Practice Partial Fraction Decomposition in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

Breaking a rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator.

Just as 712 can be split into 13+14, a complex fraction like 5x−1(x+1)(x−2) can be split into Ax+1+Bx−2. The simpler pieces are each easy to integrate.

Showing a random 20 of 50 problems.

Example 1

medium
Decompose 5(x+2)(x−3).

Example 2

hard
Compute x3+2x2−1 as a polynomial plus a proper fraction.

Example 3

medium
Decompose 4x+1(x−1)(x+3).

Example 4

medium
Decompose x+4x2+x.

Example 5

easy
Decompose 5x−1(x+1)(x−2): find B for Bx−2.

Example 6

medium
Decompose 3x−2x2−x−2.

Example 7

hard
Continue: decompose x+2x2−1.

Example 8

challenge
Decompose x2+1x(x−1)(x+1).

Example 9

medium
How many unknown constants appear in the decomposition of P(x)(x−1)2(x+2)?

Example 10

easy
Does x3x2−1 need long division before decomposing?

Example 11

easy
What is the form for 3(x−1)2?

Example 12

challenge
Decompose 4x3+x=4x(x2+1).

Example 13

challenge
Decompose x(x−1)2(x+1).

Example 14

medium
How many unknowns are needed for 1x(x2+4)2?

Example 15

easy
Decompose 5x−1(x+1)(x−2): find A for Ax+1.

Example 16

easy
Decompose 3x+5(x+1)(x+2).

Example 17

hard
Integrate ∫1x3−x dx.

Example 18

easy
Decompose 2x(x−1)(x+1).

Example 19

medium
Combine 3x−2−2x+1 into a single fraction.

Example 20

medium
Combine into a single fraction: 2x−1+3x+2.
← Back to Partial Fraction Decomposition Worked Examples Formula Explained