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.
-$$\frac { x ^ { 2 } + 4 x + 8 } { ( x - 2 ) ( x - 5 ) ^ { 2 } \left( x ^ { 2 } + 49 \right) ^ { 2 } }$$
A) $\frac { A } { x - 2 } + \frac { B } { ( x - 5 ) ^ { 2 } } + \frac { C } { \left( x ^ { 2 } + 49 \right) ^ { 2 } }$
B) $\frac { A } { x - 2 } + \frac { B } { ( x - 5 ) ^ { 2 } } + \frac { C x + D } { \left( x ^ { 2 } + 49 \right) ^ { 2 } }$
C) $\frac { A } { x - 2 } + \frac { B } { x - 5 } + \frac { C } { ( x - 5 ) ^ { 2 } } + \frac { D } { x ^ { 2 } + 49 } + \frac { F } { \left( x ^ { 2 } + 49 \right) ^ { 2 } }$
D) $\frac { A } { x - 2 } + \frac { B } { x - 5 } + \frac { C } { ( x - 5 ) ^ { 2 } } + \frac { D x + E } { x ^ { 2 } + 49 } + \frac { F x + G } { \left( x ^ { 2 } + 49 \right) ^ { 2 } }$

Question

Quizplus · Accepted Answer

For each factor in the denominator, the partial fraction decomposition must account for each power of the factor up to its highest power in the denominator. For linear factors like $x-2$ and $x-5$, we use constants $A$, $B$, and $C$ for each respective power. For the quadratic factor $x^2+49$, which is repeated, we use $Dx+E$ for the first power and $Fx+G$ for the second power, reflecting the need for a linear numerator for each power of a quadratic factor.

Choose the One Alternative That Best Completes the Statement or Answers