Page:Elementary algebra (1896).djvu/55

37 Rh 29. (3x+8)(3x-8). 33. (2x + 7 y)(2x - 5y). 30. (2x-5)(2x-5). 34. (5x + 3a)(5x - 3a). 31. (3x-2y)(3x + y). 35. (2x - 5a)(a + 5a). 32. (3x + 2y)(3x + 2y). 36. (2x + a)(2x + a).

MULTIPLICATION BY DETACHED COEFFICIENTS.

52. In the following cases we lessen the labor of multiplication by using the Method of Detached Coefficients:

(i.) When two compound expressions contain but one letter.

(ii.) When two compound expressions are homogeneous and contain but two letters.

Ex. 1. Multiply 2 x^3 - 4x^2 + 5x - 5 by 3x^2 + 4x - 2.

Writing coefficients only,

2- 4+ 5- 5 3+ 4- 2 6-12 + 15-15 + 8-16 + 20- 20 - 4+ 8 -10 + 10 6- 4- 5 + 13-30 + 10

Inserting the literal factors according to the law of their formation, which is readily seen, we have for the complete product,

6x^5-4x^4- 5x^3+ 13x^2-30x + 10.

Ex. 2. Multiply 2a^4 -- 2 a^3b + 4ab^3 + 2 b^4 by 2 a^2 -b^2.

In the first expression the term containing a^2 b^2 is missing, so we write a zero in the corresponding term in the line of coefficients. In the second expression we write a zero for the coefficient of the missing term ab.

The law of formation of literal factors is readily seen, and we have for the complete product,

6 a^6 + 4 a^5b - 3 a^4b^2 + 6 a^3b^3 + 4 a^2b^4 - 4 ab^5 - 2 b^6.

EXAMPLES IV. g.

1. Multiply x^5 + x^4 + x^2 + 2x + 1 by x^3 + x - 2. 2. Multiply a^3 + 6 a^2b + 12 ab^2 + 8b^3 by by 3 a^3 + 2 b^3. 3. Multiply 2a^4- 3a^2 + 4a + 4 by 2a^2-3a-2. 4. Multiply 3 x^5 + 2 x^4y - x^3y^2 + xy^4 by x^2 + 4 xy - 5 y^2.