Chapter 11: Q. 41 (page 765)

Write the partial fraction decomposition of each rational expression.
$\frac{x^{3}}{{(x 2 + 16)}^{3}}$

Short Answer

Expert verified

The partial fraction decomposition is $\frac{x^{3}}{{(x 2 + 16)}^{3}} = \frac{x}{{(x^{2} + 16)}^{2}} + \frac{- 16 x}{{(x^{2} + 16)}^{3}}$

Step by step solution

Step 1. Given Information.

The rational expression is,

$\frac{x^{3}}{{(x^{2} + 16)}^{3}} = \frac{A x + B}{x^{2} + 16} + \frac{C x + D}{{(x^{2} + 16)}^{2}} + \frac{E x + F}{{(x^{2} + 16)}^{3}}$

$\frac{x^{3}}{{(x^{2} + 16)}^{3}} = \frac{{(x^{2} + 16)}^{2}}{(A x + B)} + \frac{(x^{2} + 16)}{(C x + D)} + \frac{E x + F}{{(x^{2} + 16)}^{3}}$

Therefore,

$x^{3} = {(x^{2} + 16)}^{2} (A x + B) + (x^{2} + 16) (C x + D) + E x + F$

$x^{3} = x^{5} A + x^{4} B + 32 x^{3} A + x^{3} C + 32 x^{2} B + x^{2} D + 256 x A + 16 x C + x E + 256 B + 16 D + F$

Therefore,

$A = 0 B = 0 32 A + C = 132 B + D = 0 256 A + 16 C + E = 0 256 B + 16 D + F = 0$

Step 2. Partial Fraction Decomposition.

Decompose the rational fraction, we get,

$A = 0, B = 0, C = 1, D = 0, E = - 16, F = 0$

Therefore,

role="math" localid="1647535930996" $\frac{x^{3}}{{(x 2 + 16)}^{3}} = \frac{x}{{(x^{2} + 16)}^{2}} + \frac{- 16 x}{{(x^{2} + 16)}^{3}}$

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Write the partial fraction decomposition of each rational expression.
$\frac{x^{3}}{{(x 2 + 16)}^{3}}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Partial Fraction Decomposition.

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Write the partial fraction decomposition of each rational expression.x3(x2+16)3

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Partial Fraction Decomposition.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Mechanics Maths

Logic and Functions

Statistics

Study anywhere. Anytime. Across all devices.

Write the partial fraction decomposition of each rational expression.
$\frac{x^{3}}{{(x 2 + 16)}^{3}}$