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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 34 (page 794)

In Problem, Write the partial fraction decomposition of each rational expression.
$\frac{6}{x (x - 4)}$

Short Answer

Expert verified

Partial fraction decomposition of a rational expression $\frac{6}{x (x - 4)}$ is

$\frac{6}{x (x - 4)} = \frac{\frac{- 3}{2}}{x} + \frac{\frac{3}{2}}{(x - 4)}$

Step by step solution

01

Step 1. Given data

The given rational expression is

$\frac{6}{x (x - 4)}$

02

Step 2. Decomposition of rational expression

partial fraction decomposition of rational expression

$\frac{P (x)}{Q (x)} = \frac{A_{1}}{x - a_{1}} + \frac{A_{2}}{x - a_{2}} + 鈰� + \frac{A_{n}}{x - a_{n}} \frac{6}{x (x - 4)} = \frac{A}{x} + \frac{B}{(x - 4)} \frac{6}{x (x - 4)} = \frac{A (x - 4) + B x}{x (x - 4)} 6 = A (x - 4) + B x$

03

Step 3. Value of numerators of partial fractions

Rearrange the equation

$6 = A (x - 4) + B x 0 x + 6 = (A + B) x + (- 4 A)$

Equate the constants

$- 4 A = 6 A = \frac{- 3}{2}$

Equate the coefficients of x

$A + B = 0 \frac{- 3}{2} + B = 0 B = \frac{3}{2}$

So partial fraction decomposition is $\frac{6}{x (x - 4)} = \frac{\frac{- 3}{2}}{x} + \frac{\frac{3}{2}}{(x - 4)}$

04

Step 4. Verification

Simplify the partial fractions

$\frac{\frac{- 3}{2}}{x} + \frac{\frac{3}{2}}{(x - 4)} = \frac{\frac{- 3}{2} (x - 4)}{x (x - 4)} + \frac{\frac{3}{2} x}{x (x - 4)} = \frac{- 3 x + 12 + 3 x}{2 x (x - 4)} = \frac{12}{2 x (x - 4)} = \frac{6}{x (x - 4)}$

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

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