Chapter 11: Q. 37 (page 794)

In Problem, Write the partial fraction decomposition of each rational expression.
$\frac{x^{3}}{{(x^{2} + 4)}^{2}}$

Short Answer

Expert verified

Partial fraction decomposition of a rational expression $\frac{x^{3}}{{(x^{2} + 4)}^{2}}$ is

$\frac{x^{3}}{{(x^{2} + 4)}^{2}} = \frac{x}{(x^{2} + 4)} + \frac{- 4 x}{{(x^{2} + 4)}^{2}}$

Step by step solution

Step 1. Given data

The given rational expression is

$\frac{x^{3}}{{(x^{2} + 4)}^{2}}$

Step 2. Decomposition of rational expression

Partial fraction decomposition of rational expression

$\frac{P (x)}{Q (x)} = \frac{A_{1}}{x - a_{1}} + \frac{A_{2}}{x - a_{2}} + \dots + \frac{A_{n}}{x - a_{n}} \frac{x^{3}}{{(x^{2} + 4)}^{2}} = \frac{A x + B}{(x^{2} + 4)} + \frac{C x + D}{{(x^{2} + 4)}^{2}} \frac{x^{3}}{{(x^{2} + 4)}^{2}} = \frac{(A x + B) (x^{2} + 4)}{(x^{2} + 4)} + \frac{(C x + D)}{{(x^{2} + 4)}^{2}}$

Step 3. Value of numerator of partial fractions

Rearrange the equation

$\frac{x^{3}}{{(x^{2} + 4)}^{2}} = \frac{(A x + B) (x^{2} + 4)}{(x^{2} + 4)} + \frac{(C x + D)}{{(x^{2} + 4)}^{2}} x^{3} = (A x + B) (x^{2} + 4) + (C x + D) x^{3} = A x^{3} + 4 A x + B x^{2} + 4 B + C x + D x^{3} + 0 x^{2} + 0 x + 0 = A x^{3} + B x^{2} + (4 A + C) x + (4 B + D)$

Equate the coefficients and constant

$A = 1 B = 0 a n d 4 A + C = 0 4 (1) + C = 0 C = - 4 a n d 4 B + D = 0 4 (0) + D = 0 D = 0$

So partial fraction decomposition is $\frac{x^{3}}{{(x^{2} + 4)}^{2}} = \frac{x}{(x^{2} + 4)} + \frac{- 4 x}{{(x^{2} + 4)}^{2}}$

Step 4. Verification

Simplify the partial fractions

$\frac{x}{(x^{2} + 4)} + \frac{- 4 x}{{(x^{2} + 4)}^{2}} = \frac{x (x^{2} + 4) - 4 x}{{(x^{2} + 4)}^{2}} = \frac{x^{3}}{{(x^{2} + 4)}^{2}}$

It is equal to the original expression so partial fraction decomposition is correct.

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In Problem, Write the partial fraction decomposition of each rational expression.
$\frac{x^{3}}{{(x^{2} + 4)}^{2}}$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Decomposition of rational expression

Step 3. Value of numerator of partial fractions

Step 4. Verification

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In Problem, Write the partial fraction decomposition of each rational expression.x3x2+42

Short Answer

Step by step solution

Step 1. Given data

Step 2. Decomposition of rational expression

Step 3. Value of numerator of partial fractions

Step 4. Verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Pure Maths

Decision Maths

Discrete Mathematics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problem, Write the partial fraction decomposition of each rational expression.
$\frac{x^{3}}{{(x^{2} + 4)}^{2}}$