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Chapter 7: Problem 41

Give the appropriate form of the partial fraction decomposition for the following functions. $$\frac{2 x^{2}+3}{\left(x^{2}-8 x+16\right)\left(x^{2}+3 x+4\right)}$$

Short Answer

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Question: Determine the partial fraction decomposition form of the given function: $$\frac{2 x^{2}+3}{\left(x^{2}-8 x+16\right)\left(x^{2}+3 x+4\right)}$$ Answer: The partial fraction decomposition form is: $$\frac{2 x^{2}+3}{\left(x^{2}-8 x+16\right)\left(x^{2}+3 x+4\right)} = \frac{Ax + B}{x^2 - 8x + 16} + \frac{Cx + D}{x^2 + 3x + 4}$$

Step by step solution

01

Identify the factors in the denominator

In the given function, the denominator is \(\left(x^{2}-8 x+16\right)\left(x^{2}+3 x+4\right)\). These are two quadratic factors, \((x^2 - 8x + 16)\) and \((x^2 + 3x + 4)\).
02

Determine the form of the partial fraction decomposition

Since we have two distinct quadratic factors, the partial fraction decomposition form will be: $$\frac{2 x^{2}+3}{\left(x^{2}-8 x+16\right)\left(x^{2}+3 x+4\right)} = \frac{Ax + B}{x^2 - 8x + 16} + \frac{Cx + D}{x^2 + 3x + 4}$$ where A, B, C, and D are constants to be determined.

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