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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. 38 (page 765)

Write the partial fraction decomposition of each rational expression.

Short Answer

Expert verified

The partial fraction decomposition is,

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

Step by step solution

01

Step 1. Given Information.

The rational expression is,

$\frac{x^{3} + 1}{x^{5} - x^{4}}$

02

Step 2. Partial Fraction Decomposition.

Decompose the rational fraction, we get,

$\frac{x^{3} + 1}{x^{4} (x 鈭� 1)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x^{3}} + \frac{D}{x^{4}} + \frac{E}{x - 1}$

$\frac{x^{3} + 1}{x^{4} (x 鈭� 1)} = \frac{x^{4} E + x^{3} (x - 1) A + x^{2} (x - 1) B + x (x - 1) C + (x - 1) D}{x^{4} (x 鈭� 1)}$

Therefore,

$x^{3} + 1 = x^{4} E + x^{3} (x - 1) A + x^{2} (x - 1) B + x (x - 1) C + (x - 1) D$

$x^{3} + 1 = x^{4} (A + E) + x^{3} (鈭� A + B) + x^{2} (鈭� B + C) + x (鈭� C + D) 鈭� D$

Equating the coefficients with the like powers to get,

$A + E = 0 鈭� A + B = 1 鈭� B + C = 0 鈭� C + D = 0 鈭� D = 1$

03

Step 3. Solving the equations.

we get,

$A = 鈭� 2, B = 鈭� 1, C = 鈭� 1, D = 鈭� 1, E = 2$

Therefore,

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

A = 鈭� 2, B = 鈭� 1, C = 鈭� 1, D = 鈭� 1, E = 2

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