117. The actual processes of multiplication and division can often be partially or wholly avoided by a skilful use of factors. It should be observed that the formulæ which the student has seen exemplified in the preceding pages are just as useful in their converse as in their direct application. Thus the formula for resolving into factors the difference of two squares is equally useful as enabling us to write down at once the product of the sum and the difference of two quantities. Example 1. Multiply 2a +36-c by 2a-36+0. 2a+ (36–c) and 2a-(36–c). Hence the product={2a+ (36—c)} {2a – (36—c)} =(2a)2 – (36–c)2 [Art. 86.] =4a2— (962–6bc+ca) =4a2-962 +6bc--c?. Example 2. Divide the product of 2x2+x-6, and 6x2–5x+1 by 3x2 + 5x -2. Denoting the division by means of a fraction, the required quotient (2x2+x-6) (6x2–5x+1) 3x2 + 5x – 2 (3x - 1)(x+2) EXAMPLES XIV. b. Find the product of 1. 2. — 7y+3z and 2: +7y - 3z. 2. 3x2 — 4xy +7y2 and 3x2 +4xy +7y2. 3. 5x2 + 5xy - 9y2 and 5x2 – 5xy - 9y2. 4. 702 — 8xy+3y2 and 7x2 +8xy - 3y2. 5. 23+2x2y + 2xy2 + y2 and 23 – 2x2y + 2xy2 — Y8. 6. (x+y)2 +2(x+y)+4 and (x+y)2-2(x+y)+4. 7. Multiply the square of a +36 by a2–6ab+962. 1 1 CHAPTER XV. FRACTIONS. 118. THOSE portions of the present chapter given to general proofs of the rules employed in the treatment of fractions may be omitted by the student reading the subject for the first time. 119. DEFINITION. If a quantity x be divided into b equal parts, and a of these parts be taken, the result is called the fraction of x. If x be the unit, the fraction of x is called simply “the fraction,"; 4”; so that the fraction a represents a equal parts, b of b which make up the unit. a REDUCTION OF FRACTIONS. a ma By s ma 120. To prove where a, b, m are positive integers. b mb' we mean a equal parts, b of which make up the unit ...(1); b by mb. (2). mb But b parts in (1)=mb parts in (2); .. 1 part. ma .....Em... — та Hence we have the following rule: RULE I. The value of a fraction is not altered if we multiply or divide the numerator and denominator by the same quantity. An algebraical fraction may therefore be reduced to an equivalent fraction by dividing numerator and denominator by any common factor; if this factor be the highest common factor, the resulting fraction is said to be in its lowest terms. 24a3c2x2 Example 1. Reduce to lowest terms 18a3x? - 12a223 • 24a'c'r? 24acca 18a’r"-12a-x3 6a2x2 (3a - 2x) 4aca 3a - 2x 6x? – Sxy Example 2. Reduce to lowest terms 9xy - 12y2' 2x (3x – 4y) 2.c Ixy - 12y2 + 3y (3x – 4y) By Note. The beginner should be careful not to begin cancelling until he has expressed both numerator and denominator in the most convenient form, by resolution into factors where necessary. 6.x2 - 8xy a>.x2 - AC Reduce to lowest terms: ax 3. acx + c.x2 20(-2-3-y) 4. 5. 6. 100(a3 - a26) 4.22 +6xy' 5.x2 + 5xy +5y2 2 (2a2 – 3ax) 203 — 2xy2 (27-3y2) 9. 3x2 + 6x 5a3b+10a232 11. 12. 3a6% +0ab3 x?y+2x+y+4xy 3a1 + ab +6ao12 13. 14. at+ab - 2a" 2.x2 +17x+21 15. 16. 17. x+ – 2x2 - 15 23 — 43 3.6- +26.0+35° a2x2 – 16a 3.x2 + 23.0 + 14 27a+at 18. 19. 20. axa +9ax + 20a' 3x2 +41x + 26 18a - 6a: +243 10. 121. When the factors of the numerator and denominator cannot be determined by inspection, the fraction may be reduced to its lowest terms by dividing both numerator and denominator by the highest common factor, which may be found by the rules given in Chap. XII. 3.x3 – 13x2 + 23.0 – 21 Example. Reduce to lowest terms 15x3 – 38x2 – 2x +21" First Method. The H.C.F. of numerator and denominator is 3x – 7. Dividing numerator and denominator by 3. – 7, we obtain as respective quotients x2 – 2x +3 and 5x2 - x - 3. 3.x3 – 13x2 +23x – 21 (3x – 7) (x2 – 2x + 3) x2 - 2x +3 Thus 15x3 – 38.x2 – 2x + 21 (3.x – 7)(5.x2 – X – 3) 5.ro -3° This is the simplest solution for the beginner ; but in this and similar cases we may often effect the reduction without actually going through the process of finding the highest common factor. Second Method. By Art. 97, the H.C.F. of numerator and denominator must be a factor of their sum 18x3 - 51x2 + 21x, that is, of 3x (3x – 7) (2x – 1). If there be a common divisor it must clearly be 3x – 7; hence arranging numerator and denominator so as to shew 3.x – 7 as a factor, «(3x – 7) – 2x (3x – 7) + 3 (3x – 7) 5x2 (3x – 7) – x (3.c – 7) – 3 (3.x - 7) 5x2 - x - 3• 122. If either numerator or denominator can readily be resolved into factors we may use the following method. x3 + 3x2 - 4x Example. Reduce to lowest terms 7c3 – 18c+ 6x +5° The numerator=x (x2 + 3x – 4)=x (x+4) (x - 1). Of these factors the only one which can be a common divisor is x-1. Hence, arranging the denominator, the fraction = x (x +4) (x - 1) x (x+4) (30 – 1) X (2C+4) EXAMPLES XV. b. 203 – 5x2+72 – 3 1. 2. a+3a2b+ 3ab2 +263 * 203 – 3x+2 a3 + 2a2 – 13a +10 2x3+5x4y – 30xy2 + 2743 3. 4. autaa – 10a +8 4.x3 + 5 xy2 – 2143 4a3 + 12a2b - ab2 – 1563 1 + 2x2 + 23 + 2x4 5. 6. 6a3 +13a2b-4ab2 – 1563" 1+3x2 + 2x + 3x4 x2 – 2x + 1 3a3 – 3a2b + ab2 – 73 7. 8. 3.x3 +7x – 10° 4a- 5ab +62 4.x3 + 3ax2 + a3 4.2.3 – 10.x2 +4x + 2 9. 10. 24 + ax3+ac+at 3x4 - 2.0,3 – 3.0 +2 16x4 – 72.c-a2 +81a4 6x3 + x2 – 5.0 - 2 11. 12. 4x2 + 12ax +9a? 623 +522 – 3x – 2 5.23 + 2x2 – 15x – 6 4x4 + 11x2 +25 13. 14. 7.23 - 4x2 – 21.0 +12° 4.24 – 9x2 +30x – 25 303 – 27ax2 +78a2x - 7243 aX3 – 5a x2 – 99a3x +40a4 15. 16. 2x3 + 10ax2 — 4a2x - 4823 X4 — bax3 – 86ax2 + 35ax® MULTIPLICATION AND DIVISION OF FRACTIONS. a ac 123. RULE II. To multiply a fraction by an integer : multiply the numerator by that integer; or, if the denominator be divisible by the integer, divide the denominator by it. The rule may be proved as follows: (1) o represents a equal parts, b of which make up the unit; o represents ac equal parts, b of which make up the unit; and the number of parts taken in the second fraction is c times the number taken in the first; that is, ī 7. a ac XC= a |