Enter (A,B,C) in order below if A, B, and C are the coefficients of the partial fractions expansion of (2x^2 + 3x - 5)/(x(x^2 - 1)) = A/x + B/(x - 1) + C/(x + 1).
+23245 Rewrite A/x + B/(x-1) + C(x + 1) with the common denominator x(x - 1)(x + 1):
A / x · (x + 1)(x - 1) / [(x + 1)(x - 1)] = A(x + 1)(x - 1) / [x(x + 1)(x - 1)]
+ B / (x - 1) · x(x + 1) / [x(x + 1)] = B(x)(x + 1) / [x(x + 1)(x - 1)]
+ C / (x + 1) · x(x - 1) / [x(x - 1)] = C(x)(x - 1) / [x(x + 1)(x - 1)]
= [ A(x + 1)(x - 1) + B(x)(x + 1) + C(x)(x - 1) ] / [x(x + 1)(x - 1)]
= [ Ax2 - A + Bx2 + Bx + Cx2 - Cx ] / [x(x + 1)(x - 1)]
= [ (A + C)x2 + (B - C)x - A ] / [x(x + 1)(x - 1)]
Comparing the original numerator to this numerator:
2x2 + 3x - 5 = (A + C)x2 + (B - C)x - A
So: A + C = 2
B - C = 3
-A = -5
Solve these equations for A, B, and C to get the answer.
