# Partial Fraction Decomposition

How to solve partial fraction decomposition problems: examples and their solutions.

## Example 1

Write the given fraction

as the sum of the fractions

whose denominators are

the reduced factors of the given denominator:

(given) = *A*/*x* + *B*/(*x* - 1)

Your goal is to find *A* and *B*.

To add the fractions,

make each fraction's denominator

to the given fraction's denominator.

Multiply the missing factors

to both of the numerator and the denominator.

Adding and subtracting rational expressions

The numerators of both side fractions are equal.

So 3*x* - 2 = *A*(*x* - 1) + *Bx*.

To compare the terms of both sides,

arrange the right side in descending order.

Ascending order, descending order

Then switch both sides.

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

Compare the constant terms on both sides:

-*A* = -2.

Then *A* = 2.

Compare the *x* terms' coefficients on both sides:*A* + *B* = 3.*A* = 2

So 2 + *B* = 3.

Then *B* = 1.

Put *A* = 2 and *B* = 1

into *A*/*x* + *B*/(*x* - 1).

Then

(given) = 2/*x* + 1/(*x* - 1).

## Example 2

Write the given fraction

as the sum of the fractions

whose denominators are

the reduced factors of the given denominator:

(given) = *A*/*x* + *B*/*x*^{2} + *C*/(*x* - 1)

Your goal is to find *A*, *B*, and *C*.

To add the fractions,

make each fraction's denominator

to the given fraction's denominator.

Multiply the missing factors

to both of the numerator and the denominator.

Adding and subtracting rational expressions

The numerators of both side fractions are equal.

So 5*x*^{2} - 1 = *Ax*(*x* - 1) + *B*(*x* - 1) + *Cx*^{2}.

To compare the terms of both sides,

arrange the right side in descending order.

Ascending order, descending order

Then switch both sides.

(*A* + *C*)*x*^{2} + (*A* + *B*)*x* + *B* = 5*x*^{2} - 1

Compare the constant terms on both sides:*B* = -1.

Compare the *x* terms' coefficients on both sides:*A* + *B* = 0.*B* = -1

So *A* + -1 = 0.

Then *A* = 1.

Compare the *x*^{2} terms' coefficients on both sides:*A* + *C* = 5.*A* = 1

So 1 + *C* = 5.

Then *C* = 4.

Put *A* = 1, *B* = -1, and *C* = 4

into *A*/*x* + *B*/*x*^{2} + *C*/(*x* - 1).

Then

(given) = 1/*x* - 1/*x*^{2} + 4/(*x* - 1).