Q. 52 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. 52 (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.
$\frac{x^{4} + x^{3} - x + 2}{x^{2} - 2 x + 1}$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given data

The given expression is

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

Step 2. Division

Division Algorithm

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

Step 3. Partial fraction

partial fraction decomposition of the proper fraction

$\frac{6 x - 3}{x^{2} - 2 x + 1} = \frac{6 x - 3}{{(x - 1)}^{2}} \frac{6 x - 3}{{(x - 1)}^{2}} = \frac{A}{(x - 1)} + \frac{B}{{(x - 1)}^{2}} 6 x - 3 = A (x - 1) + B 6 (x - 1) + 3 = A (x - 1) + B S o A = 6 B = 3$

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

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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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