Write the partial fraction decomposition of the rational expression.
-$\frac { 6 x ^ { 3 } + 4 x ^ { 2 } + 13 x + 15 } { \left( x ^ { 2 } + 3 \right) ^ { 3 } }$
A) $\frac { x } { x ^ { 2 } + 3 } + \frac { 6 x + 4 } { \left( x ^ { 2 } + 3 \right) ^ { 2 } } + \frac { - 5 x + 3 } { \left( x ^ { 2 } + 3 \right) ^ { 3 } }$
B) $\frac { x + 1 } { x ^ { 2 } + 3 } + \frac { 6 x + 4 } { \left( x ^ { 2 } + 3 \right) ^ { 2 } } + \frac { - 5 x + 3 } { \left( x ^ { 2 } + 3 \right) ^ { 3 } }$
C) $\frac { 6 x - 4 } { \left( x ^ { 2 } + 3 \right) ^ { 2 } } + \frac { - 5 x - 3 } { \left( x ^ { 2 } + 3 \right) ^ { 3 } }$
D) $\frac { 6 x + 4 } { \left( x ^ { 2 } + 3 \right) ^ { 2 } } + \frac { - 5 x + 3 } { \left( x ^ { 2 } + 3 \right) ^ { 3 } }$

Question

Quizplus · Accepted Answer

The correct partial fraction decomposition must account for all powers of the denominator up to its highest power, without exceeding the degree of the numerator. Option D correctly decomposes the given fraction into terms that match the structure of the denominator's powers, ensuring each term is of lower degree than the corresponding power of $x^2 + 3$.

Write the Partial Fraction Decomposition of the Rational Expression 6x3+4x2+13x+15(x2+3)3\frac { 6 x ^ { 3 } + 4 x ^ { 2 } + 13 x + 15 } { \left( x ^ { 2 } + 3 \right) ^ { 3 } }(x2+3)36x3+4x2+13x+15​

Write the Partial Fraction Decomposition of the Rational Expression $\frac { 6 x ^ { 3 } + 4 x ^ { 2 } + 13 x + 15 } { \left( x ^ { 2 } + 3 \right) ^ { 3 } }$