Problem 32 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{9 x+2}{(x-2)\left(x^{2}+2 x+2\right)}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \( \frac{9 x+2}{(x-2)\left(x^{2}+2 x+2\right)} \) is \( \frac{A}{x - 2} + \frac{Bx + C}{x^{2} + 2x + 2} \), where A, B, and C are constants to be determined by equating coefficients.

Step 1: Identify the Form of the Decomposition

Recognize that the original fraction can be decomposed into the following form:\[ \frac{9 x + 2}{(x - 2)(x^{2} + 2x + 2)} = \frac{A}{x - 2} + \frac{Bx + C}{x^{2} + 2x + 2}. \] The simple linear term in the denominator leads to the \( A/(x - 2) \) term, while the quadratic term in the denominator leads to the \( (Bx + C)/(x^{2} + 2x + 2) \) term.

Step 2: Multiply Through by the Common Denominator

Multiply through by the common denominator to clear the fractions, and set this equal to the numerator of the original fraction: \[ 9x + 2 = A (x^{2} + 2x + 2) + (Bx + C)(x - 2). \]

Step 3: Solve for A, B, and C

Expand and group like terms, and equate coefficients to solve for A, B, and C. The values of these constants will give the partial fraction decomposition.