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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 39 (page 765)

Write the partial fraction decomposition of each rational expression.
$\frac{x^{2}}{x^{3} - 4 x^{2} + 5 x - 2}$

Short Answer

Expert verified

The partial fraction decomposition is

$\frac{x^{2}}{x^{3} 鈭� 4 x^{2} + 5 x 鈭� 2} = \frac{鈭� 3}{x 鈭� 1} + \frac{鈭� 1}{{(x 鈭� 1)}^{2}} + \frac{4}{x 鈭� 2}$

Step by step solution

01

Step 1. Given Information.

The rational expression is,

\frac{x^{2}}{x^{3} - 4 x^{2} + 5 x - 2}

02

Step 2. Partial Fraction Decomposition.

Decompose the rational fraction, we get,

$\frac{x^{2}}{x^{3} 鈭� 4 x^{2} + 5 x 鈭� 2} = \frac{x^{2}}{(x 鈭� 2) {(x 鈭� 1)}^{2}}$

$\frac{x^{2}}{(x 鈭� 2) {(x 鈭� 1)}^{2}} = \frac{A}{x 鈭� 1} + \frac{B}{{(x 鈭� 1)}^{2}} + \frac{C}{x 鈭� 2}$

$\frac{x^{2}}{(x 鈭� 2) {(x 鈭� 1)}^{2}} = \frac{(x 鈭� 2) (x 鈭� 1) A + (x 鈭� 2) B + (x 鈭� 1) 2 C}{(x 鈭� 2) {(x 鈭� 1)}^{2}}$

Therefore,

$x^{2} = x^{2} A + x^{2} C 鈭� 3 x A + x B 鈭� 2 x C + 2 A 鈭� 2 B + C$

$x^{2} = x^{2} (A + C) + x (鈭� 3 A + B 鈭� 2 C) + 2 A 鈭� 2 B + C$

Equating the coefficients with the like powers to get,

$A + C = 1 鈭� 3 A + B 鈭� 2 C = 0 2 A 鈭� 2 B + C = 0$

03

Step 3. Solving the equations.

we get,

$A = 鈭� 3, B = 鈭� 1, C = 4$

$\frac{x^{2}}{(x 鈭� 2) {(x 鈭� 1)}^{2}} = \frac{鈭� 3}{x 鈭� 1} + \frac{鈭� 1}{{(x 鈭� 1)}^{2}} + \frac{4}{x 鈭� 2}$

role="math" localid="1647425259262" $\frac{x^{2}}{x^{3} 鈭� 4 x^{2} + 5 x 鈭� 2} = \frac{鈭� 3}{x 鈭� 1} + \frac{鈭� 1}{{(x 鈭� 1)}^{2}} + \frac{4}{x 鈭� 2}$

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