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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. 40 (page 765)

Write the partial fraction decomposition of each rational expression.
$\frac{x^{2} + 1}{x^{3} + x^{2} 鈭� 5 x + 3}$

Short Answer

Expert verified

The partial fraction decomposition is

$\frac{x^{2} + 1}{x^{3} + x^{2} 鈭� 5 x + 3} = \frac{\frac{3}{8}}{x 鈭� 1} + \frac{\frac{1}{2}}{{(x 鈭� 1)}^{2}} + \frac{\frac{5}{8}}{x + 3}$

Step by step solution

01

Step 1. Given Information.

The rational expression is,

$\frac{x^{2} + 1}{x^{3} + x^{2} 鈭� 5 x + 3}$

02

Step 2. Partial Fraction Decomposition.

Decompose the rational fraction, we get,

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

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

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

Therefore,

$x^{2} + 1 = x^{2} A + x^{2} C + 2 x A + x B 鈭� 2 x C 鈭� 3 A + 3 B + C$

$x^{2} + 1 = x^{2} (A + C) + x (2 A + B 鈭� 2 C) 鈭� 3 A + 3 B + C$

Equating the coefficients with the like powers to get,

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

03

Step 3. Solving the equations.

we get,

$A = \frac{3}{8}, B = \frac{1}{2}, C = \frac{5}{8}$

Therefore,

$\frac{x^{2} + 1}{x^{3} + x^{2} 鈭� 5 x + 3} = \frac{\frac{3}{8}}{x 鈭� 1} + \frac{\frac{1}{2}}{{(x 鈭� 1)}^{2}} + \frac{\frac{5}{8}}{x + 3}$

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