Find the product CD of the integers C and D for which
C/(x - 3) + D/(x + 8) = (4x + 65)/(x^2 + 5x - 24)
for all real value of x except -8 and 3.
+130071 We can use the method of partial fractions
Note that (x^2 + 5x - 24) = ( x +8) (x -3) (1)
Multiply through by (1) and we have that
4x + 65 = C (x + 8) + D(x -3) simplify
4x + 65 = (C + D)x + (8C - 3D) equate terms and we have these equations
C + D = 4 ⇒ D = 4 - C (2)
8C - 3D = 65 (3)
Sub (2) into (3) and we have
8C - 3(4 -C) =65
8C - 12 + 3C = 65
11C = 77
C = 7
and
D = 4 - 7 = -3
C * D = (7) (-3) = -21
