Partial Fraction Decomposition of x/((x+1)(x-4))

Published on October 27, 2024

Algebra

The partial fraction decomposition of the rational expression x/((x+1)(x-4)) is (1/5)/(x+1) + (4/5)/(x-4).

Question

partial fraction decomposition x/((x+1)(x-4))

Answer

Let’s break down the partial fraction decomposition of the rational expression

1. Factor the Denominator

The denominator is already factored:

2. Set up the Partial Fractions

Since we have two distinct linear factors in the denominator, we set up the decomposition as follows:

where

and

are constants we need to determine.

3. Clear the Fractions

Multiply both sides of the equation by the common denominator,

4. Solve for the Constants

We can solve for

and

by strategically substituting values for

Let :
This eliminates :
Let :
This eliminates :

5. Write the Decomposition

Now that we have the values of

and

, we can write the partial fraction decomposition:

Therefore, the partial fraction decomposition of is .

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