Problem 55 - 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 fraction decomposition of \( \frac{4 x^{2}+5 x-9}{x^{3}-6 x-9} \) can be written in the form \( \frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+3} \). The coefficients A, B, and C can be found by solving equations obtained from equating coefficients of similar terms.

Step 1: Factor the Denominator

The first step in performing a partial fraction decomposition is to factor the denominator if possible. However, the denominator of the given function, \(x^{3}-6 x-9\), cannot be factorised in simple manner. So, we will leave it as it is.

Step 2: Set Up Partial Fractions

Express the given fraction as sum of simpler fractions, where each fraction has denominator equal to a factor of the original denominator. In this case, we will consider \(x^{3}-6 x-9\) as three linear factors, even though they cannot be explicitly written, and write the fraction as: \[ \frac{4 x^{2}+5 x-9}{x^{3}-6 x-9} = \frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+3}\]

Step 3: Cross-Multiply And Expand

Cross multiply to remove the denominator and then expand the expression, this gives: \[4 x^{2}+5 x-9 = A(x-3)(x+3) + B*x*(x+3) + C*x*(x-3)\]

Step 4: Group Similar Terms

Group similar terms together. This step will give an equation formed by collecting coefficients of similar terms.

Step 5: Solve for Coefficients A, B, and C

Solve the coefficients A, B, and C from the equation obtained above. In this case, coefficients can be found by substitution method by putting suitable values of x. For example, \(x = 0, x = 3, x = -3\).