I have two pre-calculus questions I need help on. They deal with decomposing partial fractions with repeating linear factors.

Question

Problem 1: x/5(x+2)^2

Problem 2: 5x^2+20x+8/2x(x+1)^2

Answer 1

Hi Amber:

Partial fractions are very important for integration in calculus. Let's decompose these 2 examples!

(a) Let x / 5(x + 2)² = A /(x +2)² + B / (x + 2) for some numbers A and B

=> x/5 = A + B (x + 2) [multiplying both sides by x +2)²

Now, we can equate the coefficients of each power of x on the LHS and RHS

x¹: 1/5 = B

x⁰: 0 = A + 2B

= A + 2/5 [since B = 1/5

=> A = -2/5

Hence, x / 5(x + 2)² = -2 / 5(x +2)² + 1 / 5(x + 2)

(b) First we need 5x²+20x+8 / 2x(x+1)² as a single fraction. The largest common denominator is 2x(x+1)²

So (5x²+20x)+8 / 2x(x+1)² = [(5x²+20x)(2x(x+1)²) + 8] / 2x(x+1)²