Q. 53 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. 53 (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.
$\frac{x^{5} + x^{4} - x^{2} + 2}{x^{4} - 2 x^{2} + 1}$

Short Answer

Expert verified

Expression $\frac{x^{5} + x^{4} - x^{2} + 2}{x^{4} - 2 x^{2} + 1}$ as the sum of a quotient and the partial fraction decomposition is $\frac{x^{5} + x^{4} - x^{2} + 2}{x^{4} - 2 x^{2} + 1} = x + 1 + \frac{1.25}{(x - 1)} + \frac{0.25}{{(x - 1)}^{2}} + \frac{0.75}{(x + 1)} + \frac{0.25}{{(x + 1)}^{2}}$

Step by step solution

Step 1. Given data

The given expression is

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

Step 2. Division

Division Algorithm

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

Step 3. Partial fraction

partial fraction decomposition of the proper fraction

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

Solving A,B,C,and D

$A = 1.25, B = 0.25, C = 0.75, & D = 0.25$

So $\frac{x^{5} + x^{4} - x^{2} + 2}{x^{4} - 2 x^{2} + 1} = x + 1 + \frac{1.25}{(x - 1)} + \frac{0.25}{{(x - 1)}^{2}} + \frac{0.75}{(x + 1)} + \frac{0.25}{{(x + 1)}^{2}}$