Write the partial fraction decomposition of the rational expression.
-$$\frac { x + 1 } { ( x - 2 ) ^ { 2 } ( x + 4 ) }$$
A) $\frac { 12 } { x - 2 } + \frac { 2 } { ( x - 2 ) ^ { 2 } } + \frac { - 12 } { x + 4 }$

B) $\frac { \frac { 1 } { 2 } } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 12 } } { x + 4 }$

C) $\frac { \frac { 1 } { 12 } } { x - 2 } + \frac { \frac { 1 } { 2 } } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 12 } } { x + 4 }$

D) $\frac { - 1 } { x - 2 } + \frac { \frac { 1 } { 4 } x } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 4 } } { x + 4 }$

Question

Write the partial fraction decomposition of the rational expression.
-$$\frac { x + 1 } { ( x - 2 ) ^ { 2 } ( x + 4 ) }$$
A) $\frac { 12 } { x - 2 } + \frac { 2 } { ( x - 2 ) ^ { 2 } } + \frac { - 12 } { x + 4 }$

B) $\frac { \frac { 1 } { 2 } } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 12 } } { x + 4 }$

C) $\frac { \frac { 1 } { 12 } } { x - 2 } + \frac { \frac { 1 } { 2 } } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 12 } } { x + 4 }$

D) $\frac { - 1 } { x - 2 } + \frac { \frac { 1 } { 4 } x } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 4 } } { x + 4 }$

Quizplus · Accepted Answer

The correct partial fraction decomposition must account for each factor in the denominator, including repeated factors. For a denominator of $(x-2)^2(x+4)$, the decomposition should include terms for $\frac{A}{x-2}$, $\frac{B}{(x-2)^2}$, and $\frac{C}{x+4}$. Option C correctly represents this structure with constants A, B, and C solved to balance the original equation.

Write the Partial Fraction Decomposition of the Rational Expression x+1(x−2)2(x+4)\frac { x + 1 } { ( x - 2 ) ^ { 2 } ( x + 4 ) }(x−2)2(x+4)x+1​

Write the Partial Fraction Decomposition of the Rational Expression $\frac { x + 1 } { ( x - 2 ) ^ { 2 } ( x + 4 ) }$