Decompose P/Q, Where Q Has Repeated Linear Factors
-$$\frac { 5 x - 5 } { ( x + 5 ) ( x + 3 ) ^ { 2 } }$$
A) $\frac { A } { x + 5 } + \frac { B } { x + 3 } + \frac { C } { ( x + 3 ) ^ { 2 } }$
B) $\frac { A } { x + 5 } + \frac { B } { x + 3 } + \frac { C } { x + 3 } + \frac { D x + E } { ( x + 3 ) ^ { 2 } }$
C) $\frac { A } { x + 5 } + \frac { B } { x + 3 } + \frac { C x + D } { ( x + 3 ) ^ { 2 } }$
D) $\frac { A } { x + 5 } + \frac { B } { x + 3 } + \frac { C } { x + 3 } + \frac { D } { ( x + 3 ) ^ { 2 } }$

Question

Quizplus · Accepted Answer

When decomposing a fraction where the denominator has repeated linear factors, each factor $x + a$ that appears $n$ times in the denominator must be represented by $n$ terms in the decomposition, each with the factor to a successively lower power until reaching 1. In this case, $x + 3$ appears twice, so the correct decomposition includes $\frac{B}{x + 3}$ and $\frac{C}{(x + 3)^2}$, along with $\frac{A}{x + 5}$ for the distinct factor.

Decompose P/Q, Where Q Has Repeated Linear Factors - 5x−5(x+5)(x+3)2\frac { 5 x - 5 } { ( x + 5 ) ( x + 3 ) ^ { 2 } }(x+5)(x+3)25x−5​

Decompose P/Q, Where Q Has Repeated Linear Factors
- $\frac { 5 x - 5 } { ( x + 5 ) ( x + 3 ) ^ { 2 } }$