Q. 25 In Problems 24聽and 25, write th... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 25 (page 796)

In Problems $24$ and $25$ , write the partial fraction decomposition of each rational expression.
$\frac{4 x^{2} - 3}{x {(x^{2} + 3)}^{2}}$

Short Answer

Expert verified

Partial fraction decomposition is $\frac{- \frac{1}{3}}{x} + \frac{\frac{1}{3} x}{(x^{2} + 3)} + \frac{5 x}{{(x^{2} + 3)}^{2}}$

Step by step solution

Step 1. Given information

An expression is given as $\frac{4 x^{2} - 3}{x {(x^{2} + 3)}^{2}}$

Step 2. Decomposition fraction

First write the expression as

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

Now multiply $x {(x^{2} + 3)}^{2}$ on both side,

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

Now simplify,

$= A (x^{4} + 6 x^{2} + 9) + (B x + C) (x^{3} + 3 x) + (D x + E) (x) 4 x^{2} - 3 = (A + B) x^{4} + C x^{3} + (6 A + 3 B + D) x^{2} + (3 C + E) x + 9 A$

Step 3. System of equations

Equate the coefficients to form new set of equations,

$\{\begin{cases} A + B = 0 \\ C = 0 \\ 6 A + 3 B + D = 4 \\ 3 C + E = 0 \\ 9 A = - 3 \end{cases}$

From above set $A = - \frac{1}{3}$ and $B = \frac{1}{3}$

Substitute the values of A and B in the equation $6 A + 3 B + D = 4$

$6 (- \frac{1}{3}) + 3 (\frac{1}{3}) + D = 4 - 2 + 1 + D = 4 - 1 + D = 4 D = 5$

Step 4. Solving set of equations

Now substitute the value in the equation

$3 C + E = 0 3 (0) + E = 0 E = 0$

Now substitute the value in first equation,

$\frac{A}{x} + \frac{B x + C}{(x^{2} + 3)} + \frac{D x + E}{{(x^{2} + 3)}^{2}} = \frac{(- \frac{1}{3})}{x} + \frac{(\frac{1}{3}) x + (0)}{(x^{2} + 3)} + \frac{5 x + 0}{{(x^{2} + 3)}^{2}} = \frac{(- \frac{1}{3})}{x} + \frac{(\frac{1}{3}) x}{(x^{2} + 3)} + \frac{5 x}{{(x^{2} + 3)}^{2}}$

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91影视

In Problems $24$ and $25$ , write the partial fraction decomposition of each rational expression.
$\frac{4 x^{2} - 3}{x {(x^{2} + 3)}^{2}}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Decomposition fraction

Step 3. System of equations

Step 4. Solving set of equations

One App. One Place for Learning.

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91影视

In Problems 24and 25, write the partial fraction decomposition of each rational expression. 4x2-3xx2+32

Short Answer

Step by step solution

Step 1. Given information

Step 2. Decomposition fraction

Step 3. System of equations

Step 4. Solving set of equations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Logic and Functions

Applied Mathematics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

In Problems $24$ and $25$ , write the partial fraction decomposition of each rational expression.
$\frac{4 x^{2} - 3}{x {(x^{2} + 3)}^{2}}$