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Chapter 5: Problem 19

Write the partial fraction decomposition of each rational expression. $$\frac{4 x^{2}-7 x-3}{x^{3}-x}$$

Short Answer

Expert verified
The partial fraction decomposition of \( \frac{4x^{2}-7x-3}{x(x-1)(x+1)} \) is : \(\frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+1}\) where A, B, and C are real numbers obtained by solving the system of equations from Step 5.

Step by step solution

01

Factorize the Denominator

Firstly, factorize the denominator. The cubic polynomial can be factored as \(x^{3}-x = x(x^2-1)= x(x-1)(x+1)\)
02

Set up the Partial Fractions

Next, set up the partial fractions with unknown coefficients. This can be written as: \(\frac{4x^{2}-7x-3}{x(x-1)(x+1)}= \frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+1}\)
03

Clear the fractions

Multiply through by the denominator \(x(x-1)(x+1)\) to get rid of the fractions. So, you get: \(4x^{2}-7x-3 = A(x-1)(x+1) + Bx(x+1) +C x(x-1)\)
04

Expansion and Simplification

Expand the right hand side and collect like terms. This will result in a polynomial of the same degree as the left hand side.
05

Comparing Coefficients

Set coefficients for terms in the resulted polynomial equal to the coefficients for the same power of x in the original polynomial. This sets up a system of linear equations.
06

Solve for A, B and C

Solve this system of linear equations for A, B and C. These will be the coefficients (numerators) of the partial fractions.
07

Write the Decomposed Fractions

Write the decomposed fractions by replacing A, B and C in the set-up from Step 2 with the values obtained in Step 6.

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Most popular questions from this chapter

In Exercises \(19-30,\) solve each system by the addition method. \(3 x+2 y=14\) \(3 x-2 y=10\)

Explain how to find the partial fraction decomposition of a rational expression with a prime quadratic factor in the denominator.

In Exercises \(19-30,\) solve each system by the addition method. \(x+y=6\) \(x-y=-2\)

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} 5 x+2 y &=0 \\ x-3 y &=0 \end{aligned} $$

Explain how to find the partial fraction decomposition of a rational expression with distinct linear factors in the denominator.
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