# Partial fraction decomposition.Write the ratio (3x+2)/(x^2+x) using partial fraction decomposition.

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### 2 Answers

The partial fractions of (3x+2)/(x^2+x) have to be determined.

(3x+2)/(x^2+x)

=> (3x+2)/x(x + 1)

This can be written as A/x + B/(x + 1)

A/x + B/(x + 1) = (3x+2)/[x(x + 1)]

Ax + A + Bx = 3x + 2

=> A + B = 3 and A = 2

=> B = 1

**The partial fractions are 2/x + 1/(x + 1)**

To decompose a fraction in partial fractions, we'll must factor the denominator. We'll factorize by x:

x^2+x = x( x + 1 )

(3x+2)/(x^2+x) = A/x + B/( x + 1 )

We'll multiply both side by the common denominator x( x + 1 ):

3x+2 = A(x+1) + Bx

We'll remove the brackets:

3x+2 = Ax + A + Bx

We'll factorize by x to the right side:

3x+2 = x(A+B) + A

We'll compare and we'll get:

A+B = 3

A = 2

2 + B = 3

B = 3 - 2

B = 1

**Therefore, the fraction (3x+2)/(x^2+x) = 2/x + 1/(x + 1)**