Decompose P/Q, Where Q Has a Prime, Repeated Quadratic Factor
-$$\frac { 2 x - 5 } { \left( x ^ { 2 } + x + 6 \right) ^ { 2 } }$$
A) $\frac { A x + B } { x ^ { 2 } + x + 6 } + \frac { C x + D } { \left( x ^ { 2 } + x + 6 \right) ^ { 2 } }$
B) $\frac { A } { x ^ { 2 } + x + 6 } + \frac { B x + C } { \left( x ^ { 2 } + x + 6 \right) ^ { 2 } }$
C) $\frac { A } { x ^ { 2 } + x + 6 } + \frac { B } { \left( x ^ { 2 } + x + 6 \right) ^ { 2 } }$
D) $\frac { A x + B } { x ^ { 2 } + x + 6 } + \frac { C } { \left( x ^ { 2 } + x + 6 \right) ^ { 2 } }$

Question

Quizplus · Accepted Answer

When decomposing a fraction where the denominator has a repeated quadratic factor, each term in the decomposition must account for each power of the repeated factor up to its highest power. Here, the denominator $(x^2 + x + 6)^2$ is a quadratic factor squared, so the decomposition must include terms with both the quadratic factor to the first power and to the second power. Each term in the numerator must be of the form $Ax + B$ to match the degree of the quadratic factor, leading to the correct form being $\frac{Ax + B}{x^2 + x + 6} + \frac{Cx + D}{(x^2 + x + 6)^2}$.

Decompose P/Q, Where Q Has a Prime, Repeated Quadratic Factor 2x−5(x2+x+6)2\frac { 2 x - 5 } { \left( x ^ { 2 } + x + 6 \right) ^ { 2 } }(x2+x+6)22x−5​

Decompose P/Q, Where Q Has a Prime, Repeated Quadratic Factor $\frac { 2 x - 5 } { \left( x ^ { 2 } + x + 6 \right) ^ { 2 } }$