Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants. $\frac { x + 5 } { x \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
A) $\frac { A } { x } + \frac { B x + C } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
B) $\frac { A x + B } { x ^ { 2 } + 6 } + \frac { C x + D } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
C) $\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 6 } + \frac { D x + E } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
D) $\frac { A x + B } { x } + \frac { C x + D } { x ^ { 2 } + 6 } + \frac { E x + F } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
E) $\frac { A } { x } + \frac { B } { x ^ { 2 } + 6 } + \frac { C } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$

Question

Quizplus · Accepted Answer

For the given rational expression, the denominator has a linear factor $x$ and a repeated irreducible quadratic factor $(x^2 + 6)^2$. The correct partial fraction decomposition includes a term for the linear factor $\frac{A}{x}$, and for each power of the irreducible quadratic factor, terms of the form $\frac{Bx + C}{x^2 + 6}$ and $\frac{Dx + E}{(x^2 + 6)^2}$ are needed, matching option C.

Write the Form of the Partial Fraction Decomposition of the Rational