a?+U-0. 3. a*-.a?-12-0. 6. 30ar»-«-l„o. 7. 12ar« + a?-l-0. 11. a^+aaj-a-^^O^ 13. 3a:«-63ar + 34-0. 16. 780a:>-73af+l-0. 17 «* + 3a:»+6 . 19 « + 2 4-ar ^, ar-l" 2a; 2a« IS"* 23. «*-2aaf + 4aA-2Aa; " 2ar-.3";T4"*^- 17 2. «"~8ar+15-0. 4. 6a:*-6a;+l«0. 6. a!«+a:-20»0. 8. 2a:>-27ar-U. 10. 3a;«-6a:-.2. • 12. 6ar>-12ar + 2«ll. U. 110a?"-21ar+l»o. 16. z + 2 L.i ar + 2 *• 18. ?l:i+l? 3ar+6 2a?+l'^ll"3;il;' 20. 12 8 32 6" T"; — — aa - -a? 4-.a? « + 2' 22 -ifJif) « 3a- 2a; "i" 24. «"-2aa? + 8ar-16aL at 26. ;:!: ^o » + 5 ^; n a> + 4 i| \ 250 QUADRATIC EQUATIONS OF ONE UNKNOWN. 27 ?fli-izi.?±?_t±? 28 '~^ ^•^l g-f7 ar-S ar + 3 a:«-9"3-ar 3+a;' ' 2«+3'2x-3"4x»-9"2^^* 29. ■3 «+3 a:'-9 « + 3' 1* *-* OA 4* 0-46 a + 46 30. — + a «-26 a: + 26* 282. It frequently happens that a quadratic expression cannot readily be factored. In such cases the following method of find- ing the roots is usually adopted : — Ex. 1. — Let «'+jo.r + 5'-0, find x. Transposing, x*+pxmm -q. Complete the square by adding the square of one-half the oo- elfioient of a: to both sides of the equations. Extracting square root of both sides, ^2 * 2 ' ••*- 2^—1 » Hence or 2^2 2 2 • The st udent will observe that the double sign ± is prefixed to '\/jf-iq The reason is that the square root of a quantity may be dither positive or negative. (See Art. 191.) 1^ QUADRATIC EQUATIONS OF ONB UNKNOWN. Bx. 2. — Solve oa:* + 6a? 4- c - 0. 261 J^;:'™"*' ""^ ■» " »« "-'e th, coefficient of ^ unit. .'. «•+-«+ --a a a «■+ -ar— — , Transposing, Completing the square, « V2a/ V2ay a i^^ Extracting square root, 1\' /M' «'-4ai; (-a- 2a • • *•■ — '——4- 2a 2a Therefore roots are - -i+ ^^'-^"^ ^„ . * ^/'^^^i^ 2« 2a *^*^ -2i 2^—. The equation x^^px^q^Q is the/rnn of all quadratic equa. t^ns which have the coefficient of .», unUy and positiverand ctnt r.""' r* '"'^"^ ^' ^ quadratics in which the o:ffi cient of ar' is not Mm, .*. 4aV + 4a6a? + 4ac-.0. Transposing, 4a V + iabx - - iae. Adding 6* to both sides, .*. 4aV+4a6;r + 6«-6«-4a«i Extracting square root, 2a« + 6 - ± \/6>_4ao <>' ^ax^ ^ b± V^6«- 4flc; -6± v'6«~4ao »-i 2a a w 0. QOADIUTIO .gUATrONS or ONS UNKNOWN. 268 Ex, £.— Solve 2ar* + 8* + 6 ■ 0. Multiplying by 8, 16x« + 64*--40. Adding (8)« to both aidei, 16a:«+64jr^64-24. Extracting .quare root^ 4* + 8 - i ^^24; . , -8±i/24 4 reduction and dmpliflltion „f '^^'T ' '^^ ""''"^ '"■• ""e «. ..e ,.n .. rr-ai^r t eis-rr- ^«.— Solve i±5+f±_* « + c o we^'oilt' "^' '^""^"^^ '^ '"^ --PO"<^-g denominator . 2a 26 2c •• ~ •\ -+. _A Dividing by 2, Clearing of fractioni, Multiplying out and collecting coefficienH ^(a + 6 + c)-x(2aA + 2ao + 26c) + 3a6c-0. Now,we knowthe roots of aar»+6ar+o=.0 are ~i.±^5^EEl^ and as a, 6. c stand for any coefficients whatever we can at" ' wnto down the roots of a iriven n„.w«..;.T_ _? T^^ ^'^^^ . ^ and c tbeir particular ^S^^^^^TZ^ ? - 254 QUADRATIC EQUATIONS OF ONE UNKNOWN. example we must write for a, a + b + e; for 6, - 2(ab + bo + ea) ; and for c, Sabc. Therefore roots are 2{ab +bc+ca) Vi{ab + bc + caf - 1 2(abc){a + 6 + r) ^ , 2(a + i + c) "*" 2(a + 6 + c) ' or simplifying, a6 + 6c + ca v'o'i' + 6V + c'a' ~ abc{a + 6 + c) a + 6 + o Solve 1. 2ar»-7a; + 3 = 0. 3. 2a:»-2a;- 5 -0. a-\-b + e EXERCISE XO. 2. 3a:»- 53a: + 34 = 0. 4. 14a; -.«»=- 33. 6. (2ar+l)(x + 2) = 3a:»-4. 8. (a;+l)» + (ar + 2)« = (ar + 3)». 6. («-!)(«- 2) » 6. 7. (a;+l)(2a; + 3) = 4.t:»-22. 9. (x - l)(a: - 2) + (a; - 2)(ar - 4) = 6(2ar - 6). 10. («-7)(a:-4) + (2a:-3)(a:-5)^103. "•(^-^)(-^)^('-a-)(-i)-(-j)(-^> ,_ a? 2 a? 3 12. o + - = «+-. 2 a; 3 « 13. «■ - 5a; a; + 3 ' .a;-3+i. 14. 16. 18. a;+2 4-a; 7 a;-l~ '^x "3* a;-2 a; + 2 2(a; + 3) « + 2'*"J^2" a;- 3 ' 8-a! 2a;-ll_ a;-2 ~2 «^^1 6~' ,^ a; + 3 a;-3 2a;-3 15. j, + -'-> -, x+2 x-2 x-l 4 17. 5 12 a;+l a; + 2 x + Z' 2a; -1 2a;+l 19. ^—t+flZl^s^ 20. 5._!(^).2«H.fci). 21. a?-3 2 2a;+l 2a;- 1 80 7a; -fl £ 33? ~ (-^-) 6| X— — 3 QUADRATIC EQUATIONS OP ONE UNKNOWN. 266 3^+8 6(12 -:r) 23 -f_+^±^=?iLli? • x-l a;+l"'4(a:-2)' 25. .03a!»-2.7a:=.30. 24. 2aa^ + (a -2)^-1-0. o-cV a:/ (a-c)» a-c\ xj 28. +— - b + x b • e. on * * 30. nx + ~ '=:na+-, X a 29. ^-^.c b+x b-x 31. ofta»-(a + 6)car + c>-0. 32. 2a;(a + x) 3a 3a + 2a; "T* »'• (1^-:)'- 1+ ex T' 34 tt(a?-^)(^-c) 6(a;-c)(a;- a) • (a-6)(a-c) "*" (b-c){b-a) "** 35. a:* + («+!)• + 1=2(3:* + a; +1)». « 7.5 36. a^ -2x- 15"«» + 2a;-35"a;»+10a; + 2r 3 37 — — — + ^^— ^ I ^ _ aj + a «+2a a? + 3a x' -- 12 + 2ar 4.r-3 4ar-l m, 1 ._ « a; + 3 l + 2a! x-l ' „- 2a:'- a; -1 2a:»-3a;-8 Sa:" - 9 «jy. -^z — h- 40. 42. a:- 2 ' a;-3 1+a ft + ar 1 + oa: 1 + 6a:* 1 1 - + 2a;- 3 • 41. + "4a'. x+a x~a 1 + o — 6 x-^a + b x — a-b x-a + b 0. 43 ""^ ^Z5 - ^ + ^ "^f'' a: + o - 26 *«-a aj-6~a: + a-26 a; + 6 - 2a* 44. 1 ^a + 6)(a?-o) (6+c)(af-a) (o + o)^a:~6) -a ',i V. 256 45. QUADBATiC EQUATIONS OP ONE UNKNOWN. —1—. 1 + 1 + 1 x + a -^b X a b' 46. (-^')'-i(«.»)(u3. 47 (a> + ft)(a?-t-o) (x+c){x + a) {x + a)(x + b) ' (x-b)(x~o) (x-o){x-ay{x-a){x-b)''^' ' Aa *-« «-a-ft x + a x+a + b *o. — jH — ■■ + : x-0 x-b — e x+b x+b + e .Q ax - be bx ~ oa ex - ab _ , 4». -^^^ + --—+——^0 when a + ft + c-0. 50. (4a« - 9ed)x^ + (4aV + 4a6cP)a! + (oo" + 6rf»)» - 0. 51. Find the value of ^±-*+^ z-a x—b when ar is a root of the equation «* + a:(a + 4) - 3a6. 285. Various artifices are employed in quadratics, as in other equations, to lessen labor. A familiar and useful one is the sub- stitution of one symbol for a number of symbols. Ex. 1. — Solve a; + 2 x-2 5 x-2 x + 2 6' Here the second fraction is the inveree or reoiprooal of the first. x+2 Let then the equation becomes y - ^^x-2* 1 5 y"6* Clearing of fractions, or S^-gy-1-0. Factoring, (y+ 3) (y- ^) "^> 8 or — - 3' iy' QUADRATIC EQUATIONS OF ONE UNKNOWN. 267 ^ + 2 Substituting for y its value — - we have two equations to •C "" i* lolve^ viz.: — x+2 3 ,x+2 x-2 2 x~2 o 3* 2 From these equations we get «« 10 or — ^r. Mx. «.— Solve 1 (x + lXx-S) 1 (a?+3)(a;-5) 2 (x + 6)(x-7) 92 + « 6 (« + 2)(a!-4) 9*(« + 4)(ar-6) 13 ' (ar + 6)(a:-8)"685* It is easily seen that each numerator and denominator con- tains «" - 2a?. Put (a? - 1)' - y, then the equation becomes 1 y-4^1 y-16 2 y-36 92 6 y-9' 9 y-25 13 y-49"586* But N 1 1 2 92 6'*"9 13"586* Subtracting (2) from (1), 15 1 9 6 y-9 9 y-25 2 13 ^ 13y-49"^ 1 or — . + — y-9 y. 1 2 ^ -26'y-49""* f From this equation y-19j 1 /. («-l)«-19j ••. x^l±VT9. (1) (2) 268 QUADRATIC EQUATIONS OP ONE UNKNOWir. I BXBROISB XOL Solve according to given examples, 1. «-! 3 z-l 2r+ . 2. g-6 a;-12 5 z~l2 «-6~6* 3 ^^~^ 3j;-6 5 ' 3a:-5"^2a;-3""2* 6 -JL. ^''•^ 13 * «+l''"~«~"T' /. r ■ =2a. 4 ^* izi 1? 6. 3.r-2 2a;- 5 8 2jr-5~3ar-2"3' x-l x+1 6 g a:+16 x-i 37 o. + — . — , «-4 a;+16 6* 9. x + 2 x+2 = 1. 10. x*+x+l-^ 42 a:»+ar* il. (2a:«-3^)«-2(2a:»-3^) = 15.12. (^-:r)'-8(ar>-a;)+ 12-0 15. :i;« + 2a:»-24-0. 16. a^+19a;»- 216-0. 17. (a^ + «-2)«-.13(a:» + ar-2) + 36-0. 18 1 1 1 • ^•'+ll«-8"^ar«+2a;-8"*"a:«-13ar-8"^' 19. «" - 7ar + 10'^^-13ar + 40'^""^^^+^^ «^0 _^ a;'-6a?-l 1 a;'-6a; -4 2 ar»-6a:-7 3 «»-6ar-4 6 ' «»-6a: - 9~9 * i»-6x-16 -11 45 ar»-.6a:-9* CHAPTER XVIII. QUADRATICS INVOLVING SURDS. 286. The methods of solving surd quadratics are in the main the same as those for solving simple equations containing surds. There are, however, two kinds of equations frequently occurring which deserve some notice. Ex. i.— Solve ^1 - 2« + f 1 + 2a?- 1?' 4. We know that (a + 6)» = a» + 6» + 3a6(a + 6). Applying this formula to the given equation we obtain, by cubing both sides, But ^T^x+^uTx^^i; /. l~2a?+l + 2a? + 3^(l-2a:)(l+2a;)i?'I-4 or Transposing, Dividing both sides by 3, Cubing both sides, from which we obtain 2 + 3i?'4(l-4a^) = 4. 3i?'4(l-4a;=^)-2. 4-16a;«-;i; 27' «-± ■ ii 'I 6 Vi \ 260 QUADRATICS INVOLVING SURDa Iffx. f.—Solve «» - 2« + 6 V^?-2a? + 6 - 1 1. If wa add 5 to both sides of the equation of a quadratic, («•- 2x + 5) + 6 Vx'-2x'T5. ■ — 'JSX + Lety-'/a:»-2ar + 5, then x>~2x + 5^y>; .-. y" + 6y-16 it will take the form 1& or Factoring, that is, Squaring both sides. y" + 6y-16-.0. (y + 8)(y-2)-0; .*. y- -8or 2, Vx'~2x+5^ -8 or 2. a:'-2a; + 5=-64 or 4; t .*. a:'-2ar«69 or -1. We have now two equations to solve, viz.;— a:"-2a?-59-0 ""'^ ««-2«r+l-0. ^'^"^(^>' x-l±2./!5. (a?-l)«-Oorar-l. Prom (2), (1) (2) ^x. A—Solve VSx^ ■i-2x + i^6x' + ix~ 622. ^+4'-622-2(3ar« + 2ar + 4)-630. Lety-'/3.r» + 2« + 4;then y»-3r» + 2ar+4; .-. y=2y>-630 *^' 2y«-y-630-0. ^ij «Irr/l^.*''' ^^^i^^i^ be found which, when in turn QUADRATICS INVOLVINO SURDS. 261 (1) (2) Solve EXERoiSB xon 1. VxT3+^/7VS'~6^/'x. 2. v'2^-'/iTl»Jv^jr3. 9 3 ^7^ + 4 -f 2 \/3^^rT ^7T2^_ ^^32^ a; v'7a? + 4_2v'3i-l '2aj J* \^x-2a+ Vx+2a 2a* b. — :=^- ^_«-p. 7. X + \/2-x^ 4 ar-v'2-a^ 3* 8. Vl + a;- a/1 -« «+ ^9^ 7 10. 1?'25 + ar+ ^25^-. 2. 12. '^sT^+i^'rri-^T: 14. ^a + «4- l?'a^-. a'' 6^ 16. a:*-3a:-6v'«»-3a:-3--2. 9. ^ar + 22- l^'^TS-l. 11. 4?'rri+^iT;-'^3: 13. '^^+T-.^i:rr«^n. 16. -^1^+ i?'8+^»3. 17. «»+ l/ir-7„i9. 18. 2x«-2\/2^^:6i-5(«+.3). 19. 2«»-2a; + 2A/2j:»-7«+6-5«-6. 20. 3a?(3-a;)-ll-4V^«»-3a? + 6. 21. ar>-3a; + 7v/ll«-2a!» + 2--«+21. 2 22. 5ar-7a;»-8\/7a^-6a;+l-8. 23. « + V'«»-ax + 6» - a-»a;> + 6. 24. 9«-4«>+ V4a:»-9«+ll-5. I 262 QUADRATICS INVOLVING SURDS. 25. -==--. 2- / 8x»+12i+l \-i 1 • Kix' + ix-l) "3' 4 4 26. 28. VSx'+x + d 3 ^4a:»-ar + l"2* (2-3g-a!^* 4 (8-6«-2«>)* ^' 29. 30. 31. 32. ^ ^12 x+Vi-a^ x-VT^~'^' 1 1 VS 4a;- 1 V'2a; +1 + 3v'2a; + l-Tv'a;. 33. .:^: jg±4 - -/2a;» + 1 _ 1^ ^^ V27?T4 + i/9?T5 35. 36. 37. l/3a:« + 4 + V2a:» + 1 7* ' V 27^^+4 - VO^T^Ts i/5.r-4+ V^-x ^Ti+ 1 ^5a:-4- Vb-x V^4^ - T V^aa: + 6 + Vaar 1+ ^ax-h ^ax + h — */ax 1-Vax—b «7 v'o + ar + Va -x x > 2a Va + x- ^a-x 2a; 38. -/!+ i/2a-x--^. Vx 39 ^^ + ^^ V^o-a;+i^6-x >/x-^h Va-x-Vb-x' 40. ii/in^-^^^^.o. 41. V^T^TT-a-v^in^n. QUADRATICS INVOLVING SURDa 263 ^W -3^-p Via-x Va - p ' 43. \^a + x+\^a-x Va + x-Va- Vx+ b+ Vx- b Vx+b-Vx-b Vx + 2a+ Vx-2a x .^ a-x x-b , Vx-2a- Vx+2a 2a Va-x Vx-b 46. . lir^-Jilf .«. 47. ^li^^J^ Sb + x Sa-x Sb-x Sa- i« llZf 1 ^-^ AQ 17+20, x+a 60. VSx' + Q- VSx* ~9^VSi + 4c. 51. V2a:» + 5+ V2a:«-5=.'/l5+ V'S. 52. 'v/3ar' + 10 + V3a:» - 10 = -/ 1 7 + V - 3. 53. *J^/^r^ + 3v/2^=^^. 54. -^-+ ** 55. -— .+ -r= Va? -a X a + x ^a + i V X + V a — X V X - V a - X V x 56. \/x+l-2^x+ 1=.4. 57. (a + a;)'-(a-a;)' -.(o'-«»)*. 58. (a + x)^ - 5(a» -«>)*=- 4(a -x)^, 69. (7 + 4 1/ 3')i:» + (2 + V 3)x - 2. 60. ar* + «-* - (1 + x)* + (1 + »)■*. L 264 QUADRATICS INVOLVING SURDa 287. It sometiinea happens that the roots of a surd quadratic apparently/ do not both satisfy the equation. For example, let it be reqii'i :' ,.> Snd the roots of a?+ -/« = 20. Ttrl posing, Squaring or or 400-40ar + a:«-ar ««-41ar + 400 = («-16)(«-25)-0; .*. ar-.16or 25. Now, if for a? in the equation we substitute 26 there is an apparent inconsistency; for 1/25 + 25 = 30, not 20. To explain this, we must bear in mind that the root ot a quan- tity may be either positive or negative; therefore 1/25 =- ±5. If we substitute the negative root> - 6, instead of the positive, + 5, the equation is satisfied. So, too, in the foUowing example, the apparent inconsistency disappears if we take the negative root instead of the positive. Solve 3a? + l/2a;-2 = 7. Transposin g and squaring, (3aj - 7)' » 2a; - 2 or 9a:»~42a; + 49 = 2ar-2 or 9a:»- 44a? + 51=0. Factorings (9a: - 1 7)(a? - 3) - ; . . a? i 17 or 3. Here the value a? =3, if substituted, gives 9+ 1/4 - 7. But 1/ 4 - ±2; therefore taking negative root, 9-2 — 7. CHAPTER XIX. But SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS. 288. Tie solution of simultaneous equations of the second cle gree can be accomplished in a variety of ways. A common but wmewhat cumbrous method is that of substitution. iKr.i.— Given «+y = 7, (1) «y-12, (2) find ar and y. Substituting this value of y in (2), iry=-ar(7-a:)=12 » «»-7ar+12 = 0; .-. ar = 3or4. ®"* y'J-x; :, y=4or3. N0TK._The student should notice that when x and y are symmetricall v ZZ " *" Tf V^« -»- °^ *«^- roots are Jtercha^^lt. ^ y^tzi xr=3;Tr ^^' ^=^ °' ' -' ^=* - ^' ^•^•' -^- -" This equation, however, can be more neatly solved by the fol- lowmg method : — "^ « + y=7, . ary»12. a:'+2ary + y«=a 49 4ary ■ 43 Ex. ;?.— Solve Squaring (1), Multiplying (2) by 4, Subtracting (4) from (3), Extracting square root, but x*-2xy + f^ 1 ar-y=±l ar + y- 7 (1) <2) (8) (*) Therefore adding and subtracting, ar - 3 or 4 y-4ora. I '. la 266 SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS. Ex, 8. — Solve « - 2/ -i 10, a^+y"-178. Squaring (1) and subtracting the result from (2) we obtain 2a;y-78; but «» + y»-178. Adding (2) to (3), (x + y)» = 256. Extracting squar<) root, ar + y = ± 1 6. From (4) and (1), by adding and subtracting, we get a:=13or -3 and y-3or -13. 0) (2) (3) (2) (4) Ex. 4. — Solve 1 1 ^^ ar'*"y~20' a;»'^y» 400* Squaring (1) and subtracting (2) from the result, 2_ 40 oy^iOO* Subtracting (3) from (2), and extracting square root, 1 1^ J_ From (4) and (1) we find ar = 4 or 6 and y=.5 or 4. Ex. 6. — Solve «> + y»=35, «+y=6. Divide (1) by (2), then x'~xy + i^>*7. 0) (2) (3) (4) (1) (2) (8) 5L i\jiii ^»j; Bfcuu ^43; wc Will mm s ana y oj x,ae ordinary methods. TICS. tain (1> (2) (8) (2) (4) SIMULTANEOUS EQUATIONS INVOLVING QUADRATIC& 267 0) (2) (3) (4) (1) (2) (3) aethoda. 1. X -y-'O, 4. X-r, .i9, a?y= 100. 7. ar + y= 12, ar» + y»=104. 10. ar-y=- 10, a^ + y'-lTS. 13. 2a:-3y=i3, «»-y»=27. 16. i-1-3. « y * -5+---21. 19.1+1-1. X y 12* 1 1 7 «* a:y 144" 22. 4y = 5aj+l, 2a!y = 33-«>. 25. *'' + y»»91, aj+y-7 Qfl .^ .J KA «v« mi — y- ^ V^ «-y-2. BXBBCI8B XOIIL 2. af + y- 13, a^-36. 6. « + y-18, a:y- 72. 8. x-y^\i^ a:' + y» = 436. 11. a: + 4y=.5, a;« + ay-6. 14. aj + 3y=10, 3y>-ar»=27. 17. 1.1-?. as y 4* j_ J^ 5 «"'^y»~16* 20. i + l=7, « y 5^^ = 26. 8. d;-y>-45. *y-250. 6. 2-y-15, «y-54. 9. ar + y = 49, a:' + y»=.1681. 12. aj-3y-l, «y+y'=6. («-l)'+(y-l)»-8. 18. i + 1-5, « y ai. ay =12, « y 12* 23. 7a!>-8xy=.159, 24. a:> - 2«y - y» = 1, 5a: + 2y-7. oj + y^l 26. a;» + y» = 341, 27. a:» + y»=1008, « + y=ll. «+y=12. 23. A--yS = 98, 30. «5_ys, 279. a5-y-2. «-y-S. i t it (1) (2) 268 SIMULTANEOUS EQUATIONS INVOLVING QUADRATICa 289. Another class of equations frequently occurs, viz., that in which both equations are of the second degree and both homo- geneous. Ex. — Solve a;* + ary + 2y» - 74, 2i;>+2a:y + y» = 73. Cross multiplying, 73(a:> + a^ + 2y») = 74(2a:» + 2ary + y«). Collecting like terms, 75x» + 75;ry-72y'=.0 w 25ar» + 25a;y-24y' = 0. Factoring (3), (5ar + 8y)(5ar - 3y) = 0; .*. 6ar + 8y = or 6a;-3y = 0. 8 3 .% «---y orar=-y. Ifx- --ythen «' + ay + 2y»= M y«- ?y«+2y'-74 (3) or from which we get and therefore 2>-7^' y-±5, ar-=Fa Two other values for x and y respectively can be found by substituting for x its value -y. 5 290. This equation might have been solved by another method, often used when the equation corresponding to (3) is not capable of being easily factored. From (3), 26a:» + 25a:y - 24y» . 0. Dividing by j/«, 25 . ^+ 26 • - - 24 - y* y This is now a quadratic in ^- j and can be solved in the ordi TICS. viz., that oth homo- 0) (2) (3) 74 )und by method, capable be ordl SmULTANKOUS EQUATIONS INVOLVINQ QUADRATICS. 269 na,7 manner. Solving we find ^ . ^ ^r - « We now proceed M before to find x and y. ^ae student should observe that any equation of the form ax' + bxy + cf^O can be treated in this fashion and the values of - readily found. 291. We have used the equation ^+«^.V + V-741 tra^ two general methods; but this equation can be soil readily by the following simple artifice :— ^ Adding the equations together 3a;» + 3a^ + 3y»-U7 ^ «*+ xy+ y«= 49 ^"^^ g'+ gy + 2y»= 74 Subtracting, ~ y^^~26 y = ±5 Substituting the values of y in one of the original equations the values of a? can be found. I^awons 292. We now give examples of equations easily solved by simple artifices. ^ ^ Ex. 1. — Solve ar» _ y« _, 26 «* + a:y + y»=13, Dividing (1) by (2), a:-y-2. Squaring (3), «»- 2;ry + y* = 4j ^""^ «« + ary + y»=13. Subtracting (4) from (2), Zxy =- 9; .'. ay = 3. Adding (5) to (2), a:« + 2ary + y« = 1 6. Extracting square root, « + y » ± 4. From (7) and (3) x and y can easily bo found. 0) (2) (3) (4) (2) (5) (6) (7) fi 4 ;i S r 270 SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS. Ex. «.— Solve y'fxy-28. 0) (2) Dividing (1) by (2), « 21 8 , 3 This value of x substituted in (1) or (2) will give a quadratic iny. Solve BZBROISB XOIV, I. 2y»-4.ry + 3x«-17, y'-«»«16. 3. x» + a;y+ 4y»=6, 3^» + 8y'-=14. 5. ar»-ary + y»=21, y»-2ary- -15. 7. ar* + xy- 15 = 0, a*y-y'-2=»0. 3«»+13jry + 8^^-162. 11. j:' + ary-=84, ary + y» = 60. 13. ar» + ,ry = 66, «"-y'^=ll. IS. ««-y> = 37, «' + «y + y' = 37. 17. ir' + ary + y> = 39, 3y*-5j-y-26. 2. «»-4y»-9»0, a:y + 2y»-3 = 0. 4. a^-ary-35 = 0, ay + y»-18 = 0. 6. J^ + ary+2y'=44, 2j:'-.ry + y»=16. 8. 2.c» + 3jry + y» = 70, 6x» + a?y-y» = 50. 10. a:' - .ry - y« == 5, 2x» + 3jry + y» = 28. 12. a:»-.Ty = 6, a:» + y» = 61. J. 4. ir + 2j:y + 3y»al7, 2«*+3jry + 5y"-28. 16. a:* + 6a-y=144, 6xy + 36y»=r:432. 18. 3jr«+42-y=»20, 6xy+2y*-12. ^■^^ nca. SniULTANEOUS EQUATIONS INVOLVING QUADRATICS. 271 (1) (2) [uadratio 19. a:» + y»« 225, «y«.108. 21. x'-xyr^SS, a?y + y»=« 18. 23. a;» + ary-6t/'=»24, x> + 3a;y-10y«-32. 26. ^±i^+l:-y«LO, «-y x+y 3' «» + y»-20. 20. ic«+9ary-340, 7«y-y«-171. 22. a:« + y>-68, a:y-16. 24. «'-xy + y«-=21, y»- 2x2/ + 16 = 0. 26. ^i-^+iry^? ar-y x + y 2' 27. y»-4ary + 20a:» + 3y-264.r==0, 6y* - Z%xy + ar» - 12y + 1056a: = 0. 28. l + l.*ty._^_ « y 12 « + y + 5* •rT • judmtA ^ ' I I I; OHAPTEB XX. PROBLEMS RESULTING IN QUADRATIC EQUATIONS. a pTbiem'orn*!""* '' ^^f'""^ ^^^^"^^^^ ^^ *^« «>-s number of oxen. 80 -— »= price of each in ^'g. If the person had bought 4 more the price of each in £'s would have been , ; hence, to satisfy the given condition, a? + 4 80 Simplifying, X ar + 4 «*+4a;=.320; .'. a: =16 or -20. (1) . ^^^"^'^^f,"'^^* ^*-« - 20 oxen, therefore the answer required ^! 4. 3'20 7 ' '' "r "'^^' ^ ^'^"^ *^« -edition It t Tf u "°^^^^«^% ^ ^ ^oot of that equation. Had Itre^tt" "^^^ ^°^^' - ^^"-« - --i«^-y would be Find an algebraic number such that when it is divided into 80 the resulting quotient will be on. more than that obtained whL 80 IS divided by the number increased by 4. BXBR0I8B XCr7, 2 Three times the product of two consecutive numbers ex oeeds four times their sum by 8. Knd the numbel ti,.lT^ ""/.?' °' '?"* """'»'=»'"» ""■nters is equa! to three times the middle number. Find the numbers. bought rr^^'fl! """■'"' "* "PP"'" •■•'' '« «»*'• Had he ^ught 4 more for the same inoney he would have paid J of . cent less for each apple. How many did he buy I '^ * "" i! il il! 274 PROBLEMS RESULTING IN QUADRATIC EQUATIONS. worth lowers the price 31^ cents per 100. * Ja ,t '^'';\^''\^^^Sht some pieces of silk for f 168.75 He ^ d the sxlk for $12 a piece and gained a. much as one piet cost him. flow much did he pay for each piece t ^ 7. A merchant bought some pieces of silk for $900 Had he bought 3 peces more for the same money he would "ha^p^d $1 5 less for each piece. How many did he bay ? ^ 1 ^:.t^v^ r^* ""^ * '^"^'^ "*^ ^ ^ovLhled by increasing, if, length by 6 inches and its breadth by 4 inche/ BeZr/e Z 1 yU'a'nd th? '' ' T'^"^"'"'' '^'' ^"^^^^^ *^« *>^«-Jth by 1 yard, and the area is 3 acres. Find its dimensions arou'ndVTh'''' ' T',' ^'"^ '"' ' ^^^'^^ ^^^^ ^^ * Path around it. The area f the path is equal to that of the Lt Find the width of the path. ^^^ 11. Divide a line 20 inchfia lnT»« ;«+« * the square on the other part. ^ ^ *^ *** 12. A vessel which has two pipes can be filled in 2 hours less ime by one than by the other, and by both together in 2 hou 5^minutes. How long will it take each pipe'alone to fiU L 13. A vessel which has two pipes can be filled in 2 hours less time by one than by the other, and by both together inThl" 52 minutes 30 seconds. How Ions will ,> fnt/ u • , to fill the vessel ? ^ * ^^^^ ^^^ P^P^ ^*>^« 14. A number is expressed by twc digits, one of which i, ih rr."' tjr rr- ^^ ^ "^^ '-'^"-"- 15. A number is composed of two digits, the first ot which ex ce^is the second b^ 2. The snn. of the sq.a™ of the nutC ^d of that which ia obtained hy reversing the doits TC What IS the number I " *• A^^"::' ONS. 62^ cents' 8.76. He one piece Had he have paid casing its srmine its ■eadth by <^ a path the plot that the equal to ours less 2 hours fiU the >urs less 1 hour >e alone 1 is the i inter- lich ex- lumber ; 4034. PROBLEMS RESULTING IN QUADRATIC EQUATIONS. 275 16. A number is composed of two digi^ the first of which ex- ceeds the second by unity, and the number itself falls short of the sum of the squares of its digits by 35. What is the number? 17. Divide 35 into two parts so that the sum of the two frac- tions formed by dividing each part by the other may be 2,^ 18 A boat's crew row 3^ miles down a river and back again m 1 hour 40 mmut^. if the current of the river is 2 mile: an hour find the rate of rowing in still water. 19. A jockey sold a horse for 8144 and gained as much per cent, as the horse cost. What did the horse cost ? 20. A merchant expended a sum of money in goods which he sold again for $24, and lost as much per cent, as the goods cost him. How much did he pay for the goods ] 21. A broker bought a number of bank shares ($100 each) when they were at a certain rate per cent, diacount, for $7 500 and afterwards, when they were at the same rate per c^nt. pre- mmm, sold all but 60 for $5,000. How many shares did he 22. A person's gross income is £1,000. After deducting a percentage for income tax, and then a percentage less by one than the income tax rato from the remainder, the income is re- duced to £912. Required the rate per cent, at which income tax IS charged. 23. If the length and breadth of a rectangle were each in- creased by 1 the area would be 48; if they were each diminished ^.4^'"^ "''''''l^^^ ^*- ^"^^ *^^ ^^''^'^ ^"d breadth. .24. The sum of the squares of the two digits of a number is 25, and the product of the digits is 12. Find the number. 25. The sum, product and difference of the squares of two numbera are all equal. Find the numbers. (Let x -^ y and ar - 1/ represent the numbers.) ^ 26. The sum of two numbers divided by their difference gives the same quotient as if the greater number were divided bv the less. Find the quotient. ^ ill m 276 PROBLEMS RESULTING IN QUADRATIC EQUATIONS. 27 The difference of two numbers is f of the greater, and the sum of the squares is 356. What are the numbers ? 28. The difference between the hypotenuse and two aides of a nght-angled tnangle is 3 and 6 respectively. Find the sides. 29. Fmd' two numbers whose sum is nine times their differ- ence rvnd whose product diminished by the greater number is equal to twelve times the greater number divided by the less. 30. A person has $13,000, which he divides into two parts, and placmg each at interest receives an eqnal income. If he placed ToaT ** *^^ "^^ ^^ ^"*^''"^* «^ *^« ««««^d he would re- ceive $360 income, and if he placed the second sum at the rate of the first he would receive $490 income. What are the two sums and what the rates of interest ? 31. A and B have each a quantity of flour, A having 4 barrels more than 5. They sell their flour to each other at different prices per barrel, and the account between them is settled by B giving U>A £7 16s. B's quantity sold at A's price would have amounted to ^28, and A's quantity at B's price to £34. Find how much waa sold by each and the rates per barrel. 32 From a sheet of paper 14 inches long a border of uniform width is cut away all around it, and the area is thereby reduced | • but had the sheet been 3 inches narrower, and a border of the ^me width been cut away, the area would have been reduced # What was the breadth of the paper 1 33. The hypotenuse of a right angle is 20, and the area of the tnangle is 96. Find the sides of the triangle. 34. The fore wheel of a carriage turns in a mile 132 times more than the hind wheel; but if the circumferences were each increased by 2 feet it would turn only 88 times more. Find the circumference of each. 36. The numerator and denominator of one fraction are each greater by 1 than those of another, and the sum of the two frac- tions 18 1^5. If the numerators were interchanged the sum of the fraction would be 1^. i^d the fractions. ONS. )r, and the • aides of a B sides. leir differ- lumber is 16 less. parts, and he placed would re- ; the rate 3 the two 4 barrels different Jed by B 'uld have 4. Find uniform duced I ; Jr of the iduced f ia of the 2 times 3re each rind the kre each fro f rac- sum of CHAPTEB XXI. THEORY OF QUADRATICS. 294. A quadratic equation cannot have more than two roots. Let the quadratic be reduced to the form x'+px + q^O Then, since x»+^x + y is of only Pu^o dimensions, it cannot have more than two factors, each of one dimension. Ux-a and ar-i be the factors of x»+px + q then x»+px + g It is evident that ^ + px + q will vanish for x^a and x^b- therefore a and b are roots of the equation. Also, a and 6 ar^ the oiily roots; for no values other than a and 6 will make {X - a)(x - 6) - 0. Hence the quadratic has only two roots, viz a and 6. *' 295. The following proof is given in many textrbooks, but it IS defective, as it proves that a quadratic cannot have three un- equal roots, but does not prove that it cannot have three roots two or more of which are equal. ' If possible let «, ^ and y be the roots of the quadratic equa. tion ax' + bx + c^^O. ^ Then, since a is a root> aa* + ba + e^O and since |3 is a rootj a^ + b^+o^O; similarly, since y is a root, af + by+c^O. Subtracting (2) from (1) and (3) from (1) we obtain «(«'-^) + A(a-/3)-0, ^4^ (1) (2) (3) 4 278 I i V THEORY OF QUADRATICS. Dividing (4) by (a-/3), which is by hypothesis not sero, and (5) by (a - y), which also is not zero, we obtain a(a + ^ + i-0, (6) o(o + y) + 6-0. (7) Subtracting (7) from (6), a(|3 - 7) - 0. (8) By hypothesis j3- 7 is not zero and a is not- 0; therefore the result is impoasible, and the equation cannot have three unequal roots. Note.— The reason why a quadratic equation cannot have more than two roota is the fact that it cannot have more than two factors, each of one dimension. For a similar reason a cubic equation cannot have more than three roots, etc. 296. If a and h are the roots of a quadratic equation then {x-a) and {x-b) are the factors of the corresponding quadratic expression. For if the expression vanishes when X'^a then x-a is a factor. Similarly ar - 6 is a factor; .'. !i? + px-\-q^{x-a){x-h) or oc^+px-\-q=^x^-x{a-\-b) + ab. Now, since this is an identity, by equating coefficients, we obtain p- -(a + 6) or a + 6= -^, ^j^ and q=^ab or ab^q. z^) Hence the important principle: — The sum of the roota of a quadratic equation of the form a^ +px + gr = w equal to the coefficient of x with its sign changed; and the product of the roots equals the absolute term, that is] the term independent of x, 297. This result may be obtained directly as follows: Solving 3?-\-px + q^0^Q find the values of rr to be 2^ _^?and -^ ^P'-"^g 2 " 2 2 • zero, and («) (7) (8) refore the le unequal more than trs, each of have more tion then quadratic bhen x-a we obtain (1) (2) the form changed; It is, the THEORY OF QUADRATICS. If o and b are the roots then 2 ^ 2 • 2 2 • Adding (1) and (2), a + b^ -p. Multiplying (1) and (2) together, 279 (1) (2) (5) ab> (-1)"-^ f^ - 4q ^_f- iq iq "4 ~4 T"^' /, abm^q. (4) 298. From the fact that if a and A are roots then (x - a) and (X - b) are factors of the corresponding quadratic expression we can at any time find the equation when its roots are givea Let a and A be given roots; to find the equation of which thev are the roots. ^ If a and 6 are roots then (x - a) and (x - b) are factors; .-. {x-a)(x-b)==0 <**" «'-a;(o + 6) + a6=iO is the required equation. Ex. i._Given 2 and 3 to be roots of an equation, find the equation. Since 2 and 3 are roots a; - 2 and a? - 3 are factors; .*. (a?-2)(a:-3)-0 f' x*-5x + 6~-0 is the required equation. £x, ^.-Given 3 and - 4 to be roots, find the equation. Here the factors are^-3 and x-(^i) or ;r + 4; therefore equation » («-3)(» + 4)-0ora:» + x-12-0. ill ' 1 IMAGE EVALUATION TEST TARGET (MT-3) A ^<' :4^r. '<^<^ V 1.0 I.I u m us 2.5 2.2 ii& MM \25 lllll 1.4 1= 1.6 'c^l ■ c*. c* ^^ ^ /S V .^ Photographic Sciences Corporation 33 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) A %^v^ Ij' '% 5a; + e » may be real and eqitaly real and unequalf or vmagi/na/ry. Solving aar" + iar -f c = we find '--2i* 2a Let h Vh^ - iao -^ + suid ar-=» —rr-- ' 2a 2a ' 2a b ^/h'-iae 2a then Xi and x^ are the roots of ax* + bx + e=>'0. (a) Fixing attention up on the e xpression Vb"* - 4ac, we see that if l^^iac the quantity l/6»-4ac = 0, and x^ and ar, each become —Y ') therefore «i and x^ are equal when ft" =» 4ac (ft) B ut il ft' > 4ac then ft" - 4ao ig a positive quantity and Vb*- 4ac is a real quantity. Let then d:V'ft»-4ac- ±rf, -i-^' and di^i therefor* ^t <^d a:, differ by 2d, and ^i and x^ are rea/ and ufMotial, 9 il THEORY OF QUADRATICS. re X -V2 jiven, sub- at remain- 3 equation may he e see that h become atity and tme^tal 281 (c) If, hov^ever, 6» < 4flc, 6» - 4ac ia ne^a 4 x 3 x 9 therefore roots are real and unequal. ' Ex. ^— What value of n will make x«-8a; + n = have equal roots! The condition of equal roots is (8)»=:4n <>' 64 = 4n/ .". n=13. Ex. 5.— What value of a will make 4a;« x aa: + 16 = have equal roots 1 To have equal roots, or a'=»4x4x 16 a«=256; a =±16. Ex. (?.— What value of 6 will make 6«» + 9a; + 20 = have equal roots! Condition required is 81-46x20 <» 81-806/ . h ®1 •• ^-so- 300. The equation ax' + bx + o^Q requires examination, (1) when the value of a approaches zero, or as is generaUy said-O- (2) when = and 6-0. ' 19 m 282 THEORY OF QUADRATICS. (1) Let as^ + bz + e^O have the coefficient of m^^zero. Divide by x\ then X ar Let - —y, then But a = 0, therefore or But y ■• - 1 therefore 6y + cy' = y(6 + cy)==0; .*. y = or --. 1 n ft -=0or --. iB Let - ■» — , then x— - -, one of the roots required. X Let - "0: then, since the value of a fraction diminishes as ^ 1 the denominator increases, the value of a? in - — must be very X great; for the quotient obtained by dividing 1 by a? is very small, or zero. In such examples as this x is said to be infinitely great, or infiniti/, the symbol for which is oc. Therefore the roots of aa^ 4- &x + e » are — -r and oc when a ■■ 0. (2) Let a B and also 5-0. cut^ + bx + c^O when divided by z* becomes ft « A X ar 1 - -y; •X As before, put be , - a+- +:5»a + oy + cy» = X ST 0. But a — and 6 « 0, therefore a + by + cy*>'^cy'=:>0; hence both values of y = 0. But y = - ■• 0, therefore x = iri/Jwt be found in textbooks on Algebra require for their solution nothing more than a clear con- ception of the fact that the sum of the roots of a quadratic, in Its simplest form, equals the coeffioient of x with its sign changed and the product of the roots equals the absolute term. Ex. L—U a and b are the roots of «> + »«; + n = 6nd the equation whose roots are - and - . a b ^* a ^'^^ b *^ **^® ^^^^ ®^ *" equation then or or Bat . ^/a + 6\ 1 '^'i-oTj-'Tb-''' a + bim - ju and ab « n; 1 . a+t m , 1 1 .. , -» and — = -. ao n ah m 1 /. a»- /a + h\ 1 ^ , -'[abj^Vb'^^^'^^ hence both 1 y. 8o that 1 or a^ + — +- =0 n n na;* + WW + 1 = 0. Otherwise^ 2 = we mean H it differs from H Divide then x^ + mx'\-n==0 by 3^, If I 1 I ■ ri m 1 7 i ' 284 THEORY OF QUADRATlCa If y -i - then the equation becomes I + my + ny* » 0, and the * 1 1 values of y will be - and r , since the .alues of y are the recipro- a cals of those of x. Therefore 1+wy + ny' — is the equation whose roots are - and ^ . a b Ex. S. — If m and n are the roots of aar* + 6a; + c «■ find the values of (o) m* + mn + n', (b) tn* - n\ (e) vn? - win + n', {d) m* + m V + n*, («) m" + n». (a) Dividing and Squaring (1), and Subtracting, oa:* + 6aj + c = by a, a a a e a m' + 2mn vn' = — a' e a (1) (2) (8) (2) wi' + mn + n' = — - 6' c b'-ae ,J » .*. m' + mn + n' =» l^-ao a' (b) To find the value of w' - 7t'. Since and a win™ — , (1) (2) squaring (1) and subtracting 4win, , 6» 4c b*-iao m' - 2»m + n' — -= = = — • a* a THEOBY OP QUADRATICS. Extracting square root^ fi» - n » ± But Multiplying together (3) and (4), m + n-i — , a S8S («) m'-n'-iq: A\/6»-4ac (c) To find the value of m*-mn + n\ But and 3mn«--. Prom (2) fn*-mn + n*^ __. o' or a' _ (y~3ac)(&'~ac) " a* ' (fl) To find the value of m» + nl m' + n?<=(m + n)(m* -mn + n»). 6 And Subtracting, (d) Since uid m + nm» — a « « ^ - 3ao ... m» + n«-- ^<^'-^^) Another Method. m»+n»-(m + n)«-3mn(m + n):==/-*y-3(^)(-*) I)' I 286 THEORY OF QUADRATIOEL Ex. S. — If one-third the sum of the squares of the roots of the equation oa!' + 6a; + c=0 is equal to the product show that i'=5ac. Let m and n be roots of the equation, then and also, by hypothesis, in + n~ — a a - (to' + n") = mn. 9 (1) We have now to express m' + n" and mn in terms of a, 6 and c. Since 1. . .. 1 /f>^-2ac\ f»^4-n^ai(m + nr-2mn=>-^ — = — ; ^ ' a* a a* 1/ « .si /6'-2ac\ also, e winss — , a 1 Substituting these values of ^ (m* + n') and mn in (1) we obtain 1 / y - 2a<; \ c or ft'— 2ac=3ac, or 6' »■ 5ae. a & + ; + x+a x+b x+e Ex. 4' — Find the roots of the equation when a + 6 + c «=» 0. Clearing of fractions, a(x + b){x + c) + b(x + a)(x + c) + e{x + a)(x + 6) = 0. Multiplying out, and collecting coefficients of s^ and x, ar'(o + 6 + c) 4 ar(2at> + 2ae + 2bc) + Zabc - 0. Since a-t b + e, the coefficient of x', bQ, one rcot of the quad- ratic is infinity and the other root is ^rr—. — ; — '■ — r . •^ ^ 2(o6 -^-bc-k-ea) pL_^ tHKORT OF QtJADRATIOd. tet Mx. S.—Jt the equation o^+px + qr.0 have equal roots show that ax'+p{a + b)x + q{a + 2b)^0 has one of them, and find the other. If the roots of a^+px + q^O have equal roots then p^'^iq or q^^; :, ax" +p{a + b)x + q{a + 26) - oa^ +p(a + b)x +^(o + 26) - 0. But <^+p{a + b)x+^(a + 2b)^^x+^yax+P-(a + 2b)}i .'. (* + |j{, and therefore <-^«^' and ao q But (a+6)« a b ^ ^ --r+- +2 and a6 n (m + n)* pi" mn jTi * (m+n)* m n ^ mn n «» ftnd, by hypothesis, a b m n or , (g + by (m + n)* I* J — =■ , ao mn • • — *" — BXBBOISB XOVI. 1. If a and b be the roots of x* +px + q = prove a' + i* = 3jt?y - jt>". 2. If a and 6 be the roots of pa^ + qx + r^O prove that the equation whose roots are - and - wiUhe pr3^+(2pr-q*)x+pr==0. 3. If m and n be the roots of aaj* + ar + 6 = t show that \ n)\ mj" ab' 4. Find the sum of the roots of the equation .0. Oi bi e, ■ + — —+ X - a X - b X - e THEORY OF QUADRATICS. S89 5. If a and /? be the roots of t*.^ + 6z + c - form the equation whose roots are ff. ts are ^ and -^; also form the equation whose roots or , SP F ^ are -^ and -4. cr 6. If a and ^ be tba roots of «" + ra; + — r*-0 form the equa- tion whose roots are a" + ^ and «• - jS". 7. If ^ and^ be the roots of the equation pa^ + qx + qm,0 show 8. If the roots of a^+px + q^O and x' + qx+p^O differ by the same quantity show that p + q + i = 0. 9. If a and b be the roots of ix^+px + q^O, and a and c those of «» + ra? + » = 0, then (b + c) and 6c respectively will satisfy the equations ar» + (jp + r)a; + 2(5- + «)=.0 and x' + {q + 8-pr)x + qa=>0. 10. If the roots of aa^ + bx + cO are in the ratio of w» to n show that — « ^ i- . ac mn 1 11. If a and ^ be the roots of x*-x(l+a)+ -(l+a + o«)»0 prove that o? + ^^a. 12. Form the equation whose roots are the square of the sum and difference of the roots of 2a;' + 2{m + n)x + m' + w' = 0. 13. If a and j3 are the roots of oaj* + 6a; + c = form the equa- tion whose roots are 0^ + ^ and — v — . a' jS* 14. If a and ^ be the roots of aa:* -i- 6a? + c = prove that the quadratic equation whose roots are — and i- is § a o^ci? + 6(6* _ 5a6»c -i- 5aV)a; + ac* - 0. 1 5. Find the .-elation between the coefficients of aar* -f- 6a? -h c -» that one root may be double of the other. 16. Form the equation whose roots are V' ST and - */'%. 17. Form the equation whose roots are m-\-n and m-n. 290 THEORY OP QUADRATICa 18. Find the condition that oar'-f 6« + c— and aix' + biX+ei'^O mcy have a common root. 19. If a3^ + bx + o>mO and a^x' + b^x + Ci^O have respectively two roots, one of which is the reciprocal of the other, prove that (ooj - eCiY - (abi - bci){a^b - b^c). 20. For what value of m will the equation 2a^ + 8:r -i- m - have equal roots t 21. Show that in every quadratic of the form aa^ + bx-^a'mO the roots are the reciprocal of each other. 22. Show that the roots of ca:* + 6a; + a = are the reciprocals of those of aa:" + 6a: + c = 0. 23. If a and ^ be the roots of the equation ax^ + 6ar + c = 0, and a' + ^ = 1, show that 2ac =» 6' - a*. 24. If a;* - 3a: - a = and a:* - 4a! - 6 =» have one root in com- mon show that o' - 14a + 40 = 0. 25. If the roots of the equation n^+px + q^O be respectively any equimultiples of the roots of the equation a:* + ra; + « — prove that sp* ■■= qr^. 26. If the diflference between the roots of the two equations «* + (j9-a)a: + 6> = and ar* + (/> ~ 6)a: + a» = be equal then shall 2j»=:5(a + 6). 97. If a and b be the roots of x* -px + 5- = 0, and a and B the 1111 roots of a:'-g'a; + » = 0, show that — + -;:+— t+ttt-I. aa ap ab pb 28. Show that the roots of the quadratic equation 3a — a: 36 — a; 3c — a; are real if a' + 6' + c* is greater than a6 + ac + 6c. 29. Find the values of m which will make the equation 3wa:' + (6m - 12)a: + 8 = have equal roots. 30. If a6 + 6c + ca = find the roots of a' 6> x-a x—b x—o 0. APPENDIX. SPBOIAL FORMS OP SIMPLE EQUATIONS. 1. By methods already given any simple -equation involving but one unknown quantity, ar, may always be reduced to the form axtmb where a represents the sum of the coefficients of all the terms containing or, and b the sum of all the other term& 1. Let a — 0, then axtmO; for if one factor =»0 the product = 0, and therefore no value of x can satisfy the equation. If, how- ever, a be not zero but a very small quantity, then the equation will be satisfied when x is very great; and by making a small enough x may be made greater than any assignable quantity. In this sense it is customary to say that when a = a;= oc. Similar remarks apply to equations (3) and (4) of the following Art. when the coefficients of x and y vanish. II. Let a = and also 6 = 0, then ax = 0, and the equation is satisfied for any value of x. In this case the given equation is an identity, emd therefore true for all values of r. (Art. 108.) 2. The following simple example will illustrate the meaning of the preceding cases : — A and B commence business with capitals of $a and $b respec- tively; A gains $?» and B $n per annum. In how many years will they have an equal amount of money 1 : II t 292 SPECIAL FORMS OF SIMPLE EQUATIONS. Let X » the number of years, then and Then a + mx *r i's money at the end of x yeara b+ nx^ £*8 money at the end of x years. a + inxm,b + nx; ,*. (m - n)x — 6 - a or « ■ b-a m — n Now, if m - n =» no value of x can satisfy the equation unless also 6 - o «= 0, and then any value of x will satisfy it, Referring to the original problem we see that these i esults are correct ; for it m = n and a is not «=• b then evidently the two men can never have the same amount of money; but ii in=^n and also ami then they have always the same amount. 3. Simultaneous equations of the first degree involving two unknown quantities may always be reduced to three terms each, and may therefore be expressed in the form ax + by ^c, (1) a'x + b'y=^c'. (2) Eliminating y and x successively in the usual way we get (ab'-a'b)x'^b'e-bc\ (3) {ab'-a'b)y==c'a~ca'. (4) I. Let ab' - a'b =« 0, then no values of x and y can satisfy equa- tions (3) and (4), and consequently no values of x and y can satisfy the given equations. In this case the given equations are said to be inc 16. 2a«6»c. 17. 3ar + 7y+ll«. 18. 6x+lSy+Qz. 19. 6a + 66 + 6a 20. EXERCISE VI. (Paob 25.) 1. {a + m)x + {b + n)i/. 2. (2a-7)a; + (10 -36)y. 3. (m-6+l)x + (n-o-l)y. 4. (5o-lOd)x + {a + 7b)y. 5. (2o + 2%. 6. 3ax. 7. nx + y. 8. -nx, 9. 0. 10. (a + 6 + c)a?. EXERCISE Vn. (Paob 27.) 1. + 8. 2. -8. 3. -30. 4. +30. 5. + 41. 6. -75. 7. 6a. 8. -6a. 9. -8m. 10. 3a;. 11. -8a6. 12. 8a«a;. 13. -Say. 14. -3a6y 15. 17mV. 16. -I2xy 17. 6«". 18. V. 19. m -n. 20. 0. 21. 476-(-763)-i229. 22. 2000 - ( - 1600) - 3600. ANSWERa 297 B, 27. niles west. EXERCISE VIII. (Page 28.) 6. -16 b + 7c + d. )debt. 136 -4c. 12. 0. 6e. 20. 3%. *-7%. • + e)x. h30. -6a. I2xy. • 3500. 1. a-36-2c. 3. 2a -26 -a?. 5. 2x^-6x*y + 6xy'-2y'. 7. 7-10ar+10ar»-6x». 9. x^-i/*-Zxy-a + b. 2. -2a- 6- 2c. 4. 6a:»-5ar»y-llary'. 6. -a:»-2ar»-.r-4. 8. 2a»~26'-2ac+26c. 10. a% + ab\ 11. 6x» + 3y''-lld;y-|-52/a-7ara. 12. 2a»6c + 2a6c» - 2a6c. 13. {a-cy- {h-d)f. 14. (a-/)a^ + (26-2m)a^ + (c-n)j/». 15. {e-by-^{a-c)xy^{h-ayf. 16. (a - 6) + 8(a; + y). 17. 2(a+6)-2(c + c?) + 10(x + y) + m-n. 18. 26a: + 2cy + 2a«. 19. -a:»+15a;«2/ + 7y'. EXERCISE IX. (Paob 30.) 2. 0. 3. 2a -6. 6. 6a; -3a. 7. 6a; -8y. 10. 3a. 11. -2x-y. 14. 7a -76. 15. 10c -6. 16. (a-6) + (c-d)-(8-/), (a-6 + c)-(rf + e-/). 17. a^{b-{c-d-e)-f}. 18. a- {(6-c) + (rf + «)-/}. 20. -a + 6-c. 21. 17. 1. 0. 5. 2c-6rf. 9. 2a-6-rf. 13. a. 4. 4a-6-c. 8. 2a -26. 12. Ix-^y. 19. a^[b-{e-{d+e-f))\ 1. 16. 5. 196. 9. 162. 13. 4816. 17. 24w»*»*p. 21. 36-0. EXERCISE X (Paob 34.) 2 -15. 3. -42. 6. 15. 7. 135. 10. -2. 11. -450. 14. -35a». 15. -80a«6c. 4. 99. 8. 0. 12. -21. 16. -36aVy». 18. -30a;»y«». 19. 60(a6a;y)'. 20. 0,0,0. EXERCISE XI. (Paob 35.) 1. 6a;* - 9a;' + 12a;. 2. 15«*y- 6a;'y'. 3. -2a»6»+2a'6»-2a6*. 4. - 8w'nV + 20wVp. 5. - 35a«6a;2/» + 25a»6y - 55a6y + 30a»6y*. 6. - 24a*y + 40j;y» + 56a;»y» + 32a;»y + Uxy* - 1 6a;y. 7. 13j^-272;' + 44»r, 8, 4a'-ll.'?6-f 156'. 10. 2a;* + 26a;»-12a;«, U. a' + 6« + c«, 12. 0. 20 298 13. 2arf+26c. 17. lax. ANSWERS. 14. 2my-2nx. 15. 0. 16. 96. 18. 2bx+2by. 19, 2Aa;-i-2cy. 20. 0. EXERCISE XII (Pagb 37.) 2. 23^-Sx*+Ux-lO. 4. -8a:'+18a:> + a;-15. 7. a«-6». 8. a» + 6». 10. 4a*-13a»i"+9A*. 1. a^-x*-2x + S. 3. 8a:»-6.T»-23.r + 21. 5. a^ + a;'-^l. 6. a:* + 4. 9. a*-4a»+12a-9. IL 6a;*-23r» + 25a;»-16a;+i5. 12. 2x'-ix'^ + 5x^ + Q3^-9x^ + Zx + 7. 13. l-4a:»+16a:»-a!*-12«». 14. a;7 + 5a^-3aH»-9a;*+10j:'-a:»-6a: + 3. 15. 2aJ»-7.i;*-5.r»-9a;»-15a;+18. 16. a:» - x« - 23^^-3;* + 2a:' + cB»-ar-l. 17. a*+2a;« + 3a;* + 2a;»+l. 19. a:^- 41a: -120. 21. 2a»6' + 26V+2a»c»"0*-6«-c*. 23. «•+ 10a; -33. 25. «»- 4a:y + 6a:y - 4a:»y + y». 27. a:" + (a + 6 + c)a;' + (ab + bc + ca)x + a be. 28. «* - (a + 6 + c)a:' + (ab + bo + ca)x - a6c. 29. a:»-9a:» + 26a:-24. 30. ar*-i0x»+9. 31. af^-a\ 32. a:» + «*y* + y». 33. 729m«-a«. 18. a:6+151.r-264. 20. a«+6» + +«»-2a:y-2ar« + 2y». 8. x' + 46' + 9c»- 4a5 + 6ac- 1 26c. 9. 4a'+6»+9c»-4a6 + 12ac-66c.lO. l + 2j:4.3ar«+2«» + a:*. 11. l-2a:+3a:'-2x» + ar*. 12. a;* + 2r»y + 3a:»y» + 23^3^+3/*. 13. 4jr*-12.r»+25x«-24a:+16. 14. x*-2x^-3x' + ix + i, 16. 4a!*-4a:»-15x» + 8a;+16. 16. a>+6«+c' + cP-2a6 + 2ac-2a(^-26c + 26<^-2crf. 17. a*+2a'6-2a»c-4a»6c + a»6»- 2a6»c + aV + 2a6c*+ W. 18. ««-2x» + 3x*-4«»+3a:«-2a: + l. 19. 4(a:» + y» + «>). EXERCISE XV. 1. a»+2a6 + 6'-c». 2. 3. ix'-ixy + y'-dz^. 4. 5. ar* + a:'+l. 7. 9a*+lla»6» + 46*. (Paqb 41.) 6>_c»+2ic. 9z^-x^-if + ixy. 6. a* + a'62 + 6*. 8. «* + 4a*. 9. a^+b'-c'-d'+2ab + 2cd. 10. c» + d'-a»- 6> + 2crf+2a6. 11. -yV-.3»*. 12. 0. 13. 2a»6'+26V+2aV-o*-6*-<^. EXERCISE XVI. (Paob 42.) 1. «" + 7ar+12. 2. x»+7a:+10. 3. a^+7« + 6. 4. a:*+21a:+108. 5. a;2 + 21ar + 90. 7. a:»-9x+14. 10. x^ + x-Q. 13. aj'-lS-r-lOO. 16. a:* -80;'^- 20. 19. r»-4ary-5y», 22. ix'~dixy + 70y\ 24. 9a*-3(6-c)a'-6(j. 6. x» + 20x + 99. 9. a;«-20x + 75. 12. a:'+ 15^-100. 15. «»-20ar-21. 18. a:« + 20j;*-126. ^t'-dx'yz+Uyh^ 21. x' + (a^b)x-^ab. 23. 25.r»-60ar + 20. 25. rtV + a(6 + c)ar + 6a 8. a^- 13a; + 30. 11. a:» + 7a?-8. 14. a:'-20ar-300. 17. a:« + 10a:»-56. 20, EXERCISE XVII. (Paob 43.) 1. a^ 4- Si:^ 4- Sr. 4. 1 . o ^J q^ S. «* + 4«» + 6a;» + 4x+l. 4. «*-4a:» + 6«'-4a! + l. -r i»* - *. y ■if ■ *t| iNI I '{ill ill ■ I 11 I 300 ANSWERS. 5. «»+6a;«+10x»+10a^+6ar+l. 6. 3^-6x*+10jc^-10t' + 5x-1, 7. si» + Gx*y+l2xy' + 8f. 8. 8t»- 12a^y + 6a-y»-y». 9. 8a»-36a»6 + 54a6«-276". 10. a*-8a'6 + 24a»6»-32ai»+166*. 9. 8a»-36a»6 + 54a6«-276". 10. 1 1. 16a* + 32a'A + 24a'i« + 8a6» + 6*. 1. 4. 6. 250. - 24a«6\ 2a«6Vrf». x* - 2x - 7. -5a»+6a6+86'. 7a:'2!' - 9xy. 11. 14. 17. 20. 2.3. 26. EXERCISE XVm. (Paob 46.) 2. -4. 3. 50. 4. -50. 7. a. 8. 2a»c'. 9. 8a«6. 12. 9ar*y». 13. - 15. 23/». 16. 18. -y^ + af-hy. 19. 21. ar*-mp.r»+;>». 22. 24. 4a5-5a6* + 66». 25. 6. -128. 10. 16aY - 25a». Uxy. a» - 2a6 - Zb\ -ix^ + Txy-%f. 2a» + 3a«6-46. 12a:"-ll;.y+10y». 27. 2-Zho + bad?. EXERCISE XIX. (Pagb 60.) 1. a?+5. 2. a;-4. 3. x + 1. 5. a:"+7ar+12. 6. «* + a;+l. 7. o+6. 9. a + 6. 10. o*-a'6 + rt26«-a6» + 6*. 12. 2a^-3ar+7. 13. 2a; + 32/. 14. 4ar-l. 16. ^3? + ixy + y\ 17. a»-2a26 + 3a6» + 46». 19. a«-a6 + 6l 20. a:* + a^+l. 4. a;-9. 8. a2 + a6 + 6«. 11. a? + Zx + 1. 15. a:»-3r« + 3.r + l. 18. a»-3a'a;+2aa;^ 21. a^ + 2/». 22. a;" + ar»y» + aj«y» + a:»y» + 2/". 23. a^+2a:»+3.r» + 2a;+ 1. 24. .i!* + a:' + a:+l. 26. 2a^-3a^ + 2aj. 28. -13a»6»-3a6-l. 30. .i^ + 3r'y + 8ajy-8y*. 32. 2a36 + 3a'6»-a6' + 46* 25. 3^ + x^y- afi'i^ - a;V - x'?/* + a;y^ + y\ 27. «*-3a;y-y». 29. a:* + 2r'y+2a;V + ary». 31. 27a:V-18.ry»-92/'. 33. «» + 4«*-3x» + a:2-6. 34. 3^ + x''y-ix^^-j(l^*-^y^ + xy'' + i^, ANSWERS 'ip^ + bx-l. 32a6»+16ft* 10a;y*. y - y« - zz. y-yz-zx. J. -128. ). 16a'y. 6 - 36*. ■7a:y-9y». ia«6 - 46. j + 5acP. 801 9. a6 + 6». • 3ar + 2. 3r2 + 3.r+l. 3a'a; + 2aa;^ + 1. r^ + ary^ + y'. -6, EXERCISE XX. (Paob 62.) 1. x*-{a-^c)x + ac, x* - {a + h)x + ah. 2. a? + a. 3. a:«-(6 + rf).r + H ar« - (a + <^a: + ad 4. ar» + aar-6. 5. (l+m).r + (l-n)y. 6. a«+i« + c«-ai + Jc + ca. 7. ai-6-c. 8. 4a» + i«:fc» + 2«6 + 6c-2ca. 9. x+2v-« 10. 2:r-3y-|4». 11. a6 + Ac + ca. 12. o6-6c + ,v. 13. x-2y + Zz. 14. a;" + y' + 2' + 2.ry-2y2-2sa:. 15. x^-ax + a\ 16. a:» + aa: + a'. 17. (:r' + x+ l)a -(j:+ 1). EXERCISE XXI. (Page 53.) 1. a?-^y^.xy^-if. % a:* + a,-»y + a- V + xy* + y«. 3. aJ» + ar*y + ar»y» + a;y + a;y* + y8, 4. a^-a^y-^-xY-s^y' + xY-xif^-i/. ^•*'-^+l- 6. :c»-a^ + ;c»-ar' + ar-l. 7. 1+ary + a-y. 8. «*-;c»y« + a;VV-a:y»a» + «««♦. 9. ar*-2rV4ar'-8x+16. 10. 9 + 3ar + a^. 11. a^-3x»+9r'-27ar + 81. 13. a:»-2ar + 4. 16. 8a:»-4.r» + 2ar-l. 17. 36 + 6ar + a:». 19. 9a:'-30xy+100y«. 21. 25a^-40jry+64y». 12. 4a;' + 6.ry + 9y». 14. a;* + 2i.-»+4a;« + 8.r+16. 16. a:' + 3j:> + 9ar + 27. 18. 4x»+14.ry + 49y'. 20. 16j;' + 28a:y + 49y>. 22. a;* + 6a:2y« + y*. 23 EXERCISE XXJI. (Paqb 54.) l-3(a^-6). 2. 5(2.r»-3a:y + 4y). 3. 7/(^-9). 4. ll(2m'-3mn- 10n«). 5. aa:(x» ~hx+ 1). 6. 27a»6«(2 + 4a»6» - ga^A"). 7. 35a:y*(ar + 2y - Zz). 8. (a + h){x + y). 9. (c- 6)(ar-y). 10. (c + (f)(2a + 36). 11. (3ar - y)(a - 6). 1 3. (x + a){x + 6). 1 4. (a: - a)(a; - h). 16. (x + a)(a;-6). 17. (ar + 6)(a - ar). 19. (ao-bd){ax-by). 20. (ar - y){ao + hd). 22. (ar+l)(a:>+l). 12. (a-f 6)(c+l). 15. (a;-. !)(ar + 6). 18. (ar + a)(6-ar). 21. ix + y)(ad-b {X+i){x^-X'+l). 25. (2a»-36c)(36»-ac). 24. (j:-y)(2a + 36-c). 26. a{x - y){bx -^ cy). SOS ANSWERS. EXERCISE XXIIT. (Paob 66.) h(x + i)(x + 3). 2. {x + 5){x + 2). 3. {x + 6)(a: + 1). 4. (ar+10)(x + 3.) 7. {x-l2){x-6). 10. {x'-18){x'-i). 13. (ar + 7)(a:-5). 16. {x + 7)lx-U). 19. (aJ'-17)(j:»4-4). 5. {x + 6){x + 5). 8. (a:-20)(;t-2). 11. {x + G){x-5). U. (x+ll)(a:-8). 17. (x + 2){x-9). 20. (jJ_4)(.r»+3). 6. (x-15)(x-2). 9. (x-172/)(x-3y). 12. (x + 7)(.r-6). 15. (ar+10)(ar-l). 18. (i:«+l)(a;»-12). 21. (a-» + 20)(a:»-l). 22. (xyH-39)(a:y-10). 23. {xhj + 20){x^y-5). 24. {xyz-20){xyz + 5). 25. 3(ar-9)(ar + 8). 27. 5(a^-10y»)(a^ + 8y»). 29. ll(x» + 20)(x*-15). 31. a»/>(l + 19a:)(l - «). 33. -(x+29)(a;-l). 34. 26. 4(a; + 3)(x-2). 28. 2a{a^-7a){3^ + 2a). ZO.x{l-Gx)(l+x). 32. -(x+6)(a;-l). (x - a)(ar - b). 35. (a - b){b - c). 1. (x + 5)'. 6. (a«a; + 4y»)». 9. V-76)». 13. (4a»-3i)'. 17. x\3a-7by. EXERCISE XXIV. (Paob 57.) 2. (a:» + 9)». 3. (.r+lOy)'. 4. (arr + 2)». 8. (Ix—myy. 12. (3jr-5y)». (5aa: - 7iy)'. Q{2x + 3i/)\ {a-b + iy. {x+yy. 16, 20 24, 28, 6. (w'-8ny. 7. (a;»-19)». 10. (9x»-l)». 11. (2j: + 3y)». 14. (a + Zby. 15. (2a -56)'. 18. a\2x-5yy. 19. (6a-4i)». 21. 3x(aa;-3y)l 22. 3y'(a;-3ay)«. 23. (a + b + cy 25. ar*. 26. 4x\ 27. a:", 29. (2a+26 + 3c + 3y'- lOz^). ~e-d). 22. (r~y + «)(ar-y-«). 24. (a + 6-c)(n-i + c). 26. la + 2h-3c){n~2b + 3c). 28. (a + 6 + o-cZ)(a + 6-c + <£). 30. (a + 6-c-c/)(a-6-o+d). 32. 12(2x+l)(x-l). 34. (x + 2)(a;-2)(j; + 4)(z-4). 36. Z{x + 3){x + 6){x+7){x-5). 23. (i + c + a)(ft + c-a), 25. (ax + 4y+l)(ax + ). 29. (a-6 + c + fl?)(a-6-c-rf). 31. {a + b + c+d){a-b + c-d). 33. (a;-l)(a:-2)(a:-3)(.r-4). 35. (ar + 5)'(a;-3)(x-7). 37. (a + c?)(a - <;)(6 + c)[b - c). 38. (a:' + y')(« + y)(a;-y)(l +n')(l +n)(l -n). EXERCISE XXVI. (Paqb 59.) 1. {x* + x+l){.r--x + l), 2. (a!* + 2a; + 4)(ic»-2ar + 4). 3. {x^ + 3x + d){x^-3x+9). 4. (a:"^ + 2a;y + 2y)(.r»-2a:y + 2y=) 5. (x' + xy + Sy^x^-xy + dy^). 6. (x' + xy-y^Xx^-xy-y^). 7. {x + y){x+27/){x-y){x-2y). 8. (2a:2 + 2ar+l)(2i-^-2a;+l). 9. (2x» + 3.Ty + 3y«)(2x»-3ry + 32/«). 10. (a^ + 3a: + 5)(a;«-3ar + 6). 11. (a?+y)(3a;+y)(a;-y)(3a:-y). 12. (a:» + a:+l)(ar»-a; + l)(a:*-a:» + l). 13. {2x+y)(:t + Zy)(2x-y)(x-3y). 14. (2a:+3y)(a: + y)(2a;-32/)(a;-y). 15. (a' + 3/>2)(6»+3a2). 16. (5a' - 2a6 + 6»)(a> - 2a6 + 5i'). 17. (a + 6 + c)(a + 6-c)(a-6 + c)(a-6-c). 4. {2x + 3y)(4a;' - 6xy + 9y»). 6. (a;-y)(a:» + a:y + y»). 8. (9ar-82/)(81a;« + 72xy + 642/') EXERCISE XXVII. (Pagjs 60.) 1. (a + bXa^-ab + b-"). 2. (a + 2)(a»-2a + 4) 3. {3 + b){d-3b + P). 5. {3x-\-iy){0x^-12xy+l6y% 7. {x-lO){x^+lOx + iKJU). 9. (5a;y - 7z){25.vY + Z5xyz + i9s?) 10. (4a6- 10c»)(16a2i2 + 40a6c»+ 100c"). 11. (.r + 2/)(.c*-ar»y + .'cV-ar2/' + y*). 12. (.i; + 3)(.f*-3r' + 9x»-27a: + 81). 13. (3+y)(81-27y + 9?/2-32/3+2/«). 14. {x-y)(x* + x^y + xY + xy* + y*). 1 5. (.1- - 2y«5)(^.4 + 2;cS2/;32 + 4^j^2^i ^ g^^^o ^ 1 gy«^^^ 16. (1 -x/«')(l +xyh'' + a^y*s^ + a^fs^ + xyii^). Hi J3I il 304 ilii: ▲NSWEB& 17. (a+6)(a-6)(a« + aA + i*)(a»-a6 + 6«). 18. (a«+6«)(a<-a>6» + 6«), 19. (a + 6)(a«-a6 + 6«)(a«-a«i» + 6«) 20. (a-o)(c:«+a6 + 6«)(a« + a»6« + 6«). 21. (a + 6)(a-6)(a> + 6»)(a>+a6 + 6«)(a«-a6 + 6»)(a*-a«6« + 6«) (a* + a?b* + 6«)(a« - a»6» + i«)(a'» - rt«6« + 6"). 22. (o + i)(a-6)(a* + a»i + a'6» + ai»+6*)(a*-o»6 + a«6»-aA+6«) 23. (a«+6«)(a«-a«i» + a*6«-a»6« + 6«). 24. (a« + 6»)(a" - a}%^ + a»i« - a«i« + a*6' - a'6 »°+ i"). 26. (o-6)(a+6)(a«+6«)(a*+6*)(a'' + i«). 26. 6(3o«+3a6 + 6«). 28. (2a-6)(a«-ai + 6«). 30. (a-6)(o»+4a6 + 76*). 32. 26(3a» + 6»). 34. (a + 6)(7a»-13a6 + 76») 37. (a~6-c)(a» + A' + c«-2a6 + ac-ic). 38. («+l)(«-2)(ar*-2a:» + 3a:»-2a; + 4). 27. (2a + 6)(a« + a6 + A»). 29. (o + i)(a»-4a6->-76»). 31. 2a(o» + 36»). 33. 9(a-6)(a»-ai + i»). 35. (a+6)». 36. (a-6)«. EXERCISE XXVIII 1. (« + 2y)(2a; + y). 3. (2z+y)(a; + 3y). 5. (3a?+y)(4ar-3y). 7. (17ar>+2)(2a:'+l). 9. (4ar + 9)(2a; + l). 11. a(6a;+y)(aj~6y). 13. («+ay)(flM;+y). (Paok 61.) 2. (ar + y)(2ar + 3y). 4. (3z-4y)(5a;-2y). 6. (14ar-y)(r+6y). 8. (6a:«-y»)(z'-2y). 10. (3a:-5y)(4a; + 6y). 12. a''(5x-lly)(2ar + 3y). 14. (2a: + y)(.r + 2y)(2x-y)(x-2y). EXERCISE XXIX. (Pagb 62.) 1. (2a;+y-3«)(ar + 2y-2«). g. (2j--y + «)(ar-2y+3«). 3. (6a:-y+«)(a;-6y-4 6. (aj+y+3)(a?+y + 2). 4, {'ii'-:v + 3)(2?:'-'^^ + 2). :.«;- ay-2)(2a:-3y-l). 7. (ar-6y-5«)(ar + 2y-3«). 8. (9ar + 8y-20)(8«-y-iy 9. (15ar«+8y« + 52»)(ar»-2y» + 3«>. 10. (a+6-c)(a» + 6» + + 6c + co). 11. (a-6-c)(a» + i» + c»+a6-6c + ca). 12. (a + 6 + 2c)(a« + i« + 4c« - a6 - 26c - 2ca) w. ^aw -r u- - o-v/^-xs* -TV- -rue--- aoo i- ooe + oea i. AN8WER& 805 '). (a -by. l(«-2y). 3«). ' + 2). -1). r-1). 1. 2. 3. 5. 6. 7. 8. 10. 11. 12. U. 16. 18. 19. 20. 22. 24. 25. 27. 29. 31. 32. 34. 36. 38. 39. 40. 41. 42. 43. 44. 45. EXERCISE XXX. (Page 63.) (a + b-i-e){a + b-o){a-b + c){-a + b-^-c). {a+b + e-d){a+b-c + d){a-b + c + d)(-a+h + c+d). (a-l)(a-2)(a-3)(a-4). 4. (a+l)(a-2)(a-7)(a-10.) (3ir + 2y)(3z-2y)(x + y)(y-ar). {{a-b)z-{a-\-b)i/}{{a + b)x + {a-b)y}. (a» + o6 + 6«)(a' - afi + 6')(a' - i» + 1 ). (a* -a'6» + 6*)(a« + 6« + 1 ). 9. (a + 6)(a - 6)«(a' + aA + 6»). (a + A)'(o-6)(a'-a6 + 6'). {x'-^f + z* + 2xy + 3xz + Syz){x^ + y» + 2> + 2.ry - Sxz - 3yz). 16aA(a»-a6 + 6»). 13. (ax + bi/){az-bi/){bx + ai/){bx-ay). {x+y){x-yy. 15. 4jry(r + y)(a?-y). 2(a' + o6 + 62)(a-i + l). 17. (.T + a)(ar-a)»(a:» + a«). {(a - 6)a: + {a + %} {(a + b)x - (a - %}. (3ar+lX3ar-l)(9x+l)(x+l). {ax-bXx*-ax-b). 21. (x + a)(ar-a)(ar-2o). (a» + i»)(c* + oP). 23. {x+l){x'l)lx-^p){x-p). {a+b-\-c){a + h-c'){a-b + c){-a + b + c), (a + b + c){a* + b* + c'+2ab-bc-ca). 26. (a + 6)(a'+6« + l). {x-y-z){x'+xij + f). 28. (a;*+ary + y»)(.r'-a:y + y«+l). (a^+a:y+y»)(z*-ary + y» + aj-y). 30. (a'-a6 + 6')(a + 6+l). (a+6 + c+cf)(a-6-c + rf)(a+6-c-rf)(a-6+c-(£), (' + y)(^ + ^y+y')(«'-a:y+y'). 33. (a-i/(c-a)(c-6). {a-b)(b + \)b. 35. (x-y)(y-»)(2-.r). («-y)(y-«)(«-ar)(ar + y + «). 37. (a:» + y2)(a» + 6« + c»). (c - a)(«' + i' + c* + aft + 6c + ca). {a-b){b-c){c-a){a* + b'* + c^ + ab + bc + ca). (3a;* + 7xy + dy*){x* + 5.r»y' + y*). (a + 2/>» + c){a > 26 - c){a -2b + c)(a-2b- c). (a + 26 + 2c)(a + 26 - 2c)(o - 26 4 2c)( -a + 26 + 2c). {(a-6)x-(a + 6)y}{(a + 6)a; + (a-6)y} {(a-6)ar + (a + 6)y}{(o + 6)x-(a-6;y}. 1 6(wur + ny)(wia: - ny){my + nx){rny - nx). (2x+l)V+«+6)(2a:+3)(2a:-l)(ar + 2)(:r-l). , H 806 ANSWEBa EXERCISE XXXI. (Paob 68.) 1. aj'=2. 2. a:=2. 6. ar« -4. 7. x = 2. 3. ^•=1. 8. a: = 7. 4. .r=6. 9. ;f = 8. 5. ar — 3. 10. a: = a 11. ar-6. 12. a: = 4. 16. a?=-34. 17. ar = 3. 61 21. a: = ~. 22. ar = 2. 13. ar=10. 18. ar=15. 23. xr^a + b. 14. a;=-3| 19. a;=3. 24. x = 2a. . 15. x=|. 20.x=2/„. 2f5..= -« 26. a; = 0. 27. x = a+6 28. x = c - a -h. 9 29. a; =1. 30. ar=.-l. 31. a — b x = . a-2b 32. ar = a + 6 -c-d' 36. x=?^^-"*-*^ 37 a + c-2b •^'• a + h+t 3 ar= -1. '. 35. x = < 38. x = a^b -1. EXERCISE XXXII. (Page 70.) l.^«a, 6. 2.a:=-«, 6. 3. x = a, -J. 4. a:^-a 5. a: = 6, -c. 6. ar= -6, -c. 7. x=2, 6. 8. ^ = 2, 2. -6. 9. x=*a, by c. 10. a: =1, 2, 3. 12, x^a-b, -(a-b). 13. x= -a, 2a. 18. .-I.-!. 5' 6 21. a:=l, -1, -1. 11.0^ = 3, 1. 14. .r = 0, 36, -6. 17. a: = 3, -3. 19. ar=l, 1, 1. 20. a:=l, 3, 6. 1. 55, 45. EXERCISE XXXIII. (Page 72.) . 22 13* 3. 68. 6. 15 at lis. Gd.; 5 at 7s. 6d. 6. 538, 441. 8- 9. 9. 1400, $200, $100. 11.19,9. 12.40,20. 13.35,25. ^^- '^- 16. 177i*V miles from Toronfn «>. ^r.A «* -9)(x>-16). 14. (x-a){x-fA(,-cu 18. (ar»-l)(x»-4)(x»-9). 19. 4a»6«-(a» + i«-c»)« 20. 120.»y'(^-y)(a: + y).. 21. ^^-1). 22. a(.i :.«*)(. -a) EXERCISE XXXVII. (Page 86.) 1. (*-l)(«-2)(jr-3)(jr-4). 3. (2.r + 3)(3ar-4)(x» + 3x-l). 5. (9.r»-4y=')(4T»-9/)(:r + y). 7. (a:»+l)(.c»-a:-6). 9. (a^-l)(2a:»+a?-l)(i. + 3). 11. (.r+l)(x + 2)(a- + 3)(.r + 4). 13. (x+l)(a:»+l)(x-l)». 2. (ar-l)(x-4)(ir»-5,r-l). 4. (a;-l)»(a-' + .c+l)(ar>-3a:+l). 6. {3:-ay(x + 2a)(z + ia). 8. (ar»-l)(x»-9)(.r + 7). 10. x«-64. 1 2. aa:(a - ar)(a» - ax - Qx^. M t' If ' 1 ± • 2b- 1 11. 56»* a a + b' •«•£• 21. 25. 1 EXERCISE XXXVIII. (Page 88.) 2 1* 3c 7. — . o 2/n 4r 8. ^. 3;? 4. 9. 2aA« 3c • 5ry • 66c- ^6- 1<^- 3-^^- 12. :r— :: — — . 13. 1 17. 3a»-462 XJ/ ^ 14 ^ oa; - 5y a6 15 f^ 2(x + yy 18. -__. 19. 2a 2(x-y) a + 6 • -'-'« :: — . Zo, 1^' x' + a" ^^. 26. x'-2ax + 2a\ 27. ^ 1 S4. j-1— . 35. f. 0+c-a 6 3. 20. 24. 28. 53. a 2a»-3// ""• c"(^TI)* 1 a + 6-c a-b+c' /-'•)(a«+6«). -a*){x-a). r-1). :«-3ar+l). ia). ). 5yy 66c* ab' a c{x+\)' 1 a^ + a»* a + 6-c a-6+c' a; ANSWERa BXEROISE XXXIX. (Paob 90.) 809 «-4 2ar + 3 3. 1 4. x»-4' 6 ^^'"^Q^^^^ 7 *-2 ' a(5/^»-2a-6y iTi* 11. 15. X -6 J+5* 1 8. 12. c'--4a6* 3«3-2p* 19. 23. 27. r'-2x + 2 . 16. X-y a^-ax+b' ar' + aar-i* x-b bc-ac-ax' 20. . 24. ^-M-6 a:»+47~' 2a:+3 37^' 3ox»+ 1 4aV + 2aa;'-r + l (a~l)-ar'-(a+l)a:-l 33. (a + d){b+oy 30 ^(^ -'*''> ^1 34 %+^)+«('g-g) * b{y + z)-a{x-%y 32. 1. 35.?. EXERCISE XL. (Pagb 92.) 1. 6aj-3 + x+2' 3. 3(ar' + a:y + y») + , x-y 6. 2a:' + a;- 1 - 7. ar-l- 9. * - 1 - 2 2. a - 2a: + . a+x 4. a:' + a:y + y»H.-iL. jj-y 1 a:«-a?+r 7a?-4 3aj2-4a?+3* i^V+T' 6. 3a- 1 + 8. a--.l + 3a+r 6ar + 4 6a:«+4ar-l* 10. 1 l-«* 11. T-i-. 11 310 ANSWEBa 12. a+b' 16. ^J^tn, a-b 20. 24. 27. x + y 111 1-y- 13. 17. 21. t+y' x-S' 14. 18. 15. 19. 4ah a+b' l-x + x* -.. 22. 2a'-a6- y a+i ~* a:'-23ry-y« «+y a;"+ary+y2 a; + 2 aj-: x+a 23. ar-2' 3a + 05- 25. 5a 2a' 3x a x* 1111 29. -. + i4.i d + - 26. 28. 30. a l+b' a » J ^' «' ^ +- + -+-. z X y a 20 5'*"4~2" liC 31. (a - b)(m + n)~(a + b){m-n) -T. 32. -^ x-2y x + 1/' EXERCISE XLI. (Paob 94.) 20a» 24a; 21a: 30 ' 30 ' lO' abc* abo* abo' (a-i)« ai 24ar-9 12.r» ' .Q3» Ux" 2 + 2 7i* a j_ 6" a»-6»' a»-T» a;'- 16 a;y(a?-y)' xy{x - y)' ~~. 8. ^^(^ •*• ^) 70(a:-2) 48 30(a:»-4)' 30(a:'-4)' ZOi?^) X !«_, (a!-2)(a:-3)(a!-.4)' (a: - 2)(ar - 3)(a: - 4)' 10. •^('^+^)(^ + 2) (« + 1 )(ar + 3) a; + 4 (l-a:)(l+ar)' ar(l - a:)(l + x)' ^(iT^XTT^' 11. 12. 6'- 5. 2ab. 8 1 ^•26* 10. oc 11. (^^z%. 12. ?f^(izy). 13 6-^. 'x-3' 16. 20. 25. cue a*x h -a x^ + y^ x+y' 17. y 2x + y* 18. a" 2^* 6rf* 14. 1. 15. 1. 1 21.-1. 22.1. 23. m' - n*' 19. 24. ar» 12 + ar - 6r»' 26. "iJu^^t g' + aa; - 23;" a* + oar + a:* ' 1 *' ^ EXERCISE XLVL (Paob 104.) O, iu -j- ij -f- — -, 4.««-2 + 1 5. a:» + 64.1. 818 6. a:»+l - 7.aAa^-(a«-6V^. S-*'-^. 9. «• + ^ «:» a: 1 J* b* 1 90 10*4.*j.*j.* 11** 1«3 1 »4:-4 6"' c" a' a 17. a*+~. a* 18. -+-.+^. 1 I^ ■ 3ay* EXERCISE XLVII. (Paob 106.) 6. 2. 6. 8a;y X 12j»a:'w'* 4 '- a' 6' c* a6 be ca 15 1 ^ b\x + a) 18. a ax xy y g(a + 2a;) J( o+ft) a" • • 3(a - &)«• ll.J-l+^. 12.«*-1. U.^-6:r»+f+9. 16. -:+:3. 37. a + ft - « 19. 1. 20. a - 6 4 c* 21. 3(a: + y). ah (a' - b^)x a + b 1. 6. 15 -4a; 20 -3a; 2x-25* EXERCISE XLVm. (Paob 108.) 2.Ji-. 3. ^^^-^^y . 4.?fezi). 7 -2a;* ^- 9, I. 10. *» %-y) 7 -(^±i).' • 2(a-6)«* 8. 12-a; 1+a* 2a; * M. 1_ a-6~( 12. a« + a;». 21 11! !: n 314 13. 16. arj/* AlfSWEBS. ae -hd {x-y)\x + yy 14. CLC + bd' ''■¥.- ai»-o* 17. l.X: 6. «> 10. ar. 15. Z' EXERCISE LIII. (Paob 124.) •19. 2.a:-2. 3.x-12f 4.:r-^^. 6.^-151. '3. 7. a:=3f. 8. x=-l. 9. ar = 2. 10. a: = 0. * -a 12.2.-3^. 13. a:-?. U.a: = 6. 15. :r = 3. EXERCISE LIV. (Paob 127.) '3J. 2. ar-2. 3. a; = 3. 4. or = 4. = g . ( A cu. equation.) 7. a? - - 6. 8. a: = 2. 3 '2- 6 0. ar=a -. a ®- ^=29" 11. ar-0. 12. a:- 12. 13. ar=-l. 14. a:=7. 2 -2. 16.a? = 8. 17. a:--. 18. a: = 2. 19. a:- 2^. 1. x> i. «. 7. «. 10. x^ 13. «« 16. a?- 19. ». a6 'a + 6* a» + 6» • c-a EXERCISE LV. (Paob 129.) cd-ab 2. a;: 3. a; = a>-6« a + b-c-d' "' ■" 4a-6' 5. ar-o + 6 + c. 6. ar = ^fell±i) a 8. a;->: a + 6 • g6(a + 6-2c) 9. a:«^!:Jli?. cp - ar 12. x=i-(a + 6 + c). 2a6 7« a + 6 14. a;i m-n 15. ar = a. a'c - ^. 17. ^„^!!it^!f±^-_«::ft^c «+*+« *' ab + bc + ca-l ' ^^- ^ 3 — • ■ or V-{ab + bc + ca). (This is not a simple equation.) EXERCISE LVI. (Paob 130.) «-»^. 2(fir + a6 + 5*) 3(a + 6) 3-2a 3a-4* 4. «. 2a6 ANSWERa 1 1. (Not a simple equation.) 6. : — 2|. 7. x 4 3* 5 3' 5 4" 9. ar--J. 10. ar - ab{a + b-2c) a} •{- b'* — ac — be ' 11. ar = - 11 3* 13. T: bc + ca — ab 14. X: 17. X: b-2c ' .5J. 18. x-a + ft. 19. X 6a ' 2" 45 . 20. a; -9. b + c 22. r« - a + 6 6c(w - n) + ca{n -p) + ab{p - m) a(7u - n) + i(w - /?) + c{p - m) ab-¥bc + ca a + bfc 2 ' :a+b + c. 24. X'^a + h + c. 25. j; = nbc + be* + a*c - 2a'i - ah* be + ac - ab + c^ -a^-U^ Sabe - a'ft -b'c — c^a a^ + b'^ + c^ — ab-bc- ca b^c\b + c) + c^a\c + a) + a'/>'((i + fi) - 2abc{ab + be + ca) '' 2abc{a + b + c)- ac(a» + c^) - bc{l/' + c") - ab{a^ + i-*) " m(&, y- l-a6' 26. a; » abc(ab +ac — be) ' a^^ + a*<^-b^c** abc(ac —ab- be) 29. x = a6(a + b) a« + 6» ' a6(a - A) 6« + c* - a« 27. » macb* ]. 2a c* -f o' - 6' Vb • 30. X b*c- a» ma^b bh- a? c{a- b) a» + b' c{a-\- *) ae a 32. X' 16, 33. a = 5, 34. a; = y = 4. y = 4. 383 "29"' a '+6» y = 93 29' 154 47' 66 y = 47- '^^•^ = 63' 37. a; = 8g, 38. a: = 2, 39. a; = a''' + 6* 35. a: =25, .v=13. 40. a: = 6, ' = 8. 42. x-=2. y = 6. 60. y = 8. 43. a; = y 44. X y 47. a: 18. p+l mp - V w+1 mp - I' m+1 ~mn+ r m(n + 1) y- 45. a; y- m* + mn + n' wi' - wn + n" w-n m4-7» 48. y X p m-n V a+1 6 • 6+1 46. a; = a + 6 - c, y = a + - 6. 49. a:: mn+ 1 V m + n m — » -6, a + 6. macb* l?c- a*' ma*b b'c- a»* c(a- 6) a» + 6»- c{a + *) ;5. a: =25, y=13. to. a: = 6, mn + n* % + 7i * mn + n^ -6. ANSWERa 819 50. x> m-n n ~ m y- . pn — qm 52. X'ma-{-b — o^ y^a-b + e. M ^_ ft(rf'-c>)-c(a«>^.») bd—ac * ^rf( a»-y) -a(cf»-c«) bd—ac 53. x=»7?» + n, 54. a:-(;,'fl)(^«_l), y-(/''-i)(?'+i). EXERCISE LIX. (Paok 149.) 1. ar=«2, 2. x=5, 3. «=!, 4. a: = 5, 5. «-20 2. y X- y y = 8. 7. a:=-4. 8. 11 *=1. y= - ga. _ - y=2, «-3. aj«=9, 4=13. y=6, y=10, 9. X' a + b ab + ac+bc-b*-2<^ 2(6»-c») 3be -ab -ac-b* 10. a: = i' + c' - a' y = « = • 2bc ' c' + g' - />' a» + 6» - c' 11. a;-6, 2/=-12, «-18. 2(6»-c») 12. a;> 468 T » 13. a; = 1 y=- 335 14. 2aA 15. « = ■ 16 a;: y^ + 6' a-\-c 16. a;=lj, 17. a;=-, 18. a;=L y=-3|, «=2t\,. y=i, 6+< y=o. «=-- :r. 1 ;i n 19. a; = 2, y=3, 20. X 21.a: = 5, 22. a: = 3, 23. a: = 4. y- y -4, y=-2, 2=1. y = 5» z- n 3X0 ANSWEllS. 24. a; y 12, «->24. 25. x»2, «»■ -6. 26. a;> y= a 6c 28. «. 2a5c 6c + ca - a6* 2a6e ea + ah — be* 2abc ab + bc — ca' 31. a;--ll, 9 29. aj = a6 + 6c + ca' aic ab + bc + ca* abe ab + bc + ca' 27. a: = a, y = ^ z = c. c + a a + 6 30. ae^n+p-mj y = m+p-n^ z^m + n — p. z> Z' 7 6* 32..= ^. 1 3 « = 1. y-y 33. x = a, a== -ft EXERCISE LX. (Paqb 156.) 1. 8 and 4. 2. 12Uand90. 3. 42 and 12. 4. f. 5- 63. 6. 26. 7. 54. 8.' 18 and 8. 9.^4 and $2. 10. 4 and 2f. 11. 25 and 55. 12. 5 and 6. 13. 90 cents per kilo, for butter, 75 cents per kilo, for soap. 14. $3.60 and $3. 15. $900 and $400. 16. 2400 and 1800. 17. $12.40 and $1.60. 18. 140, 90 and 130. 19. 854. 20. 640, 720 and 840. 21. 133 J lbs. and 100 lbs. 22. A\ 24s., and B's, 16s. 23. 90 min by A, 1 hr. by B, and 3 hrs. by all together. 24. 2 gals, from A, 14 from B. 25. Tea, 5s.: sugar, 4d. ^g^nm(^_-_l)daya. np — m 28. 25 from 1st, 75 from 2nd. 21? days, woman in 50 days. 27. A'8, £40; B\ £28. 32. H. 34. £21/v, £8U. 29. 65 31. 28 barley, 20 rye 30. Man in 2 wheat. 33. A, £6; B, £18; C, £36: A £48= 35. A, 5f hrs.: B, 6^ hrs.; C, 7^ hrs. ANSWERS. 821 36. Rate of pulling, 8 miles an hour; distance, 20 miles. 37. Time down stream, 4 hrs.; time up, 6 hrs.; rate of stream, ^ mile an hr. 38. 20 lbs. cheaper tea, 10 lbs. dearer tea. 39. 40 lbs. and 90 lbs. 40. 20 bush, of rye and 52 bush, of wheat. 41. ^ in If hrs., B in 3^ hrs., C in 7 hrs. 42. A in 20 days, B in 30 days, C in 60 days. ^"*'' min ^in_J^^^£_ • n- 2aic mm., n m -j— — mm., C in 43. A in ac-^hc- ah -•' -" he Arab- ac ' ^ '" ab + ac^c ™^^- 44. 12 ft. long, 9 ft. broad. 45. 70 yds. long, 38 yds. wide. 46. 14 ft. long, 10 ft. wide. 47. 4 gallons and 10 gallons. 48. 10 gallons and 4 gallons. EXERCISE LXI. (Paqb 166.) 1. 2a6, 4a''6*c*, bab\ iz^z, Ua'h*c\ 5a'6» ■l6cc« 25a 7a6V 3. x + 6, a?- 4, 2ax + b, 2. Ua^y^' 17y»' 186' 8c*cP' 4. x' + x+l, 2a;«-a;-l. 6. a^-5aa: + 4a«. 6. y«' + 4j;y + 7/». 7. x'-xy + y^ 9. ix'-2ah + 2b\ 10. 2.r»-5.r + 3. 12. 2a* + 4aV-4c«. 13. 2a3-a»-3a + 2. 15. bx^-Zx^y-iry^ + y^. 17. a + 26 + 3c. 18. a' + a^ + ah^ + b\ 20. 3a-26 + 4c. 21. 5xV-3ary» + 2y». 22. 2y^x-3yir + 2x^. 23. 5x-2y + 3z 24. 2.i!» + y»-y. 25. »»*-2w« + 3m'-4rn + 6. 26. 2a6 + a» + 6% 8. 2a'-3a»a;-oa^. 11. x^-2x* + 3x-i. 14. x^-2x^y\-2xy*-y^, 16. 22r»-3y2 + 4««. 19. 3-4a; + 7a;2-10r'. 1 EXERCISE LXII. (Paob 167.) 7. ^'•' + 4 + ^. ^^•3-4^5-3- 2. «'-2«^ + y»-^. 5..^-! + -. o a 8.1-?.?. X y z q 1^2 3 4 3. 1 + -+- + - a; «* a:* 6. 2a-36 + 6« 2a: « 3y 9. 1 . 322 ANSWERS. EXERCISE LXin. (Paob 169.) 1. 3a*-ab + 6b\ 2. o»-3a6 + 6». 3. a» + 6'. 4. aV+3ox+l. 6. a^ + ab + ac + bc. 6. a' + 6» + c» - 3a6e. 7. (a' + ^.')(c« + c/^). 8. a'-b^ + c'-* + 4c*. y 24. 4+yM--iL. a^ 2y i" 25. X-2-X-K 28. ar" 1 x'+V xeM + ye* 29. 3a-h - 4 27. -(a6)-H'. a* + 1 - yi 30 31. 34. 37. a'-6» x^. cc* + 2 - 3y* 32. (a'-iy. 33. 4(aiz»*+26). 40. 44. 35. a?*^«-n - y9(p-»). 38 ?fl^!±l) aVO-'aV-aV + l ' (l+a:)^' • 42. a:'»+»+«. 8 9' a^x- 1 45. 11. 46. 2(.r+l) x' + ar + r 36. a"-i. 39. 1+x*. 43. 1 + 2-*. 47. I. EXERCISE LXVII. (Page 187.) 1. xyW7, 2a V2^, Say ^2^', 2^6, ba^dV^kl 2. 10^ a, 2xi/*\'207^/, 3mVv'4^, 7a'b'^Ib. 3. {a - b) ^a, 5(a - 6) V 2. 6. Vis. 1?'66. 1^406, f I125, ^96. cy. Nt i»i '^ I i 324 ANSW£B8, 7. V9a, Vi8a\ ^, Vs^. 8. -/^TT^, ^lll*, iZL ' Va-6' Njor + y* 9. iacv'dabV, 27y\^S^, 55 v'?. 10. 1^2^^, ~^8^, ii.. * Jo « 11. ahxVx, 2a^*x4'b\ EXERCISE LXVIII. (Paob 189.) 1. ^HE, aTTg^ 2. iTe, 1^49. 3. '^I6, V'27. 4. 'l?'256, 'f 5l2. 5. '^64, •i?'625, '^2l6. 6. '1?'7^ 'v5^ •^(120)'. 7. '^S^' T6^^ VW. EXERCISE LXIX. (Paob 190.) 1. 3 VT is the greater. 2. Gv/y, 5\/l0, 9VT. (Descending order of magnitude.) 3. 5 I?' 3, iv'i, 3v'5. (Descending order of magnitude.) 4. 61?^ 7, 4^9, 3v^l8. (Descending order of magnitude.) EXERCrSE LXX. (Page 104.) 1. 5\^3, 7VU. 2. 10 a/ 7", ISv'lO. 3. 2^/11, 6^/3. 4. 3^3, 4\/2. 5. 6v/4, 5i?^2. 6. IIV'7, 1^5, 7. -^2. 8. 12v/3. 9. 129 i/5. 10. -4V^3: 11. (0 + 5)1;^^ 12. (a~b+2)v^I. 13. (a^ + b'-2ab)\^z 15. (4a7>»- 2406 + 36)^26. 73 31 ^^•I8 + I5^^- 1«-^^1^- 2a» 14. 10«7V7a/>. 16. -^iJ^I. 19.^^6: 20. a — x a/ '1 -• V a- — ar. 21. (i +u -♦- 2x) V''a + J 'b*x4'b\ ^27. 216. tude.) tude.) tude.) 6v^3. r. Ix) Va + x ANSWERS. EXERCISE LXXI. (Paok 195.) 325 1. 9, 10, 6. 6, 72, 25. 9. VI -i. 12. 36-15^/6. 2. 3,4,8. 3. 3,6,2. 4. 11,27. 7. 10. 8. 4. 10. 10-2^/2. 11. -15-19V/3: 13. 6v/2'-3v'r5 + 8v'3-6v'l0. 6. 60, 1 4 14. 8-8v^l2+ v^l8. 15. \/2l. 16, a/7. 18. 5. 19. 4. 20. 4. 21. 5. 47 — 23. 86 + — \/6. 24. x*-x'+l. 25. 2ai + 2fic + 2ca-a«-5»-c«. 17. a/111. 22. 30. 26. 1?'2000, 5 a/30 Ifij 1 2o/7 \/l44' N/~ 28 x625 729 27. a/40, 125A^7». 29. 5" [2401x3' \23x3''' EXERCISE LXXII. (Paob 197.) 1. 9. 2, 3. 2. 3, 2, 5. 3. I, ? ? _ 6' 3M' 4. A/2 + 5A/6. 5. 7A/3-3A/5. 6. 4a/35 + 8a/I4. 7. 2A/42-3A/n + 5A/l0. 8. 2^25-6 V^4 + 5f 36. 9. 5 iJ'tB + 3 v'l8 - 4 a/36. 10. a, 2y. 11. a/6+1, A/a+ a/*; 1 +j/^ 12. a-A/^+6, axV7-hyVy EXERCISE LXXIII. (Page 201.) 1.2- A/a 2.1+ a/3. 3. 5+V2: 4. 5-V/2. 6. l+v/F. 6. I-a/S. 7. 2+ a/6. 8. 4- a/5. 9. A/a+v'! IO.o+a/6: 11. A/iTi+A/^Ti. 12. Vo2 + a6+6*+ Vo'-o6+d«. 13. |.^V3. I 826 14. ?-ivTo 5 10 16. ANSWERa VT+2 16. ?.^V5: Vs ' ^^' *-_;^*'- 18. (a-b)-2Vah. EXERCISE LXXIV. (Page 202.) 1. VY+XTS, VU+V5, 2\/3-\/2. 2. VlO-VS, 2^5"- 3V2, 3V'5'-2V3. 3. 3 + V6; 2VT-2V'2; 3-V:2. 4. VY-V'l Wf-W% 2V3 + V5. EXERCISE LXXV. (Paqb 204.) 1 ^ + ^^^ 8 + V55 28^5 - 60^/2 +15VI0- 35 3 •_ 3 ' 55- . 2. ?i±llVl^ 20V^3-12v/5- ?1^.-_11VI 10 2 • 3 4-.\/l5 ^^ - ^^1^ 10VT+5y 6 - 6\/2 - 6 ;_ 3 • -57— • 4. ^^+^^^^ 4+\/r5 ^^ + 20\/3 +9\/2+ 15^/6 5. 6. V2 + \/6 -2 , V2 + \/3, \/5 + 2l/2-\/l0-2. 195 + 137\/6 - 75^/3 - 187\/2 94 8V/10+ 8V/15 - 22 -5VJ0 -joyj + I9i/T+ Tfi^T o + Va» - «« o» + \/a* - // X i» 8. i; + v/r»-9a* 2a» + \/ar'(4a» - a^) 3a ««-2a» 9. ANSWERS. , : 327 2 3i|'4V _ 4V.3i ^ 64.3* - iKz^ + 4^.3 - 16.3* + 4».3» \ i_ -4*.3^ +4x9- 4^.3* + 4*.3* - 3^^ 229 10. (2\/2-\/3)(3^ + 3.5^ + 3i5U 5I) 4 18 - 34\/5^ ^j ^ ^^ • 57\/2- 2 5v/3-9VT 11 13. 6. « + « ' 13i-5a "• EXERCISE LXXVI. (Paob 208.) .bo ' 7' 6- 4. 10. 8. 1. 2. a> i. a* 5. I + V2. 3.2.^. /4a -d l~l" 6. .2679492 10 7.5^-2 • -2{^y+-)- 11- c, \. xy} EXERCISE LXXVII. (Paor 210.) 1. 25\/^, 5\/3T, 9\/"^, 12^", feV"^. 2. 4^"^, 8\/^, 3r'V"^, 9mV^. 3. -1,1, -\/"rT, 1, V"^. 4. \W~^. 6. 16 4-4V''=1. 6. 18 + (3 + 26)\/3l. 7. _i2, _35, - 122/s». 8, 47 -V"^, 9x'+24-15.rV^. 9. 2ac+36c?+(36c-2ac?)\/~, mn + 6»+ A(n- m)v/^. 10. -aV'^, -V~, V^xV'2. 11. 0. 12.0. 13. - 7 wmmmmm'mmm 328 9. x = 3J. 13. a; = 25. 17. a;=25. 21. a: =25. 26.«=?. 4 29. ar-2^. 33. xm.-2a. ANSWERS, EXERCISE LXXVIII. (Paob 215.; 2.^ = 3. e.x^-. 10. x = 8 37. « = a i 5' 14. ar = 9. 18. a; = 9. 22...'. 26. a; = 0. 30. a:=l. 34. ar = 25. V3 o. ar = — -. 4 7. ar»=;r;r. 20 11. ar = 5. 15. a; = 5. 19. ar = 64. 23. ar=100. 2275 1 4. a:.-. 8. «=» 1 3* 12. ar = 49. 13 16. x = 20. ar = 9' 36 27. ar=:- 88 31. ar= -. 4 35. a; = 4. 24. a: = 64. 28.. = ^'. 2a -6 32. a;»2a. 36. a: = 4. a* +2 38. a; = --— . (Pure quadratic.) 39. a:= -. 40. af«i3. (Pure quadratic.) 41. ar 43.«--^». 3- *2.a:=--. EXERCISE LXXIX. (Pagb 221.) 1. 2(a» + 6' + c»+a6 + 6c + ca). 2. 2(a'+A» + c»-a6-ic-ca). 3. a' + 6'+c' + 3(a6 + Ac + ca). 4. ab + bc + ca-a^-b'-c\ 6. a6 + 6c + ca-a»-6»-. 6. a^ + b^ + c^-ab-bc-ca. 7. 3(a' + 6» + c«)-2(a6 + ic + ca). 8. 3(a' + 6» + c'')-2(a6 + 6<; + ca). 9. 0. 10. 2{ab+bc+ca). 11. 0. 12. 0. 13. 0. 14. 0. 15. 3. 16. 0. 17. 0. 18. 0. 19. ^ 20. Sx' + 2(a + b + c)x+ab + bc + ca. 21. Sx*-2(a + b + c)x + ab + bc + ca. 22. (o + 6 + c)x' - 2{ab + be -t ca)x + 3a6o, 1 1 r = 49. 13 36 = 64. 2a- 6' »2a. = 4. 9 8 ic — ca). —ca. b + bc + ca). 13. 0. 19. a^ ANSWEBa 329 28. 2(a+b+o)x'-{ab(a + b)+&c.}. " ^ 30. a6c{ic(6-c) + +6>+ + 6«+c>+a6 + 6c + c(/) 2. -(«-*)(6-c)(c-a)(a»+6»+o«+a6 + 4c + ca). 3. a> + 6« + o«+a6 + 4c + ca. 4. «'+^' + c'+ a6 + &c + ca 5. «*-Hy + c* + a6 + &c.fca (« + 6)(6 + c)(c + a) 6. (a + b + e)\ \v -r «/(f + C/{C + a) "V" -r- •■' -r o/.r + a- + 0' + c* l-T. abe (a + b + e)* nil in 1. Ill lUliHlHIIpp I 330 ANSWERS. 10. a6 + fto+ca-o*-6'-c*. 11. ^^(a' + i' + c'+aft + ftc+co). 12. \x. ' a+6+c 13. X'^ -{a*-¥l^-\-o^-\-db-\-be+ea). , 16. «-o" + 6* + o"+a6 + 6-\rbc-k-ca EXERCISE LXXXIV. (Paok 231.) 1. 3a6(o+6). 2. 5a6(a + 6)(a* + aft + 6«). 3. 7a6(a + A)(a" + a6 + 6«)«. 4. - 5a6(a - 6)(a« - aft -f 6»). 6. -7a6(a-6)(a«-a6 + ft«)«. 6. 7a6(a - 6)(a« - a6 + ft^. 7. 3(«-y)(y-«)(«-x). 8. 7(« - y)(y -«)(«- «)(a^ + J/" +s«-«y-y« -«*)". 9. (a-6)(6-c)(c-a)(a + 6 + c). 10. {a-b){b-e)(o-a){ab-k-bc-{-ca). 11. 2(a!» + a^ + y»)". 12. 2(«" + y» + a» - «y - ya - «ar)». 13. 3(a + 6)(6 + c)(c + a). 14. 5(a + 6)(6 - c){a - c)(a' + 6' + c* + oft - 6c - ca). 15. 24arya 16. 3(2a! + y + «)(« + 2y + «)(« + y + 2»). 17. -3(a-c)(6-d)(»-* + «-^- JtlHTft ^^\yff vr\y^ ^vf\"v •-■ • »- ».^-.«- - " - « • -t w»v w v^v -tm^^r* 24. 16(6 - c)(c - a)(a - b){x - a){x - 6)(« - c). + be-¥ca). ab+be+oa). h + be+ca. f ca • > + e){e + a) • be + ea. a + b + c Ut + bc + ca' 6 + 6»). - ab -f 6«). h + b')\ 15. ^^xyz ANSWEBa ^1 EXERCISE LXXXV. (Paok 232.) 1. (a f 6)(6 + c)(c + a). 2. (a + 6 + c)(a6 + 6c + ea). 3. (a4-6)(6+c)(c + a). 4. (a + 6)(6 + c)(c + a). ^' 6a6«- 6. (a + i + fl + a6c)(l+a6 + 6c + ca). 7. (a + 6 + c - aftc)(l - aft - 6c - ca). 8. 2(a + A 4- c - aJc)(l - aft - Jc - ca). 9. 2(a + ft + c)(a» + />2 + c''-aft-ftc-ca). 10. (a + ft + c)(a« + ft' + c' - aft - ftc - ca){x + l)(i-» - a; + 1). EXERCISE LXXXVI. (Paob 236.) 1. a- -6, 6 = 9. 2. a = 4, 6-2. 3. a-20, ft-85. 4. ar=10. 5. ar = 4, 6. 6. w»=.12. 7. «■ W 8. x-0, -4a, ^^^^' + ^') 2^(«J-^'') 7ft' -g' 2wj'-J9' 76«-a» ' 9. ar. c-q{a-p) 'p(a-p)-{b-qy 17. 8c = a(46-a'). 64rf = (46 -«»)«. 20. a= ~6, a=. -14; 6 = 2§, 6 = 0. V3- ' 12a '*°- 10. 6«=-o " • c Va* 4- i V c 1±V73) 8 SSd 83. a-a±2j— . 34. ar-o, &. 85. «-0, -1. 36. x-6, - 37. x^l(-ll±VU). 38. x = 0, 37. 39. a; . 40. «-l, . 41. a;- ±a. 42. a?-0, ±Va* + b'. 43. «- a+6 a+6 2 ' 2 * 44. a:»{a« + 6» + c» + 3(a5 + 6c + ca)} - 3x(a + 6)(6 + c)(c + a) + a»6» + 6 V + c»a» + 3o6c(a + 6 + c) = 0. 46. «a -a, -ft. 47. a;-0, -r- — . a + b-^-e 49. a:-0. -T^^. a» + ft* + c* 61. 1. 4S -±V^+2^. ±J^'. 48. a.=o. ± JEMBEEE5 ' S/ 2a-6-c 50. ac^ + b

T A 4, g- -. 23. a;-*0, a, a±V5a'-8a6 1, - = 3, 11 24' 2 3* 27. X: 26. ar = 2, - 16|. 1 5 2' 7" 30. a;=±2. 2a '±-7=. V6 ±26. 28. ar = 3, -3?. 31. a:-±i. 34. ^=±3. 33. a;=±2. 36. a: = ^(l±Vr+l6^). ^a 38. ^^^ ^±V2) ^ ^g _^^^^^___ a fa' -4 .^ 41. x=^±-^-^—j. 42. a; = 0, a. 2a 44. a:«=±— 7==. 46. a; — a, 6. V-3 ANSWERa S36 50 21* -30 ) V357. :, ±V23. ■ 663 -Sab -3?. \/l)'-(U>. g. a-b , (a + b)e 2 2Vc»+4 Q a + b a-b ,^ — - *®' '-- 2-=*=-27^'^+*- 47. :r-^*±i^Z*^. ^ 2\/c'-4 2ab 60. ar=± 62 V3' 49. « -. a + b 61. a:-±V6. 63. 2;a>9a, -a. 54. «= = where A= - - ± . 2 2 A»±ifcVA:' + 4a 2 6±V6*-2a6 65. X 67. aj»-^(lTV5). 66. x-55±24'/5. »n *» 63 68. :r-iO, 65 69. a; = 2-V3; 2(\/3-2). 60. ar- ^(- liVT). EXERCISE XCIII. (Paob 267.) 1. «=10, 30; y-30, 10. 4. a;«25, 4; y-4, 25. 7. a;=10, 2; y-2, 10. 10. a;«13, -3; y-3, -13. 12. «-4, -H; 13. x = 6, -^; 2. a; = 4, 9j y = 9, 4. 5. iB«12, 6; y = 6, 12. 8. aj=.20, -6; y-6, -20. 3. a: = 50, -6; y-5, -50. 6. «=18, -3; y-.3, -18. 9. aj-40, 9; y-9, 40. 11. x--|(5± 1/266), 42 35±V266 24 — 14. x=- 10±VT38 y-ii - j- „ 33 y-3. --. y 30±Vl38 6 • i ■I 886 15. c.S, 3; y-3, 3. ifi 1 1 1 1 y"3' 2* ANSWEBS. ,- 3±\/l77 ,, 16. *- -^ , 17. a.„4^ 2; y- 42 9±Vl77 t;^; y = 2, 4. 19. ». 24 • 6± 61/67 y--3±V67. 20..= ^,!; 1 1 y=4»3- EXERCISE XCIV. (Paot 270.) 1. a; = ±x, ±3] y s. « 13 .« -±3, T 2. a^-iS^^ 4. «=±7, q: V2 5. a;< y 7. «. Ji ±n, ±2 ±3\/3. ±4j ±^3, ±5. ±3, ±5 y-±2, ± N/2' y-±2, ±--=. V2 6. a5=±2, ±1/2; y=±4, ±31/2. 8. a;=±3, 9. x=±2, ±2Hj y=±4. y=±3, ±2A. 10. X' y 13. «. y ±3, ±1. ±6. ±5. 11. «-±7, 12 aj-±6, ± y-±5. y=-±5, q:ll ^2' U. aj-±l, ±-T^; 15. a;-4, -3j y-3, -4. y-±2, ± V2 3 V2 ANSWERS. 337 16. X: y 19. X' y 22. X: y- 25. X. y- 2 J. X-- y ±6, ±3. 17. «= ±2, ±^; 18. aj« ±2, ±-3->J-7 J y=±5, ± 7 5 y-±l, t6 N 7* ±12, ±9; 20. a;=±4, ±-5-; 21. «-±-— , ±7; + 9 +12 ^ ^2 * • ^ 19 9 y=±9, ±-3. y«±__, ±2. ±8, ±2; 23. rc=±6, ±2, ±8. 2/= ±2. ±4, ±2; 26. x=±W\ ±2, ±4. y=±V2". 3, 4, -6±2V'6; 4, 3, -6:f2V6. 24. a;=±3V3, ±4; y^±V\ ±6. 27. a; = 0, 15; y = 0, 45. EXERCISE XCV. \2' 1. 10, 11, 12. 2. 3, 4. 5. 31^ cents. 6. $11.25. 9. 121 and 120 yards. 11. 30-10^/5 and 10>/5'-10. 13. 5 hours and 3 hours. 3. 7. 10. 12. 14. 18. 16. 78. 17. 15, 20. 20. $60. 21. 100 shares. 22. 24. 34,43. 25. \{2>±V'^\ -^ 27. 16, 10. 28. 9, 12, 15. 30. $6000, $7000; 7 per cent, and 31. 20 barrels by i4, 16 by 5. A's 32. 12 inches. 33. 12, 16,20. 34. 28 (Page 273.) 1, 2, 3. 4. 12. 12 pieces. 8. 12 inches. 1^ yards. 7 hours and 5 hours. 39. 15. 35. 5 miles per hour. 19. $80 5 per cent. 23. 7, 5. (1±V^5). 26. \±Vi. 29. 5, 4. 6 per cent. I price, &\ 15s.; E's, £1 14h 8 and 10 ft. 35. 2 '^ ■J r V 838 ii ANSWERS. EXERCISE XCVI (Page 288.) 4 «i(^ + c) + bi{c + a) + ci(a + b «i + 6i + c, »-M(^)'-^}^'-''.^-4^^-^^'-^} + 1 = 6. a;« + --ar--— =0. 12. -«»_ 32 512 «"- 47»nar - (m»-n')» = 0. .5^ ^ J (^'-2acX^ l+c;)l /6'-2acy^^^ 15. 26»=9ac. 16. ^»-3 = 0. 17. a^ - 2mar + m« - n^ = 18. {cHc-ac,y = {aib-cb,){b,c-bc,). 20. m = 8. 29. m=6 - ' 3 30. Roots of x^a + b + c)^ + Zx{abc) + abc{a + 6 + c) = 0. EXERCISE XCVII. (Page 294.) 1. An identity, true for any value of x. 2. No value of x can satisfy the equation; it is impossible uu- 3. X:= a = b or c. a + b +c 4. True for any value of x. 5. Impossible unless (m + n)(a + b) = 0. 6. Impossible unless a + b + c = 0. 7. An indefinite number of solutions may be given. One solu tion is :r = 5, 2/ = 10, « = ,5. 8. x = 3, but 2/ and z are indeterminate and may have any values so that 2y+3z^ 7. One solution is ;r ^ 3, y = 2, « = 1 . 9. Impossible. 10. Impossible unless m + n+p = 0, and then an indefinite num- ber of solutions may be given. 11. The second equation is simply the cube of the first U + 1 = 0. ssible uu- One solu ly values ito nuni- I