Chapter 11: Q. 31 (page 765)

In Problems 13–46, write the partial fraction decomposition of each rational expression.
$\frac{x}{(3 x - 2) (2 x + 1)}$

Short Answer

Expert verified

The partial fraction decomposition of a rational expression is

$\frac{x}{(3 x - 2) (2 x + 1)} = \frac{\frac{2}{7}}{(3 x - 2)} + \frac{\frac{1}{7}}{(2 x + 1)}$

Step by step solution

Step 1. Given information

Given rational expression is

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

Step 2. Partial fraction decomposition

partial fraction decomposition of a rational expression

$\frac{P (x)}{Q (x)} = \frac{A_{1}}{x - a_{1}} + \frac{A_{2}}{x - a_{2}} + \dots + \frac{A_{n}}{x - a_{n}} \frac{x}{(3 x - 2) (2 x + 1)} = \frac{A}{(3 x - 2)} + \frac{B}{(2 x + 1)} \dots (i) \frac{x}{(3 x - 2) (2 x + 1)} = \frac{A (2 x + 1)}{(3 x - 2) (2 x + 1)} + \frac{B (3 x - 2)}{(3 x - 2) (2 x + 1)} x = A (2 x + 1) + B (3 x - 2) x + 0 = (2 A + 3 B) x + (A - 2 B) \dots (i i)$

Step 3. Values of coefficients and constants of the numerator

Compare the constants in equation ii

$0 = A - 2 B A = 2 B$

Compare the coefficient of xin equation ii and Substitute the expression for A

$1 = 2 A + 3 B 1 = 2 (2 B) + 3 (B) B = \frac{1}{7}$

$A = 2 (\frac{1}{7}) = \frac{2}{7}$

Step 4. partial fraction decomposition of a rational expression

Substitute the value of Aand B in the equation i

$\frac{x}{(3 x - 2) (2 x + 1)} = \frac{A}{(3 x - 2)} + \frac{B}{(2 x + 1)} \frac{x}{(3 x - 2) (2 x + 1)} = \frac{\frac{2}{7}}{(3 x - 2)} + \frac{\frac{1}{7}}{(2 x + 1)}$

So the partial fraction decomposition of a rational expression is $\frac{x}{(3 x - 2) (2 x + 1)} = \frac{\frac{2}{7}}{(3 x - 2)} + \frac{\frac{1}{7}}{(2 x + 1)}$

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In Problems 13–46, write the partial fraction decomposition of each rational expression.
$\frac{x}{(3 x - 2) (2 x + 1)}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Partial fraction decomposition

Step 3. Values of coefficients and constants of the numerator

Step 4. partial fraction decomposition of a rational expression

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In Problems 13–46, write the partial fraction decomposition of each rational expression.x(3x-2)(2x+1)

Short Answer

Step by step solution

Step 1. Given information

Step 2. Partial fraction decomposition

Step 3. Values of coefficients and constants of the numerator

Step 4. partial fraction decomposition of a rational expression

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Mechanics Maths

Calculus

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.

In Problems 13–46, write the partial fraction decomposition of each rational expression.
$\frac{x}{(3 x - 2) (2 x + 1)}$