Decompose P/Q, Where Q Has a Prime, Repeated Quadratic Factor
Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the
constants.
-$$\frac { 3 x - 1 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } }$$
A) $\frac { A x + B } { x ^ { 2 } + x + 1 } + \frac { C x + D } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } }$
B) $\frac { \mathrm { A } } { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } + \frac { \mathrm { Bx } + \mathrm { C } } { \left( \mathrm { x } ^ { 2 } + \mathrm { x } + 1 \right) ^ { 2 } }$
C) $\frac { \mathrm { A } } { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } + \frac { \mathrm { B } } { \left( \mathrm { x } ^ { 2 } + \mathrm { x } + 1 \right) ^ { 2 } }$
D) $\frac { A x + B } { x ^ { 2 } + x + 1 } + \frac { C } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } }$

Question

Quizplus · Accepted Answer

For a repeated quadratic factor of the form $(ax^2 + bx + c)^n$, the partial fraction decomposition includes terms for each power of the factor up to $n$, each with a numerator of the form $Ax + B$. Since $(x^2 + x + 1)^2$ is a repeated quadratic factor, the correct decomposition includes $\frac{Ax + B}{x^2 + x + 1}$ for the first power and $\frac{Cx + D}{(x^2 + x + 1)^2}$ for the second power, matching option A.

Decompose P/Q, Where Q Has a Prime, Repeated Quadratic Factor 3x−1(x2+x+1)2\frac { 3 x - 1 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } }(x2+x+1)23x−1​

Decompose P/Q, Where Q Has a Prime, Repeated Quadratic Factor $\frac { 3 x - 1 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } }$