Write the partial fraction decomposition of the rational expression.
$$\frac { x ^ { 2 } } { x ^ { 4 } - 21 x ^ { 2 } - 100 }$$
A) $\frac { 1 } { 29 } \left( \frac { 4 } { x ^ { 2 } + 4 } + \frac { 5 } { 2 ( x - 5 ) } - \frac { 5 } { 2 ( x + 5 ) } \right)$
B)
$\frac { 1 } { x ^ { 2 } } - \frac { 1 } { 2 } - \frac { x ^ { 2 } } { 20 }$
C)
$\frac { 1 } { 29 } \left( \frac { 4 } { x ^ { 2 } + 4 } + \frac { 5 ( x + 4 ) } { 2 ( x - 5 ) } + \frac { 5 } { 2 ( x - 5 ) } \right)$
D)
$\frac { 1 } { 29 } \left( \frac { 4 } { x ^ { 2 } + 4 } - \frac { 5 } { 2 ( x - 5 ) } + \frac { 5 } { 2 ( x + 5 ) } \right)$
E)
$\frac { 1 } { 29 } \left( \frac { 1 } { x ^ { 2 } + 4 } + \frac { 1 } { 2 ( x - 5 ) } - \frac { 1 } { 2 ( x + 5 ) } \right)$

Question

Quizplus · Accepted Answer

The denominator $x^4 - 21x^2 - 100$ can be factored into $(x^2 + 4)(x^2 - 25)$, which further factors into $(x^2 + 4)(x - 5)(x + 5)$. The correct partial fraction decomposition must account for all these factors, and option A does this correctly by including terms for $x^2 + 4$, $x - 5$, and $x + 5$ with appropriate coefficients to balance the original expression.

Write the Partial Fraction Decomposition of the Rational Expression x2x4−21x2−100\frac { x ^ { 2 } } { x ^ { 4 } - 21 x ^ { 2 } - 100 }x4−21x2−100x2​

Write the Partial Fraction Decomposition of the Rational Expression $\frac { x ^ { 2 } } { x ^ { 4 } - 21 x ^ { 2 } - 100 }$