Q. 51 In Problems, use the division al... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 51 (page 765)

In Problems, use the division algorithm to rewrite each improper fraction as the sum of a quotient and proper fraction. Find the partial fraction decomposition of the proper fraction. Finally, express the improper fraction as the sum of a quotient and the partial fraction decomposition.
$\frac{x^{4} - 5 x^{2} + x - 4}{x^{2} + 4 x + 4}$

Short Answer

Expert verified

Expression $\frac{x^{4} - 5 x^{2} + x - 4}{x^{2} + 4 x + 4}$ as the sum of a quotient and the partial fraction decomposition is $\frac{x^{4} - 5 x^{2} + x - 4}{x^{2} + 4 x + 4} = x^{2} - 4 x + 7 - \frac{11}{x + 2} - \frac{10}{{(x + 2)}^{2}}$

Step by step solution

Step 1. Given data

The given expression is

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

Step 2. Division

Division Algorithm

So $\frac{x^{4} - 5 x^{2} + x - 4}{x^{2} + 4 x + 4} = x^{2} - 4 x + 7 - \frac{11 x + 32}{x^{2} + 4 x + 4}$

Step 3. Partial fraction

partial fraction decomposition of the proper fraction

$\frac{11 x + 32}{x^{2} + 4 x + 4} = \frac{11 x + 32}{{(x + 2)}^{2}} \frac{11 x + 32}{{(x + 2)}^{2}} = \frac{A}{(x + 2)} + \frac{B}{{(x + 2)}^{2}} 11 x + 32 = A (x + 2) + B 11 (x + 2) + 10 = A (x + 2) + B S o A = 11 B = 10$

so $\frac{x^{4} - 5 x^{2} + x - 4}{x^{2} + 4 x + 4} = x^{2} - 4 x + 7 - \frac{11}{x + 2} - \frac{10}{{(x + 2)}^{2}}$

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Short Answer

Step by step solution

Step 1. Given data

Step 2. Division

Step 3. Partial fraction

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