Write the form of the partial fraction decomposition of the rational expression. Do not find the numerical values of the constants. $\frac{x^{2}-x-8}{2 x^{3}-3 x^{2}+4 x-6}$
A) $\frac{A}{2 x-3}+\frac{B x+C}{x^{2}+2}$
B) $\frac{A}{2 x-1}+\frac{B x+C}{x^{2}+2}$
C) $\frac{A}{2 x+1}+\frac{B x+C}{x^{2}-2}$
D) $\frac{A}{2 x+3}+\frac{B x+C}{x^{2}-2}$

Question

Quizplus · Accepted Answer

The denominator can be factored as $2x(x-1)(x+2)$, so the form of the partial fraction decomposition is $\frac{A}{2x}+\frac{B}{x-1}+\frac{Cx+D}{x+2}$. However, since the numerator has degree 2 and the denominator has degree 3, we need to multiply the numerator by $x$ to get a common denominator. This gives us $\frac{x(x^2-x-8)}{2x^3-3x^2+4x-6}$, which can be decomposed as $\frac{A}{2x-3}+\frac{Bx+C}{x^2+2}$.

Write the Form of the Partial Fraction Decomposition of the Rational