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Chapter 5: Problem 32

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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: \[ 9x + 2 = A (x^{2} + 2x + 2) + (Bx + C)(x - 2). \]
03

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.

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