Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants. $\frac { 5 x - 4 } { x ^ { 3 } + 5 x }$
A) $\frac { A } { 5 x - 4 } + \frac { B x + C } { x ^ { 3 } + 5 x }$
B) $\frac { A } { x } - \frac { B x + C } { x ^ { 2 } + 5 } + \frac { D } { x ^ { 3 } + 5 }$
C) $\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 5 }$
D) $\frac { A } { x } + \frac { B x - C } { x ^ { 2 } + 5 } + \frac { D } { 5 x - 4 }$
E) $\frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { B x + C } { x + 5 }$

Question

Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants.  $\frac { 5 x - 4 } { x ^ { 3 } + 5 x }$  
A) $\frac { A } { 5 x - 4 } + \frac { B x + C } { x ^ { 3 } + 5 x }$
B) $\frac { A } { x } - \frac { B x + C } { x ^ { 2 } + 5 } + \frac { D } { x ^ { 3 } + 5 }$
C) $\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 5 }$
D) $\frac { A } { x } + \frac { B x - C } { x ^ { 2 } + 5 } + \frac { D } { 5 x - 4 }$
E) $\frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { B x + C } { x + 5 }$

Quizplus · Accepted Answer

First, factor the denominator $x^3 + 5x = x(x^2 + 5)$. The correct partial fraction decomposition must account for each factor: $\frac{A}{x}$ for the $x$ factor and $\frac{Bx + C}{x^2 + 5}$ for the quadratic factor $x^2 + 5$, which cannot be factored further over the real numbers.

Write the Form of the Partial Fraction Decomposition of the Rational