Problem 65 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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: \(\frac{4x^2 +5x -9}{x^3 - 6x -9} = \frac{A}{x} + \frac{Bx + C}{x^2 - 6x - 9}\)

Step 3: 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.