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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. 42 (page 765)

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

Short Answer

Expert verified

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

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

Step by step solution

01

Step 1. Given information.

Consider the given question,

$\frac{x^{2}}{{(x^{2} + 4)}^{3}}$

02

Step 2. Decompose the rational expression.

The denominator has a repeated non-reducible quadratic factor $(x^{2} + 4)$ . Then we decompose the rational expression as follows,

$\frac{x^{2}}{{(x^{2} + 4)}^{3}} = \frac{A x + B}{(x^{2} + 4)} + \frac{C x + D}{{(x^{2} + 4)}^{2}} + \frac{E x + F}{{(x^{2} + 4)}^{3}} . . . . . . (i)$

Where, A,B,C,D,E and F are to be determined.

Multiply every side by ${(x^{2} + 4)}^{3}$ ,

$x^{2} = (A x + B) {(x^{2} + 4)}^{2} + (C x + D) (x^{2} + 4) + (E x + F) . . . . . . (i i)$

03

Step 3. Expand the identity.

On expanding the identity in equation (ii),

$x^{2} = x^{5} (A) + x^{4} (B) + x^{3} (C) + x^{2} (8 A - 8 B + D) + x (16 A + 4 C + E) + 16 B + 4 D + F$

Equate the coefficients of like powers of x to get,

$A = 0 B = 0 C = 0 8 A + 8 B + D = 1 10 A + 4 C + E = 0 16 B + 4 D + F = 0$

04

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

On solving the system of equations, we get,

$A = 0, B = 0, C = 0, D = 1, E = 0, F = - 4$

Substitute the values in equation (i),

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

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Most popular questions from this chapter

If the lines that make up a system of two linear equations are coincident, then the system is __________ and the equations are ___________.

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

$\{\begin{array}{r} \begin{array}{rcl} 4 x - z & = & 7 \\ 8 x + 5 y - z & = & 0 \end{array} \\ - x - y + 5 z = 6 \end{array} x = 2, y = - 3, z = 1; (2, - 3, 1)$

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

$\{\begin{matrix} 4 x + 5 y = - 3 \\ - 2 y = - 8 \end{matrix}$

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

$\{\begin{matrix} \begin{array}{rcl} 3 x + 3 y + 2 z & = & 4 \\ x - 3 y + z & = & 10 \end{array} \\ 5 x - 2 y - 3 z = 8 \end{matrix} x = 2, y = - 2, z = 2; (2, - 2, 2)$

Graph the equation: $y^{2} = x^{2} - 1$

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