Question

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))

Accepted Answer

Identify the factors in the denominator: $$x^{2} - 8x + 16$ and x$$^{2}$$ + 3x + 4. $$Notice that $$x^{2} - 8x + 16$ is a perfect square trinomial and can be factored as (x$$ - $$4)^{2}. $$The other quadratic $$x^{2} + 3x + 4$ does not factor nicely over the reals, so it remains as is. Since (x$$ - $$4)^{2} is a $$repeated linear factor, set up terms for both the first power and the second power of $$(x - 4) in $$the decomposition. For each linear factor, the numerator is a constant: $$\frac{A}{x - 4} + \frac{B}{(x - 4$$)^{2}$$}. $$For the irreducible quadratic factor $$x^{2} + 3x + 4$, the numerator must be a linear expression of the form Cx + D$ because the degree of the numerator is one less than the degree of the denominator factor. Combine all parts to write the partial fraction decomposition setup as: $$\frac{A}{x - 4} + \frac{B}{(x - $$4)^{2}$$} + \frac{Cx + D}{$$x^{2} + 3x + 4$$}$$. This setup is the appropriate form for the partial fraction decomposition of the given expression. The next step (not required here) would be to multiply both sides by the denominator and solve for the constants A$, B$, C$, and D$.

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))

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Key Concepts

Partial Fraction Decomposition

Factoring Quadratic Expressions

Form of Partial Fractions for Quadratic Denominators