Chapter 11: Q. 40 (page 765)

Write the partial fraction decomposition of each rational expression.
$\frac{x^{2} + 1}{x^{3} + x^{2} - 5 x + 3}$

Short Answer

Expert verified

The partial fraction decomposition is

$\frac{x^{2} + 1}{x^{3} + x^{2} - 5 x + 3} = \frac{\frac{3}{8}}{x - 1} + \frac{\frac{1}{2}}{{(x - 1)}^{2}} + \frac{\frac{5}{8}}{x + 3}$

Step by step solution

Step 1. Given Information.

The rational expression is,

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

Step 2. Partial Fraction Decomposition.

Decompose the rational fraction, we get,

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

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

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

Therefore,

$x^{2} + 1 = x^{2} A + x^{2} C + 2 x A + x B - 2 x C - 3 A + 3 B + C$

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

Equating the coefficients with the like powers to get,

$A + C = 1 2 A + B - 2 C = 0 - 3 A + 3 B + C = 1$

Step 3. Solving the equations.

we get,

$A = \frac{3}{8}, B = \frac{1}{2}, C = \frac{5}{8}$

Therefore,

$\frac{x^{2} + 1}{x^{3} + x^{2} - 5 x + 3} = \frac{\frac{3}{8}}{x - 1} + \frac{\frac{1}{2}}{{(x - 1)}^{2}} + \frac{\frac{5}{8}}{x + 3}$

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Write the partial fraction decomposition of each rational expression.
$\frac{x^{2} + 1}{x^{3} + x^{2} - 5 x + 3}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Partial Fraction Decomposition.

Step 3. Solving the equations.

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Write the partial fraction decomposition of each rational expression.x2+1x3+x2−5x+3

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Partial Fraction Decomposition.

Step 3. Solving the equations.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Probability and Statistics

Logic and Functions

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Pure Maths

Study anywhere. Anytime. Across all devices.

Write the partial fraction decomposition of each rational expression.
$\frac{x^{2} + 1}{x^{3} + x^{2} - 5 x + 3}$