Find the product CD of the integers C and D for which (C)/(x-3) + (D)/(x+8) = (4x-23)/(x^2 + 5x - 24) for all real values of x except -8 and 3.
+37157 Common denominator for the first two terms is
(x-3)(x+8) = x^2+5x-24
so (cross multiplying the first two terms) results in
{c(x+8) + d(x-3) }/(x^2+5x-24) = (4x-23) / (x^2+5x-24)
Both left and right sides have the same denominator ....so the numerator is equal
c(x+8) + d(x-3) = 4x- 23
cx + 8c + dx - d3 = 4x- 23
so cx + dx = 4x (1) and 8c-d3 = -23 (2)
c+d = 4
d = 4-c substitute this into (2)
8c -3(4-c) = -23
8c-12+3c = -23
11c= -11
c = -1 d=4-c = 4-(-1) = 5 The product of c x d = (-1) x 5 = -5
