DIVISION OF A COMPOUND EXPRESSION BY A SIMPLE EXPRESSION. 57. Rule. To divide a compound expression by a single factor, divide each term separately by that factor, and take the algebraic sum of the partial quotients so obtained. This follows at once from Art. 40. Ex. 1. (9x12 y + 3 z) ÷ − 3 = − 3 x + 4 y — z. Ex. 2. (36 a3b2 — 24 a2b5 — 20 a+b2) ÷ 4 a2b = 9 ab - 6 b4 — 5 a2b. Ex. 3. (2 x2 – 5 xy + 3 x2y3) ÷ − † x − − 4 x + 10 y — 3 xy3. = 58. We employ the following rule: Rule. 1. Arrange divisor and dividend according to 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 as may be necessary. Repeat these operations till all the terms from the dividend, are brought down. Ex. 1. Divide x2 + 11x+30 by x + 6. Arrange the work thus: x+6)x2 + 11 x + 30( divide x2, 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 x2 + 6x under the dividend. We then have 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: x+6)x2 + 11x + 30(x + 5 6x+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 a2+11x+30 is divided by the above process into two parts, namely, x2+6x, and 5x +30, and each of these is divided by x+6; thus we obtain the complete quotient x + 5. Ex. 2. Divide 24 x2 - 65 xy + 21 y2 by 8x-3y. 8x-3y\24x2 - 65 xy + 21 y2(3 x − 7 y. 24x2 9 xy -56 xy + 21 y2 56 xy + 21 y2 16. 12 a2-11 ac-36 c2 by 4 a-9 a 8. 2x2 + 17 x + 21 by 2x + 3. 18. 7x+96x2 - 28x by 7x-2. 20. 27 x3+9x2-3x-10 by 3x-2. 19. 100 x3-3x-13x2 by 3+25x. 21. 16 a3-46a2+39a-9 by 8a-3. 59. The process of Art. 58 is applicable to cases in which the divisor consists of more than two terms. Ex. 2. Divide a3 + b3 + c3 - 3 abc by a+b+c. a+b+c) a3 − 3 abc + b3 + c3 (a2 — ab — ac + b2 — bc + ♣ ab2 - 2 abc NOTE. The result of this division will be referred to later. 60. Sometimes it will be found convenient to arrange the expressions in ascending powers of some common letter. Ex. Divide 2 a3 + 10 — 16 a — 39 a2 + 15 a1 by 2-4 a-5 a2. 2 - 4 a − 5 a2)10 — 16 a − 39 a2 + 2 a3 + 15 a1(5 + 2 a − Sa2 61. When the coefficients are fractional, the ordinary process may still be employed. Ex. Divide 3 + √21⁄2 xy2 + 11⁄2 y3 by }x+ }y. } x + {v} } x2 + √y xy2 + + z v3 ( } x2 - } xy + y2 In the examples given hitherto the divisor has been exactly contained in the dividend. When the division is not exact, the work should be carried on until the remainder is of lower dimensions [Art. 29] than the divisor." 7. aa + Ba3 + 13 a2 + 12 a + 4 by a2 + 3 a + 2. 10. 25 - 424 + 3x3 + 3x2 -3x+2 by x2 - x-& 11. 30+11 x3- 82 x2 5x+3 by 2x− 4 + 3x2. 12. 30y+9 71 y3 + 28 y1 — 35 y2 by 4 y2 – 13 y + 6. 14. 15+2 ma . 31 m + 9 m2 + 4 m3 + m5 by 3 - 2m - m2. 15. 2x3-8x + x2 + 12 − 7 x2 by x2 + 2 − 3 x. 18. 14x2 + 45 x3y + 78 x2y2 + 45 xy3 + 14 y1 by 2 x2 + 5 xy + 7 y2. 19. x - x1y + x3y2 − x3 + x2 - 20. x + x1y — x3y2 + x3 − 2 xy2 21. a9 b9 by a3 - b3. ys by x3 x -y. + y3 by x2 + xy — y2. 22. xy by x2 + xy + y2. 23. x7-2 y147xby+ — 7 xу12 + 14 x3ys by x-2y2. 34. 36 x2 + y2+1-4 xy-6x+y by 6x-y-. 62. Important Cases in Division. The following examples in division may be easily verified; they are of great importance and should be carefully noticed. |