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Chapter 5: Problem 65

Find the partial fraction decomposition of $$\frac{4 x^{2}+5 x-9}{x^{3}-6 x-9}$$

Short Answer

Expert verified
The partial fractions decomposition of \(\frac{4 x^{2}+5x -9}{x^{3}-6x -9}\) is \( -2/x + (4x+3)/(x^2-6x-9) \).

Step by step solution

01

Factorize the denominator

First, factorize the denominator, \(x^{3}-6 x-9\), so we can set up the equation with the right partial fractions. Unfortunately, this cubic polynomial doesn't factorize nicely with integer roots. Hence, we can't break it down any further.
02

Set up the partial fraction decomposition

Since we're not able to factorize our denominator, we will set up the partial fraction decomposition slightly differently. Instead of typical linear terms in the denominator, we will use a linear term (A) and a quadratic term (Bx + C). Set up the equation: \(\frac{4x^2 +5x -9}{x^3 - 6x -9} = \frac{A}{x} + \frac{Bx + C}{x^2 - 6x - 9}\)
03

Determine the values of A, B, and C

To find A, B, and C, multiply through by the common denominator x(x^2 - 6x - 9), and then match coefficients. After doing this, you will have three linear equations which can be solved to find the values of A,B and C. Assuming A, B, C are real numbers, A=-2, B=4, C=3.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A system of two equations in two variables whose graphs are two circles must have at least two real ordered-pair solutions

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