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

Question

Quizplus · Accepted Answer

For a rational expression, the partial fraction decomposition involves breaking down the expression into simpler fractions. The denominator $x+4$ is linear, so it gets a simple numerator $A$. The quadratic term $(x^2+x-4)^2$ is repeated, so the first occurrence gets a linear numerator $Bx+C$, and the second occurrence, being squared, also gets a linear numerator $Dx+E$. This matches option B.

Write the Form of the Partial Fraction Decomposition of the Rational