Practice Partial Fraction Decomposition in Math

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

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Quick Recap

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

Just as 712\frac{7}{12} can be split into 13+14\frac{1}{3} + \frac{1}{4}, a complex fraction like 5x−1(x+1)(x−2)\frac{5x-1}{(x+1)(x-2)} can be split into Ax+1+Bx−2\frac{A}{x+1} + \frac{B}{x-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)\dfrac{5}{(x+2)(x-3)}.

Example 2

hard
Compute x3+2x2−1\dfrac{x^3 + 2}{x^2 - 1} as a polynomial plus a proper fraction.

Example 3

medium
Decompose 4x+1(x−1)(x+3)\dfrac{4x+1}{(x-1)(x+3)}.

Example 4

medium
Decompose x+4x2+x\frac{x+4}{x^2+x}.

Example 5

easy
Decompose 5x−1(x+1)(x−2)\frac{5x-1}{(x+1)(x-2)}: find BB for Bx−2\frac{B}{x-2}.

Example 6

medium
Decompose 3x−2x2−x−2\dfrac{3x-2}{x^2 - x - 2}.

Example 7

hard
Continue: decompose x+2x2−1\dfrac{x+2}{x^2-1}.

Example 8

challenge
Decompose x2+1x(x−1)(x+1)\frac{x^2+1}{x(x-1)(x+1)}.

Example 9

medium
How many unknown constants appear in the decomposition of P(x)(x−1)2(x+2)\dfrac{P(x)}{(x-1)^2(x+2)}?

Example 10

easy
Does x3x2−1\frac{x^3}{x^2-1} need long division before decomposing?

Example 11

easy
What is the form for 3(x−1)2\frac{3}{(x-1)^2}?

Example 12

challenge
Decompose 4x3+x=4x(x2+1)\frac{4}{x^3+x}=\frac{4}{x(x^2+1)}.

Example 13

challenge
Decompose x(x−1)2(x+1)\frac{x}{(x-1)^2(x+1)}.

Example 14

medium
How many unknowns are needed for 1x(x2+4)2\frac{1}{x(x^2+4)^2}?

Example 15

easy
Decompose 5x−1(x+1)(x−2)\frac{5x-1}{(x+1)(x-2)}: find AA for Ax+1\frac{A}{x+1}.

Example 16

easy
Decompose 3x+5(x+1)(x+2)\dfrac{3x+5}{(x+1)(x+2)}.

Example 17

hard
Integrate ∫1x3−x dx\displaystyle\int \frac{1}{x^3 - x}\,dx.

Example 18

easy
Decompose 2x(x−1)(x+1)\dfrac{2x}{(x-1)(x+1)}.

Example 19

medium
Combine 3x−2−2x+1\dfrac{3}{x-2} - \dfrac{2}{x+1} into a single fraction.

Example 20

medium
Combine into a single fraction: 2x−1+3x+2\dfrac{2}{x-1} + \dfrac{3}{x+2}.
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