CASE II. 51. When only one factor is a monomial. 1. Multiply 8 + 5 by 3. 24; but it is not the sum of 3 repetitions of 8 that is required, but of a number 5 units less than 8; 24, therefore, will have in it the sum of 5 units repeated 3 times, or 15, too much; the product required, therefore, is 24 Therefore, 15. The product of the sum is equal to the sum of the prod ucts, and the product of the difference to the difference of the products. Multiply each term of the multiplicand by the multiplier, and connect the several results by their proper signs. 6. Multiply 5 m n + 4 m2 — 6 n2 by 4 ab. 7. Multiply 16 a2 x 8 x z + 4y by 3xy. 8. Multiply b3-cx2 + dx by — x3. C 9. Multiply 63 x y - Ans. 252 x y z +56 x z + 24 z2. 10. Multiply 14 a 13 a3 + 12 a2-11 a by. 4 a3. 11. Multiply x2a+14 by ax. 12. Multiply 17 ax 14 by + 11 cz by 4abcxyz. 13. Multiply 21 a2 b2 — 3 x y2 — 4bc by 9axy. CASE III, 52. When both factors are polynomials. 1. Multiply 74 by 53. Multiplying 7+ 4 by 5 is taking the multiplicand 3 too many times; therefore, the true product will be found by subtracting 3 (74) from 5 (7+4). a times x х ay; but - y is to be taken, not a times only, but ab times; therefore, a (xy) is too small by b (x − y) ; and the product required is a x — ay + b x - by. Hence, RULE. Multiply each term of the multiplicand by each term of the multiplier, and find the sum of the several products. Ans. aa2b2 + b2 c2 — c1. 6. Multiply 2-2xy + y2 by x2+ y2. Ans. x2xy + 2x2 y2 - 2 x y3 + y*. 7. Multiply 4 aa — 2 a3 b + 3 a2 b2 by 2 a2-2 b2. Ans. 8 a 4 a5 b 2 a1 b2 + 4 a3 b3 8. Multiply -6 a2 b4. + 2 x3 +3x2+2x+1 by x2-2x+1. 9. Multiply + y2 + z2 — x y —xz-yz by x + y + z. Ans. +y+23 — 3 x y z. 13x2 10. Multiply 4x57x4103-13 2 by 3x-2. 11. Multiply a + a2-1 by a2 1. a2 12. Multiply x2 + 7 a x Ans. x3 14 a3 by x-7 a. 49 a2 x 14 a3 x + 98 a*. 13. Multiply ya by x −y + a. x + 15. Multiply 7xy - 14 x2 y2+21 x3 3 by 6xy-3. Ans.21 xy84 x2 y2 — 147 x3 y3 + 126 x* y*. a2 16. Multiply 6 ab - 9 a b-12 a2 b2 by 2 ab- 3 b2. Ans. 12 ab 36 a2 b3 24 a3 b3 + 27 a b + 36 a2 b*. 17. Multiply — x2 + x2 — x + 1 by x + 1. 1. Ans. +1. 18. Multiply — x3 + x2 — x + 1 by x 1. Ans. x5 2x+2x3 2x2 + 2 x 19. Multiply 1 + x3 + x2 + x + 1 by x + 1. 20. Multiply x2 + x23 + x2 + x + 1 by x-1. SECTION VII. DIVISION. 53. DIVISION is finding a quotient which, multiplied by the divisor, will produce the dividend. the Rule in when the when the In accordance with this definition and Art. 48, the sign of the quotient must be divisor and the dividend have like signs; divisor and the dividend have unlike signs; i. e. in division as in multiplication we have for the signs the following RULE. Like signs give +; unlike, CASE I. 54. When the divisor and dividend are both monomials. 1. Divide 6 ab by 2b. OPERATION. 6ab2b = 3 a The coefficient of the quotient must be a number which, multiplied by 2, the coefficient of the divisor, will give 6, the coefficient of the dividend; i. e. 3: and the literal part of the quotient must be a quantity which, multiplied by b, will give a b; i. e. a: the quotient required, therefore, is 3 a. Hence, for division of monomials, RULE. Annex the quotient of the literal quantities to the quotient of their coefficients, remembering that like signs give + and unlike, -. 2. Divide a by a2. OPERATION. a5 ÷ a2 = a3 For a3 X a2 = a3. (Art. 50.) Hence, Powers of the same quantity are divided by each other by subtracting the exponent of the divisor from that of the 18. Divide 747 a3 b4 c5 d by 83 a2 b c do. 19. Divide 12 (x + y)2 by 4 (x+y). Ans. 3 (x + y). 20. Divide 27 (a - b) by 9 (ab)2. 21. Divide (b — c)5 by (b — c)2. 22. Divide 14 (x − y)' by 7 (x — y)2. Ans. (b — c)3. |