College Algebra

Partial Fractions

Lesson

Partial fraction decomposition reverses the common-denominator move: it splits a complicated rational expression into a sum of simpler ones.

Distinct linear factors

\frac{p(x)}{(x - r_1)(x - r_2)} = \frac{A}{x - r_1} + \frac{B}{x - r_2}

The cover-up method

To find $A$ , cover up $(x - r_1)$ in the original fraction and evaluate everything that’s left at $x = r_1$ . Same idea for $B$ : cover up $(x - r_2)$ and evaluate at $x = r_2$ .

Worked example

\frac{5}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3}

For $A$ : cover $(x - 2)$ and set $x = 2$ :

A = \frac{5}{2 - 3} = -5

For $B$ : cover $(x - 3)$ and set $x = 3$ :

B = \frac{5}{3 - 2} = 5

Decomposition: $-\dfrac{5}{x-2} + \dfrac{5}{x-3}$ .

When does this work cleanly?

The numerator’s degree must be less thanthe denominator’s. If not, do polynomial long division first.
The denominator must factor into distinct linear factors for the simple cover-up rule. Repeated or quadratic factors need more work — but the principle is the same.

Real-world use: partial fractions are the standard tool for integration in calculus and for solving differential equations via Laplace transforms.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\frac{1}{(x - 1)(x - 2)} = \frac{A}{x - 1} + \frac{B}{x - 2}; \ A = ?

Problem 2

\frac{1}{(x - 1)(x - 2)} = \frac{A}{x - 1} + \frac{B}{x - 2}; \ B = ?

Problem 3

\frac{1}{x(x - 3)} = \frac{A}{x} + \frac{B}{x - 3}; \ A = ?

Problem 4

\frac{1}{x(x - 3)} = \frac{A}{x} + \frac{B}{x - 3}; \ B = ?

Practice

Standard problems matching the lesson.

Problem 5

\frac{2}{(x - 1)(x - 3)} = \frac{A}{x - 1} + \frac{B}{x - 3}; \ A = ?

Problem 6

\frac{2}{(x - 1)(x - 3)} = \frac{A}{x - 1} + \frac{B}{x - 3}; \ B = ?

Problem 7

\frac{5}{(x + 2)(x - 3)} = \frac{A}{x + 2} + \frac{B}{x - 3}; \ A = ?

Problem 8

\frac{5}{(x + 2)(x - 3)} = \frac{A}{x + 2} + \frac{B}{x - 3}; \ B = ?

Problem 9

\frac{3}{x(x + 1)} = \frac{A}{x} + \frac{B}{x + 1}; \ A = ?

Problem 10

\frac{3}{x(x + 1)} = \frac{A}{x} + \frac{B}{x + 1}; \ B = ?

Problem 11

\frac{1}{(x - 2)(x + 4)} = \frac{A}{x - 2} + \frac{B}{x + 4}; \ A = ?

Problem 12

\frac{1}{(x - 2)(x + 4)} = \frac{A}{x - 2} + \frac{B}{x + 4}; \ B = ?

Problem 13

\frac{6}{(x - 1)(x + 1)} = \frac{A}{x - 1} + \frac{B}{x + 1}; \ A = ?

Problem 14

\frac{6}{(x - 1)(x + 1)} = \frac{A}{x - 1} + \frac{B}{x + 1}; \ B = ?

Problem 15

\frac{4}{x(x - 2)} = \frac{A}{x} + \frac{B}{x - 2}; \ A = ?

Problem 16

\frac{4}{x(x - 2)} = \frac{A}{x} + \frac{B}{x - 2}; \ B = ?

Problem 17

\frac{8}{(x - 3)(x + 5)} = \frac{A}{x - 3} + \frac{B}{x + 5}; \ A = ?

Problem 18

\frac{8}{(x - 3)(x + 5)} = \frac{A}{x - 3} + \frac{B}{x + 5}; \ B = ?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

\frac{10}{(x - 5)(x - 1)} = \frac{A}{x - 5} + \frac{B}{x - 1}; \ A = ?

Problem 20

\frac{10}{(x - 5)(x - 1)} = \frac{A}{x - 5} + \frac{B}{x - 1}; \ B = ?

Problem 21

\frac{12}{(x + 1)(x - 3)} = \frac{A}{x + 1} + \frac{B}{x - 3}; \ A = ?

Problem 22

\frac{12}{(x + 1)(x - 3)} = \frac{A}{x + 1} + \frac{B}{x - 3}; \ B = ?

Problem 23

\frac{7}{x(x - 7)} = \frac{A}{x} + \frac{B}{x - 7}; \ A = ?

Problem 24

\frac{7}{x(x - 7)} = \frac{A}{x} + \frac{B}{x - 7}; \ B = ?

Problem 25

\frac{2}{(x - 1)(x - 2)(x - 3)} = \frac{A}{x - 1} + \cdots; \ A = ?

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