CHAPTER VI. DIVISION. 46. THE object of division is to find out the quantity, called the quotient, by which the divisor must be multiplied so as to produce the dividend. Division is thus the inverse of multiplication. quotient x divisor=dividend, dividend - divisor=quotient. It is sometimes better to express this last result as a fracthus dividend quotient. or tion; DIVISION OF SIMPLE EXPRESSIONS. Example 1. Since the product of 4 and x is 4x, it follows that when 4x is divided by a the quotient is 4, or otherwise, 4x = x=4. 27a5 27ααααα Example 2. 27a5 : 9a3 = = 3aa, by removing from 9a3 9aaa the divisor and dividend the factors common to both, just as in Arithmetic. Therefore 27a5 = 9a’= 3a?. 35aaabbccc Example 3. 35a3b2c3 = 7ab2c2= 7abbcc =5a2c. 47. RULE. To divide one simple expression by another, divide the coefficient of the dividend by that of the divisor, and subtract the index of any letter in the divisor from the index of that letter in the dividend. a ax-5 a - a= Example 4. 84a x3 = 12a4x=7a5-423–1 =7axa. Example 5. 77a-x3y4 =- Tax-y=11axy3. NOTE. If we apply the rule to divide any power of a letter by the same power of the letter we are led to a curious conclusion. Thus, by the rule a3 = a=a3–3=a'; a3 but also a3 = a:= =1, a3 .. a?=1. This result will appear somewhat strange to the beginner, but its full significance will be explained in Chapter XXXI. 48. It is easy to prove that the rule of signs holds for division. ab axb Thus ab ; a= =b. a - ab -6. -ax-b - a =b. -- a Hence in division as well as multiplication like signs produce +, unlike signs produce Examples. (1) 6ab: 2a=36. (2) – 15xy =- 3x=- 5y. (4) 45a662,4 : - 9a3bx2= - 5a3bx?. EXPRESSION. 49. RULE. To divide a compound expression by a single factor, divide each term separately by that factor. This follows at once from Art. 34. (2) (36a3b2 - 24a15 – 20a4b) --4a+b=9ab – 664 - 5a2b. - a - ab - ab ; - a= -a EXAMPLES VI. a. - Ax. Divide 1. 3.2.3 by x2 2. 27x4 by - 9x3. 3. - 3526 by 7.x3. 4. abx2 by 5. XPy3 by xay. 6. a423 by - a223. 7. 4a2b2c3 by ab%ca. 8. 12a666c6 by – 3a4b2c. 9. - a5c9 by - ac. 10. 15xby74 by 5x2y2z2. 11. - 16.0"ya by – 4.xy 12. - 48a9 by-8a3. 13. 35ali by 7a". 14. 63a7b8c3 by 9a5b5c. 15. 7a2bc by - 7a-bc. 16. 28a183 by - 4a3b. 17. 166oy.x2 by - 2xy. 18. – 504323 by – 5.xoy. 19. 22 – 2xy by X. 20. 2.3 – 3x2 + x by x. 21. 26 – 7x5 +4.x* by x2. 22. 10x7 – 826 + 3x4 by 23. 23. 15.20 – 25x* by – 5x3. 24. 27.26 – 36.205 by 9x”. 25. – 24x6 – 32x4 by - 8x3. 26. 34.cy - 51.0y3 by 17.xy. 27. a? - ab – ac by 28. a3 – aub – aj2 by a . 29. 3.23 – 9.x2y – 12xy2 by – 3x. 30. 4x+y+ -- 8.x*?y2 +6.cy: by - 2.cy. 31. - 3a + ab-Gac by - a. 32.cy-3.c°yt by - coy. 33. - 22 + xy + 12x by- &c. 34. - 205x3 + Zafc4 by ja’x. - a. DIVISION OF COMPOUND EXPRESSIONS. RULE. 1. Arrange divisor and dividend in ascending or descending powers of some common letter. 2. Divide the term on the left of the dividend by the term on the left of the divisor, and put the result in the quotient. 3. Multiply the WHOLE divisor by this quotient, and put the product under the dividend. 4. Subtract and bring down from the dividend as many terms be necessary. Repeat these operations till all the terms from the dividend are brought down. as may Example 1. Divide za +112 +30 by x+6. 2C+6)x2 + 11x + 30 ( divide æ, the first term of the dividend, by x, the first term of the divisor; the quotient is x. Multiply the whole divisor by x, and put the product 2% + 6x under the dividend. We then have 2C+6)x2 +11x + 30(xc 2? + 6x by subtraction 5x + 30. On repeating the process above explained we find that the next term in the quotient is +5. The entire operation is more compactly written as follows: 2C+ 6 )ca +11x + 30 (2C+5 5x + 30 The reason for the rule is this: the dividend may be divided into as many parts as may be convenient, and the complete quotient is found by taking the sum of all the partial quotients. Thus 22+11x+30 is divided by the above process into two parts, namely x2 +62, and 5x+30, and each of these is divided byx+6; thus we obtain the complete quotient x+5. Example 2. Divide 2422 – 65xy +21y2 by 8x – 3y. 8x – 3y ) 24x8 – 65xy + 2172 ( 32 – 7y 24x2 – 9xy - 56xy +2172 EXAMPLES VI. b. Divide 1. x2 + 3x + 2 by x+1. 2. 22 — 7x+12 by x-3. 3. a? – lla+30 by a - 5. 4. a?– 49a +600 by a – 25. 5. 3.x2 +10x +3 by X+3. 6. 202 + 11x +5 by 2x+1. 7, 5x2+11x + 2 by x+2, 8. 2x2 + 17x+21 by 2x+3. 9. 5x2 + 16x +3 by X+3. 10. 3.x2 +34x+11 by 3x+1. 11. 4.x2 +23x + 15 by 4x+3. 12. 6x2 — 73 — 3 by 2.0 – 3. 13. 3x2 + x – 14 by x - - 2. 51. The process of Art. 50 is applicable to cases in which the divisor consists of more than two terms. Example. Divide 6x5 -- x4 +4.23 – 5x2 – X – 15 by 2x2 – X+3. 2x2 – x+3)6x5 – 24 + 4x2 – 5x2 – – – 15(3.x3 + x2 – 2x – 5 6x5 – 3x4 + 9x3 2x4 – 5x3 – 5.x? - 4x3 - 8.02 - 10.x2 + 5x - 15 52. Sometimes it will be found convenient to arrange the expressions in ascending powers of some common letter. Example. Divide 223 +10 – 16a – 39a2 +15a4 by 2 - 4a – 5a2. 2 - 4a – 5a2,10 – 16a – 39a2 +223 +15a4 (5+24 – 3a2 10 - 20a - 25a 40 – 14a2+ 2a3 6a2 + 12a3 + 15a4 |