Page:Elementary algebra (1896).djvu/103

85 Rh 46. 1-(x- y)^2. 52. (a + b)^2 + a + b 47. 250 (a - b)^2 + 2. 63. a^2 + b^2 + a + 5. 48. (c + d)^3 (c - d)^3. 64. a^2 - 9 b^2 + a +3 b, 49. 8(x + y)^2-(2x- y)^2. 65. 4 (x - y)^2- (x - y). 50. x^2 - 4 y2 + x - 2 y. 56. x^4y - x3y3 - x3y2 + xy4. 51. a^2 - b^2 + a - b. 57. 4 a^2 - 9 b^2 + 2 a - 3 6. 58. Resolve x^16 - y^16 into five factors.

CONVERSE USE OF FACTORS.

104. 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 formulae 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 at once the product of the sum and the difference of two quantities.

Ex. 1. Multiply 2 a + 3 b - c by 2 a - 3 b + c.

These expressions may be arranged thus : 2 a + (3 b - c) and 2 a - (3 b-c).

Hence the product = {2 a + (3 b- c)}{2 a - (3 b - c)} = (2 a)2 - (3 b- c)2 [Art. 98.] = 4 a^2 - (9 b^2 - 6 bc + c^2) = 4 a^2 - 9 b^2 + 6 bc - c^2.

Ex. 2. Divide the product of 2 x^2 + x - 6, and 6 x^2 - 5 x + 1 by 3 x^2 + 5 x - 2.

Denoting the division by writing the divisor under the dividend (Art. 53), with a horizontal line between them, the required quotient

= {(2x^2 + x-6)(6x^2-5x+1) } { 3 x^2 + 5 X - 2 } ={(2x - 3)(x + 2)(3x - 1)(2x - 1) } {(3x-1)(x + 2) } = (2x-3)(2x-1).