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.
-$$\frac { 7 x ^ { 5 } + 5 x ^ { 3 } + 7 x ^ { 2 } + 6 } { x ( x - 5 ) ^ { 3 } \left( x ^ { 2 } + 2 x + 5 \right) ^ { 2 } }$$
A) $\frac { A } { x } + \frac { B } { x - 5 } + \frac { C } { ( x - 5 ) ^ { 2 } } + \frac { D } { ( x - 5 ) ^ { 3 } } + \frac { E x } { x ^ { 2 } + 2 x + 5 } + \frac { F x } { \left( x ^ { 2 } + 2 x + 5 \right) ^ { 2 } }$
B) $\frac { A } { x } + \frac { B } { x - 5 } + \frac { C } { ( x - 5 ) ^ { 2 } } + \frac { D } { ( x - 5 ) ^ { 3 } } + \frac { E } { x ^ { 2 } + 2 x + 5 } + \frac { F } { \left( x ^ { 2 } + 2 x + 5 \right) ^ { 2 } }$
C) $\frac { A } { x } + \frac { B } { x - 5 } + \frac { C } { ( x - 5 ) ^ { 2 } } + \frac { D } { ( x - 5 ) ^ { 3 } } + \frac { E x + F } { x ^ { 2 } + 2 x + 5 } + \frac { G x + H } { \left( x ^ { 2 } + 2 x + 5 \right) ^ { 2 } }$
D) $\frac { A } { x } + \frac { B } { ( x - 5 ) ^ { 3 } } + \frac { C x + D } { \left( x ^ { 2 } + 2 x + 5 \right) ^ { 2 } }$

Question

Quizplus · Accepted Answer

For the given rational function, the correct partial fraction decomposition must account for each factor in the denominator to its respective power and for quadratic terms, a linear numerator (Ax + B) is required. The denominator has a linear term $x$, a cubic term $(x - 5)^3$, and a quadratic term $(x^2 + 2x + 5)^2$. Each of these requires a separate fraction in the decomposition. For the cubic term, we need separate fractions for each power from 1 to 3. For the quadratic term raised to the second power, we need two fractions, each with a linear numerator to account for all possible linear combinations. Therefore, option C correctly sets up the partial fraction decomposition with the appropriate terms for each factor in the denominator.

Choose the One Alternative That Best Completes the Statement or Answers