Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants. $\frac { x - 4 } { x \left( x ^ { 2 } + 7 \right) ^ { 2 } }$
A) $\frac { A } { x } + \frac { B } { x ^ { 2 } + 7 } + \frac { C } { \left( x ^ { 2 } + 7 x ^ { 4 } \right) ^ { 2 } }$
B) $\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 7 } + \frac { D x + E } { ( x + 7 ) ^ { 2 } }$
C) $\frac { A } { x } - \frac { B x + C } { \left( x ^ { 2 } + 7 \right) ^ { 2 } } - \frac { D x + E } { x \left( x ^ { 2 } + 7 \right) ^ { 2 } }$
D) $\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 7 } + \frac { D x + E } { \left( x ^ { 2 } + 7 \right) ^ { 2 } }$
E) $\frac { A } { x - 4 } + \frac { B x + C } { x ^ { 2 } + 7 } + \frac { D x + E } { \left( x ^ { 2 } + 7 \right) ^ { 2 } }$

Question

Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants.  $\frac { x - 4 } { x \left( x ^ { 2 } + 7 \right) ^ { 2 } }$  
A) $\frac { A } { x } + \frac { B } { x ^ { 2 } + 7 } + \frac { C } { \left( x ^ { 2 } + 7 x ^ { 4 } \right) ^ { 2 } }$  
B) $\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 7 } + \frac { D x + E } { ( x + 7 ) ^ { 2 } }$  
C) $\frac { A } { x } - \frac { B x + C } { \left( x ^ { 2 } + 7 \right) ^ { 2 } } - \frac { D x + E } { x \left( x ^ { 2 } + 7 \right) ^ { 2 } }$  
D) $\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 7 } + \frac { D x + E } { \left( x ^ { 2 } + 7 \right) ^ { 2 } }$  
E) $\frac { A } { x - 4 } + \frac { B x + C } { x ^ { 2 } + 7 } + \frac { D x + E } { \left( x ^ { 2 } + 7 \right) ^ { 2 } }$

Quizplus · Accepted Answer

The correct form for the partial fraction decomposition of the given rational expression involves breaking it down according to the factors in the denominator. The denominator has a linear factor $x$ and a repeated quadratic factor $(x^2 + 7)^2$. For the linear factor $x$, we use $\frac{A}{x}$. For the quadratic factor $x^2 + 7$, we use $\frac{Bx + C}{x^2 + 7}$ for the first occurrence and $\frac{Dx + E}{(x^2 + 7)^2}$ for the second occurrence, since each term in the quadratic factor's decomposition needs to account for the degree of the polynomial (hence the $Bx + C$ and $Dx + E$ forms).

Write the Form of the Partial Fraction Decomposition of the Rational