Question

Write the partial fraction decomposition for the expression. $$ \frac{8 x^{2}+15 x+9}{(x+1)^{3}} $$

Short answer

The partial fraction decomposition of the given expression is \[ \frac{-6}{x + 1} + \frac{22}{(x + 1)^2} + \frac{8}{(x + 1)^3} \]

Step-by-step solution

Set up the equation

Everyone can write out the partial fractions decomposition form as follows: \[ \frac{8 x^{2}+15 x+9}{(x+1)^{3}} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)^3} \]

Clear the denominator

Multiply through by the common denominator \((x+1)^3\) to eliminate the fractions. This results in the following equation: \[ 8x^2 + 15x + 9 = A(x+1)^2 + B(x+1) + C \]

Solve for A, B, and C

The parameters can be determined by picking values for \(x\). A convenient choice is \(x = -1\), as this will cancel out terms to solve for \(C\). This gives \(C=8\). Next, differentiate the equation twice to solve for \(A\) and \(B\). Differentiating the original expression yields \[ 16x + 15 = 2A(x + 1) + B, \] and setting \(x = -1\) gives \(A = -6\). Differentiating a second time gives \[16 = 2A,\] and substititing \(A = -6\) gives \(B = 22\).

Write out the partial fractions decomposition

Now that the parameters \(A\), \(B\), and \(C\) have been found, one can substitute them into the decomposition from step 1 to obtain the result: \[ \frac{-6}{x + 1} + \frac{22}{(x + 1)^2} + \frac{8}{(x + 1)^3} \]