x2+ bx+c=(x+r)(x+8); rs = c, r + s = b. 52. ax2 + bx + c = (px + q) (rx+8), wherein pr = a, ps+qr = b, qs = c. = By this method p, q, r, s must be found by trial, so that the three conditions just named will be satisfied. If possible, we must find numbers p, q, r, s, such that pr = 6, qs =— 6, ps + gr 5. 6, until a combination Try the different sets of values of p and r, whose product is 6, with each pair of values of q and s, whose product is is found which satisfies ps + qr =-3. Try the sets of values p = 3, r = 2, q = 2, 8 = — We see that the sum of the two cross products + 4x and -9x is — 5x, which is the middle term of 6 x2 - 5 x 6. 53. Factor: Hence ba a2 + a2b2 + b1 = a1 +2 a2b2 + b1 — a2b2 = (a2 + b2)2 — a2b2 54. Factor: 1. x2 + x2 + 1. = (a2 + ab + b2)(a2 — ah + b2). = 2. 9 m2 + 8 m2n2 + 4 n1. — 6. x14x2y2 + y^. 7. ms 38m*n* +n 3. 16 a 20 a2b2 + 9 b1. 8. 4 a1 — 13 a2b2 + 9 ba. 4. 78 +244 + s8. 5. a1- 5 a2b2 + 4 ba. 9. 25c+26 c2d2 + 9 d1. 10. 49 a 110 a2b2 + 81 b4. 55. In factoring, first take out monomial factors. Then inspect the resulting polynomial, ascertaining to which type form it belongs, and factor it accordingly. In the final form, all factors should be prime. 46. 3 a2-10 ab — 8 b2 + 3 ad + 2 bd − 9 ac — 6 bc. 47. x-3x2 + 4. 48. 4a+81 b1. 49. 4 a4-37 a2b2 + 9 b1. 50. 64 * +128 y^2 +81 *. FRACTIONS 58. A fraction is the indicated quotient obtained by dividing one number by another. The fundamental principle of operations with fractions isBoth numerator and denominator of a fraction may be multiplied or divided by the same number, without changing the value of the fraction. 60. From the laws of signs in multiplication and division it is 1. The signs in both numerator and denominator may be changed without changing the value of the fraction. 2. The sign of the numerator and of the fraction may be changed without changing the value of the fraction. 3. The sign of the denominator and of the fraction may be changed without changing the value of the fraction. 61. The following principles may be used effectively in operations with fractions. 1. The signs of an even number of factors may be changed without changing the sign of the product. Explain. Thus, a. b. c = (-a) · (— b) · c = abc. 2. The signs of an odd number of factors may be changed, provided the sign of the product is changed. Explain. Thus, a b c = − (− a) ⋅ b · c = abc; a · ⋅ b · c ‡ (− a) ( — b)(— c). |