When writing the solution to the partial fraction decomposition of $\frac{x^{3}}{x^{2}-x-2}$ what is the first step of this solution?
A) $\frac{x^{3}}{x^{2}-x-2}=\frac{A}{x-2}+\frac{B}{x+1}$
B) $\frac{x^{3}}{x^{2}-x-2}=\frac{A x+B}{x-2}+\frac{C}{x+1}$
C) $\frac{x^{3}}{x^{2}-x-2}=x+1+\frac{3 x+2}{x^{2}-x-2}$
D) $\frac{x^{3}}{x^{2}-x-2}=x+\frac{3 x+2}{x^{2}-x-2}$
E) $\frac{x^{3}}{x^{2}-x-2}=1+\frac{3 x+2}{x^{2}-x-2}$

Question

Quizplus · Accepted Answer

The first step in the partial fraction decomposition of $\frac{x^{3}}{x^{2}-x-2}$ involves long division since the degree of the numerator is greater than or equal to the degree of the denominator. This results in $\frac{x^{3}}{x^{2}-x-2}=x+1+\frac{3 x+2}{x^{2}-x-2}$, where $x+1$ is the quotient and $\frac{3 x+2}{x^{2}-x-2}$ is the remainder over the original denominator.

When Writing the Solution to the Partial Fraction Decomposition Of \frac{x^{3}}{x^{2}-x-2}