Q. 54 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. 54 (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.
$\frac{x^{5} - x^{3} + x^{2} + 1}{x^{4} + 6 x^{2} + 9}$

Short Answer

Expert verified

Expression as the sum of a quotient and the partial fraction decomposition is $\frac{x^{5} - x^{3} + x^{2} + 1}{x^{4} + 6 x^{2} + 9} = x + \frac{- 7 x + 1}{(x^{2} + 3)} + \frac{12 x - 2}{{(x^{2} + 3)}^{2}}$

Step by step solution

Step 1. Given data

The given expression is

$\frac{x^{5} - x^{3} + x^{2} + 1}{x^{4} + 6 x^{2} + 9}$

Step 2. Division

Division Algorithm

So $\frac{x^{5} - x^{3} + x^{2} + 1}{x^{4} + 6 x^{2} + 9} = x + \frac{- 7 x^{3} + x^{2} - 9 x + 1}{x^{4} + 6 x^{2} + 9}$

Step 3. Partial fraction

partial fraction decomposition of the proper fraction

$\frac{- 7 x^{3} + x^{2} - 9 x + 1}{x^{4} + 6 x^{2} + 9} = \frac{- 7 x^{3} + x^{2} - 9 x + 1}{{(x^{2} + 3)}^{2}} \frac{- 7 x^{3} + x^{2} - 9 x + 1}{{(x^{2} + 3)}^{2}} = \frac{A x + B}{(x^{2} + 3)} + \frac{C x + D}{{(x^{2} + 3)}^{2}} - 7 x^{3} + x^{2} - 9 x + 1 = (A x + B) (x^{2} + 3) + (C x + D) (- 7 x + 1) (x^{2} + 3) + (12 x - 2) = (A x + B) (x^{2} + 3) + (C x + D) S o A = - 7, B = 1, C = 12, & D = - 2$

so $\frac{x^{5} - x^{3} + x^{2} + 1}{x^{4} + 6 x^{2} + 9} = x + \frac{- 7 x + 1}{(x^{2} + 3)} + \frac{12 x - 2}{{(x^{2} + 3)}^{2}}$

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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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