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,

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Mechanics Maths

Pure Maths

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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

Decision Maths

Probability and Statistics

Mechanics Maths

Pure Maths

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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