How do you write the partial fraction decomposition of the rational expression (x+10)/(x^2+2x-8)?

1 Answer
Dec 9, 2015

The equivalent partial fraction is :
(-1)/(x+4) + 2/(x-2)

Explanation:

Given (x+10)/(x^2 +2x-8)

Step 1: Factor the denominator

(x+10)/((x+4)(x-2)

Step 2: Set up the partial faction as follows:

(x+10)/((x+4)(x-2)) = A/(x+4) + B/(x-2) " " " " " (1)

Step 3: Multiply both sides by the LCD, (x+4)(x-2):

(x+10) = A(x-2) +B(x+4)
x+ 10 = Ax - 2A + Bx+ 4B

Step 4: Set up a system like this
1x: " " " " A+ B= 1 " " " "(2)
10: " " " -2A+4B= 10 " " " "(3)

Step 5. You can solve the system by the elimination method:

2(A+B= 1) => 2A + 2B= 2

+ -2A + 4B= 10
6B = 12 => B= 2

Solve for A by substituting B = 3 into (2):

A+(2) = 1
A = -1

Step 6. Substitute A and B back into (1):

(x+10)/((x+4)(x-2))= (-1)/(x+4) + 2/(x-2)