Example 1 from Algebra and Trigonometry, 11.4 Partial Fractions

Decomposing a Rational Function with Distinct Linear Factors

Decompose the given rational expression with distinct linear factors.

\frac{3 x}{(x + 2) (x −1)}

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We will separate the denominator factors and give each numerator a symbolic label, like $A, B,$ or $C .$
$\frac{3 x}{(x + 2) (x −1)} = \frac{A}{(x + 2)} + \frac{B}{(x −1)}$

Multiply both sides of the equation by the common denominator to eliminate the fractions:

$(x + 2) (x −1) [\frac{3 x}{(x + 2) (x −1)}] = (x + 2) (x −1) [\frac{A}{(x + 2)}] + (x + 2) (x −1) [\frac{B}{(x −1)}]$
The resulting equation is

$3 x = A (x −1) + B (x + 2)$
Expand the right side of the equation and collect like terms.

$\begin{array}{l} 3 x = A x - A + B x + 2 B \\ 3 x = (A + B) x - A + 2 B \end{array}$

Set up a system of equations associating corresponding coefficients.

$\begin{array}{l} 3 = A + B \\ 0 = - A + 2 B \end{array}$

Add the two equations and solve for $B .$

$\begin{array}{l} \underline{\begin{array}{l} 3 = A + B \\ 0 = - A + 2 B \end{array}} \\ 3 = 0 + 3 B \\ 1 = B \end{array}$
Substitute $B = 1$ into one of the original equations in the system.

$\begin{array}{l} 3 = A + 1 \\ 2 = A \end{array}$

Thus, the partial fraction decomposition is

$\frac{3 x}{(x + 2) (x −1)} = \frac{2}{(x + 2)} + \frac{1}{(x −1)}$
Another method to use to solve for $A$ or $B$ is by considering the equation that resulted from eliminating the fractions and substituting a value for $x$ that will make either the A- or B-term equal 0. If we let $x = 1,$ the
$A -$ term becomes 0 and we can simply solve for $B .$
$\begin{array}{l} 3 x = A (x - 1) + B (x + 2) \\ 3 (1) = A [(1) - 1] + B [(1) + 2] \\ 3 = 0 + 3 B \\ 1 = B \end{array}$
Next, either substitute $B = 1$ into the equation and solve for $A,$ or make the B-term 0 by substituting $x = −2$ into the equation.
$\begin{array}{l} 3 x = A (x - 1) + B (x + 2) \\ 3 (- 2) = A [(- 2) - 1] + B [(- 2) + 2] \\ - 6 = - 3 A + 0 \\ \frac{- 6}{- 3} = A \\ 2 = A \end{array}$
We obtain the same values for $A$ and $B$ using either method, so the decompositions are the same using either method.

$\frac{3 x}{(x + 2) (x −1)} = \frac{2}{(x + 2)} + \frac{1}{(x −1)}$

Although this method is not seen very often in textbooks, we present it here as an alternative that may make some partial fraction decompositions easier. It is known as the Heaviside method, named after Charles Heaviside, a pioneer in the study of electronics.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the partial fraction decomposition of the following expression.

\frac{x}{(x −3) (x −2)}

How did it go?

Solution

Your turn

Solution

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