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.
-$$\frac { 7 x + 2 } { ( x - 4 ) \left( x ^ { 2 } + x + 2 \right) ^ { 2 } }$$
A) $\frac { A } { x - 4 } + \frac { B x + C } { x ^ { 2 } + x + 2 } + \frac { D x + E } { \left( x ^ { 2 } + x + 2 \right) ^ { 2 } }$
B) $\frac { A } { x - 4 } + \frac { B } { x ^ { 2 } + x + 2 } + \frac { C } { \left( x ^ { 2 } + x + 2 \right) ^ { 2 } }$
C) $\frac { \mathrm { A } } { \mathrm { x } - 4 } + \frac { \mathrm { B } } { \mathrm { x } ^ { 2 } + \mathrm { x } + 2 } + \frac { \mathrm { Cx } + \mathrm { D } } { \left( \mathrm { x } ^ { 2 } + \mathrm { x } + 2 \right) ^ { 2 } }$
D) $\frac { A } { x - 4 } + \frac { B x + C } { x ^ { 2 } + x + 2 }$ Write the partial fraction decomposition of the rational expression.

Question

Quizplus · Accepted Answer

For a repeated quadratic factor like $(x^2 + x + 2)^2$, the correct decomposition includes terms for each power of the repeated factor up to its highest power, and each of these terms has a linear numerator because the denominator is quadratic.

Decompose P/Q, Where Q Has a Prime, Repeated Quadratic Factor 7x+2(x−4)(x2+x+2)2\frac { 7 x + 2 } { ( x - 4 ) \left( x ^ { 2 } + x + 2 \right) ^ { 2 } }(x−4)(x2+x+2)27x+2​

Decompose P/Q, Where Q Has a Prime, Repeated Quadratic Factor $\frac { 7 x + 2 } { ( x - 4 ) \left( x ^ { 2 } + x + 2 \right) ^ { 2 } }$