Choose the one alternative that best completes the statement or answers the question.
Set up the form for the partial fraction decomposition. Do not solve for A, B, C, and so on.
-$\frac { 2 x ^ { 2 } - 7 x + 2 } { 4 x ^ { 3 } + 64 x }$
A) $\frac { A } { 4 x } + \frac { B } { x ^ { 2 } + 16 }$
B) $\frac { A } { 4 x } + \frac { B } { x + 4 } + \frac { C } { x - 4 }$
C) $\frac { A } { 4 x } + \frac { B x + \mathrm { C } } { x + 4 } + \frac { D x + E } { x - 4 }$
D) $\frac { A } { 4 x } + \frac { B x + \mathrm { C } } { x ^ { 2 } + 16 }$

Question

Quizplus · Accepted Answer

The denominator $4x^3 + 64x$ can be factored as $4x(x^2 + 16)$. The term $x^2 + 16$ cannot be factored further over the real numbers, so the correct partial fraction decomposition must account for the linear term $4x$ and the irreducible quadratic term $x^2 + 16$. This leads to the form $\frac{A}{4x} + \frac{Bx + C}{x^2 + 16}$.

Choose the One Alternative That Best Completes the Statement or Answers