Write the partial fraction decomposition of the improper rational expression.
$$\frac { x ^ { 3 } - 4 x ^ { 2 } + 3 x + 2 } { x ^ { 2 } + x - 2 }$$
A) $x - 5 + \frac { 1 } { 3 } \left( \frac { 2 } { x - 1 } + \frac { 28 } { x + 2 } \right)$
B) $x - 5 + \frac { 1 } { 3 } \left( \frac { 18 } { x - 1 } + \frac { 28 } { x + 2 } \right)$
C) $x - 5 + \frac { 1 } { 3 } \left( \frac { 18 } { x + 1 } + \frac { 28 } { x - 2 } \right)$
D) $x - 5 + \frac { 1 } { 3 } \left( \frac { 2 } { x - 1 } - \frac { 28 } { x + 2 } \right)$
E) $\frac { 1 } { 3 } \left( \frac { 2 } { x - 1 } + \frac { 28 } { x + 2 } \right)$

Question

Quizplus · Accepted Answer

First, perform polynomial long division or synthetic division to simplify the given expression into a polynomial plus a proper fraction. The denominator $x^2 + x - 2$ factors into $(x - 1)(x + 2)$. After simplification, the correct partial fraction decomposition is $x - 5 + \frac{1}{3}\left(\frac{2}{x - 1} + \frac{28}{x + 2}\right)$, which matches option A.

Write the Partial Fraction Decomposition of the Improper Rational Expression x3−4x2+3x+2x2+x−2\frac { x ^ { 3 } - 4 x ^ { 2 } + 3 x + 2 } { x ^ { 2 } + x - 2 }x2+x−2x3−4x2+3x+2​

Write the Partial Fraction Decomposition of the Improper Rational Expression $\frac { x ^ { 3 } - 4 x ^ { 2 } + 3 x + 2 } { x ^ { 2 } + x - 2 }$