Find partial fraction decomposition

Question

(7x²+2x-19)/ (x-3)(x+2)²

Answer 1

(7x²+2x-19) / [(x-3)(x+2)²] = A / (x-3) + B / (x+2) + C / (x+2)²

7x² + 2x - 19 = A(x+2)² + B(x-3)(x+2) + C(x-3)

7x² + 2x - 19 = (A + B)x² + (4A - B + C)x + (4A - 6B - 3C)

Therefore:

A+B = 7

4A-B+C = 2

4A-6B-3C = -19

Solve for A, B, and C.

Answer 2

desmos.com/calculator/aap8m3svnd

Answer 3

(7x² + 2x - 19) / (x -3)(x + 2)² = [A / (x - 3)] + [B / (x + 2)] + [C / (x - 3)(x + 2)] + [D / (x + 2)²] + [E / (x - 3)(x + 2)²]

7x² + 2x - 19 = A(x + 2)² + B(x - 3)(x + 2) + C(x + 2) + D(x - 3) + E

7x² + 2x - 19 = Ax² + 4Ax + 4A + Bx² - Bx - 6B + Cx + 2C + Dx - 3D + E

7x² + 2x - 19 = Ax² + Bx² + 4Ax - Bx + Cx + Dx + 2C - 3D + E

7x² + 2x - 19 = (A + B)x² + (4A - B + C + D)x + 2C - 3D + E

A + B = 7

4A - B + C + D = 2

2C - 3D + E = - 19