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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. 44 (page 765)

Write the partial fraction decomposition of the rational expression.
$\frac{4 x}{2 x^{2} + 3 x - 2}$

Short Answer

Expert verified

The partial fraction decomposition of the given rational expression is given below,

$\frac{4}{5 (2 x - 1)} + \frac{8}{5 (x + 2)}$

Step by step solution

01

Step 1. Given information.

Consider the given question,

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

02

Step 2. Decompose the rational expression.

The denominator can be factorized as $2 x^{2} + 3 x - 2 = (x + 2) (2 x - 1)$ .

The denominator has a repeated non-reducible quadratic factors. Then we decompose the rational expression as follows,

$\frac{4 x}{2 x^{2} + 3 x - 2} = \frac{A}{(x + 2)} + \frac{B}{(2 x - 1)} . . . . . . (i)$

Where, Aand B are to be determined.

Multiply every side by $(x + 2) (2 x - 1)$ ,

$4 x = A (2 x - 1) + B (x + 2) . . . . . . (i i)$

03

Step 3. Find the value of A and B.

Assume $x = - 2$ in equation (ii),

$4 (- 2) = A (2 (- 2) - 1) + B ((- 2) + 2) - 8 = A (- 5) A = \frac{8}{5}$

Assume $x = 0$ in equation (ii),

role="math" localid="1647593570391" $4 (0) = A (2 (0) - 1) + B ((0) + 2) 0 = - A + 2 B B = \frac{A}{2} . . . . . . (i i i)$

Substitute the value of Ain equation (iii),

$B = \frac{\frac{8}{5}}{2} B = \frac{4}{5}$

04

Step 4. Substitute the values in equation (i).

On substituting the values of A and Bin equation (i),

$\frac{4 x}{2 x^{2} + 3 x - 2} = \frac{\frac{4}{5}}{2 x - 1} + \frac{\frac{8}{5}}{x + 2} \frac{4 x}{2 x^{2} + 3 x - 2} = \frac{4}{5 (2 x - 1)} + \frac{8}{5 (x + 2)}$

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Most popular questions from this chapter

The difference of two numbers is the same as their product, and the sum of their reciprocals is $5$ . Find the numbers.

Solve each system of equations. If the system has no solution, say that it is inconsistent

x + 2y - z = -3

2x - 4y + z = -7

-2x + 2y - 3z = 4

In Problems 5鈥�24, graph each equation of the system. Then solve the system to find the points of intersection.

$x^{2} + y^{2} = 4; y^{2} - x = 4$

In Problems 5鈥�24, graph each equation of the system. Then solve the system to find the points of intersection.

$x = 2 y; x = y^{2} - 2 y$

In Problems 23鈥�34, graph each system of linear inequalities by hand. Verify your results using a graphing utility:

$\{\begin{cases} 2 x + 3 y 鈮� 6 \\ 2 x + 3 y 鈮� 0 \end{cases}$

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