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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. 43 (page 765)

Write the partial fraction decomposition of the rational expression.
$\frac{4}{2 x^{2} - 5 x - 3}$

Short Answer

Expert verified

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

$\frac{4}{7 (x - 3)} - \frac{8}{7 (2 x + 1)}$

Step by step solution

01

Step 1. Given information.

Consider the given question,

$\frac{4}{2 x^{2} - 5 x - 3}$

02

Step 2. Decompose the rational expression.

The denominator can be factorized as $2 x^{2} - 5 x - 3 = (2 x + 1) (x - 3)$ .

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

$\frac{4}{2 x^{2} - 5 x - 3} = \frac{A}{(x - 3)} + \frac{B}{(2 x + 1)} . . . . . . (i)$

Where, Aand B are to be determined.

Multiply every side by $(2 x + 1) (x - 3)$ ,

$4 = A (2 x + 1) + B (x - 3) . . . . . . (i i)$

03

Step 3. Find the value of A and B.

Assume $x = 3$ in equation (ii),

$4 = A (7) A = \frac{4}{7}$

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

role="math" localid="1647589482779" $4 = A (2 (0) + 1) + B ((0) - 3) 4 = A (1) + B (- 3) 4 = A - 3 B . . . . . . (i i i)$

Substitute the value of Ain equation (iii),

$4 = \frac{4}{7} - 3 B B = - \frac{8}{7}$

04

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

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

$\frac{4}{2 x^{2} - 5 x - 3} = \frac{\frac{4}{7}}{x - 3} + \frac{\frac{- 8}{7}}{2 x + 1} \frac{4}{2 x^{2} - 5 x - 3} = \frac{4}{7 (x - 3)} - \frac{8}{7 (2 x + 1)}$

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Most popular questions from this chapter

Graph each equation of the system. Then solve the system to find the points of intersection.

$\{\begin{cases} y = \sqrt{36 - x^{2}} \\ y = 8 - x \end{cases}$

Verify that the values of the variables listed are solutions of the system of equations.

$\{\begin{matrix} \begin{array}{rcl} 3 x + 3 y + 2 z & = & 4 \\ x - y - z & = & 0 \end{array} \\ 2 y - 3 z = - 8 \end{matrix} x = 1, y = - 1, z = 2; (1, - 1, 2)$

In Problems, use the division algorithm to rewrite each improper fraction as the sum of a quotient and proper fraction. Find the partial fraction decomposition of the proper fraction. Finally, express the improper fraction as the sum of a quotient and the partial fraction decomposition.

$\frac{x^{4} - 5 x^{2} + x - 4}{x^{2} + 4 x + 4}$

Verify that the values of the variables listed are solutions of the system of equations.

$\{\begin{matrix} x - y = 3 \\ - 3 x + y = 1 \end{matrix} x = - 2, y = - 5; (- 2, - 5)$

Graph each inequality by hand.

$y > x^{2} + 2$

See all solutions

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