Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants.
-$$\frac { 3 x + 4 } { \left( x ^ { 2 } + x + 5 \right) ^ { 2 } }$$
A) $\frac { A x + B } { x ^ { 2 } + x + 5 } + \frac { C } { \left( x ^ { 2 } + x + 5 \right) ^ { 2 } }$
B) $\frac { A x + B } { x ^ { 2 } + x + 5 } + \frac { C x + D } { \left( x ^ { 2 } + x + 5 \right) ^ { 2 } }$
C) $\frac { \mathrm { A } } { \mathrm { x } ^ { 2 } + \mathrm { x } + 5 } + \frac { \mathrm { Bx } + \mathrm { C } } { \left( \mathrm { x } ^ { 2 } + \mathrm { x } + 5 \right) ^ { 2 } }$
D) $\frac { A } { x ^ { 2 } + x + 5 } + \frac { B } { \left( x ^ { 2 } + x + 5 \right) ^ { 2 } }$

Question

Quizplus · Accepted Answer

For a repeated irreducible quadratic factor $(x^2 + x + 5)^2$, the partial fraction decomposition includes terms for each power of the factor up to its multiplicity. Each term in the decomposition for an irreducible quadratic factor should have a numerator in the form $Ax + B$, where $A$ and $B$ are constants to be determined. Thus, the correct form includes both $\frac{Ax + B}{x^2 + x + 5}$ and $\frac{Cx + D}{(x^2 + x + 5)^2}$ to account for the repeated quadratic factor.

Write the Form of the Partial Fraction Decomposition of the Rational