Decompose P/Q, Where Q Has Repeated Linear Factors
Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the
constants.
-$\frac { 48 - 8 x } { x ( x - 4 ) ^ { 2 } }$
A) $\frac { 3 } { x } + \frac { - 3 } { x - 4 } + \frac { 4 } { ( x - 4 ) ^ { 2 } }$
B) $\frac { 3 } { x } + \frac { 4 } { x - 4 } + \frac { - 3 } { ( x - 4 ) ^ { 2 } }$
C) $\frac { 3 } { x } + \frac { - 3 } { x - 4 } + \frac { 8 } { ( x - 4 ) ^ { 2 } }$
D) $\frac { - 3 } { x } + \frac { 3 } { x - 4 } + \frac { 4 } { ( x - 4 ) ^ { 2 } }$

Question

Quizplus · Accepted Answer

The denominator has a repeated linear factor, so the partial fraction decomposition will have the form:$\frac { A } { x } + \frac { B } { x - 4 } + \frac { C } { ( x - 4 ) ^ { 2 } }$Multiplying both sides by $x ( x - 4 ) ^ { 2 }$, we get:$48 - 8 x = A ( x - 4 ) ^ { 2 } + B x ( x - 4 ) + C x$Setting $x = 0$, we get:$48 = 16 A$$\Rightarrow A = 3$Setting $x = 4$, we get:$0 = 16 B$$\Rightarrow B = 0$Setting $x = 1$, we get:$40 = C$$\Rightarrow C = 40$Therefore, the partial fraction decomposition is:$\frac { 3 } { x } + \frac { 0 } { x - 4 } + \frac { 40 } { ( x - 4 ) ^ { 2 } }$Simplifying, we get:$\frac { 3 } { x } + \frac { 40 } { ( x - 4 ) ^ { 2 } }$

Decompose P/Q, Where Q Has Repeated Linear Factors Write the Form

Decompose P/Q, Where Q Has Repeated Linear Factors
Write the Form