Example 2 from Algebra and Trigonometry, 11.4 Partial Fractions

Decomposing with Repeated Linear Factors

Decompose the given rational expression with repeated linear factors.

\frac{- x^{2} + 2 x + 4}{x^{3} −4 x^{2} + 4 x}

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
The denominator factors are $x {(x −2)}^{2} .$ To allow for the repeated factor of $(x −2),$ the decomposition will include three denominators: $x, (x −2),$ and ${(x −2)}^{2} .$ Thus,

$\frac{- x^{2} + 2 x + 4}{x^{3} −4 x^{2} + 4 x} = \frac{A}{x} + \frac{B}{(x −2)} + \frac{C}{{(x −2)}^{2}}$
Next, we multiply both sides by the common denominator.

$\begin{array}{l} x {(x −2)}^{2} [\frac{- x^{2} + 2 x + 4}{x {(x −2)}^{2}}] = [\frac{A}{x} + \frac{B}{(x −2)} + \frac{C}{{(x −2)}^{2}}] x {(x −2)}^{2} \\ - x^{2} + 2 x + 4 = A {(x −2)}^{2} + B x (x −2) + C x \end{array}$
On the right side of the equation, we expand and collect like terms.

$\begin{array}{l} - x^{2} + 2 x + 4 = A (x^{2} - 4 x + 4) + B (x^{2} - 2 x) + C x \\ = A x^{2} - 4 A x + 4 A + B x^{2} - 2 B x + C x \\ = (A + B) x^{2} + (- 4 A - 2 B + C) x + 4 A \end{array}$
Next, we compare the coefficients of both sides. This will give the system of equations in three variables:

$- x^{2} + 2 x + 4 = (A + B) x^{2} + (−4 A −2 B + C) x + 4 A$

$\begin{array}{r} A + B = −1 & (1) \\ −4 A −2 B + C = 2 & (2) \\ 4 A = 4 & (3) \end{array}$
Solving for $A$ , we have

$\begin{array}{l} 4 A = 4 \\ A = 1 \end{array}$
Substitute $A = 1$ into equation (1).

$\begin{array}{l} A + B = −1 \\ (1) + B = −1 \\ B = −2 \end{array}$
Then, to solve for $C,$ substitute the values for $A$ and $B$ into equation (2).

$\begin{array}{r} −4 A −2 B + C = 2 \\ −4 (1) −2 (−2) + C = 2 \\ −4 + 4 + C = 2 \\ C = 2 \end{array}$
Thus,

$\frac{- x^{2} + 2 x + 4}{x^{3} −4 x^{2} + 4 x} = \frac{1}{x} - \frac{2}{(x −2)} + \frac{2}{{(x −2)}^{2}}$

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 expression with repeated linear factors.

\frac{6 x −11}{{(x −1)}^{2}}

How did it go?

Solution

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Solution

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