Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants. $$\frac { 5 x + 4 } { ( x + 2 ) ^ { 4 } }$$
A) $\frac { A } { ( x + 2 ) ^ { 2 } } + \frac { B } { ( x + 2 ) ^ { 3 } } + \frac { C } { ( x + 2 ) ^ { 4 } }$
B) $\frac { A } { ( x + 2 ) ^ { 4 } }$
C) $\frac { 5 } { x + 2 } - \frac { 4 } { ( x + 2 ) ^ { 2 } } - \frac { 2 } { ( x + 2 ) ^ { 3 } } - \frac { 4 } { ( x + 2 ) ^ { 4 } }$
D) $\frac { A } { ( x + 2 ) ^ { 2 } } + \frac { B } { ( x + 2 ) ^ { 2 } }$
E) $\frac { A } { x + 2 } + \frac { B } { ( x + 2 ) ^ { 2 } } + \frac { C } { ( x + 2 ) ^ { 3 } } + \frac { D } { ( x + 2 ) ^ { 4 } }$

Question

Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants.​ $$\frac { 5 x + 4 } { ( x + 2 ) ^ { 4 } }$$ ​
A) $\frac { A } { ( x + 2 ) ^ { 2 } } + \frac { B } { ( x + 2 ) ^ { 3 } } + \frac { C } { ( x + 2 ) ^ { 4 } }$
B) $\frac { A } { ( x + 2 ) ^ { 4 } }$
C) $\frac { 5 } { x + 2 } - \frac { 4 } { ( x + 2 ) ^ { 2 } } - \frac { 2 } { ( x + 2 ) ^ { 3 } } - \frac { 4 } { ( x + 2 ) ^ { 4 } }$
D) $\frac { A } { ( x + 2 ) ^ { 2 } } + \frac { B } { ( x + 2 ) ^ { 2 } }$
E) $\frac { A } { x + 2 } + \frac { B } { ( x + 2 ) ^ { 2 } } + \frac { C } { ( x + 2 ) ^ { 3 } } + \frac { D } { ( x + 2 ) ^ { 4 } }$

Quizplus · Accepted Answer

For a repeated linear factor $(x + a)^n$, the partial fraction decomposition includes all powers of the factor from 1 up to $n$. Thus, each power of $(x + 2)$ from 1 to 4 needs its own term in the decomposition.

Write the Form of the Partial Fraction Decomposition of the Rational