Write the partial fraction decomposition of the improper rational expression. $\frac { x ^ { 3 } - 2 x ^ { 2 } - 14 x + 48 } { x ^ { 2 } + x - 20 }$
A) $\frac { 1 } { 3 } \left( \frac { 8 } { x - 4 } + \frac { 19 } { x + 5 } \right)$
B) $x - 3 + \frac { 1 } { 3 } \left( \frac { 8 } { x - 4 } - \frac { 19 } { x + 5 } \right)$
C) $x - 3 + \frac { 1 } { 3 } \left( \frac { 8 } { x - 4 } + \frac { 19 } { x + 5 } \right)$
D) $x - 3 + \frac { 1 } { 3 } \left( \frac { 16 } { x + 4 } + \frac { 19 } { x - 5 } \right)$
E) $x - 3 - \frac { 1 } { 3 } \left( \frac { 16 } { x - 4 } - \frac { 19 } { x - 5 } \right)$

Question

Write the partial fraction decomposition of the improper rational expression. $\frac { x ^ { 3 } - 2 x ^ { 2 } - 14 x + 48 } { x ^ { 2 } + x - 20 }$
A) $\frac { 1 } { 3 } \left( \frac { 8 } { x - 4 } + \frac { 19 } { x + 5 } \right)$
B) $x - 3 + \frac { 1 } { 3 } \left( \frac { 8 } { x - 4 } - \frac { 19 } { x + 5 } \right)$
C)​ $x - 3 + \frac { 1 } { 3 } \left( \frac { 8 } { x - 4 } + \frac { 19 } { x + 5 } \right)$
D) $x - 3 + \frac { 1 } { 3 } \left( \frac { 16 } { x + 4 } + \frac { 19 } { x - 5 } \right)$
E) $x - 3 - \frac { 1 } { 3 } \left( \frac { 16 } { x - 4 } - \frac { 19 } { x - 5 } \right)$

Quizplus · Accepted Answer

The given expression is an improper rational expression because the degree of the numerator is higher than the degree of the denominator. To find the partial fraction decomposition, we first perform polynomial long division or synthetic division to simplify the expression into a polynomial plus a proper fraction. The denominator factors as $(x-4)(x+5)$, and after division, we get a quotient of $x-3$ and a remainder. The remainder, when divided by the original denominator, gives the proper fraction part of the decomposition. The correct decomposition includes both the polynomial part $(x-3)$ and the proper fraction part, which matches option C.

Write the Partial Fraction Decomposition of the Improper Rational Expression x3−2x2−14x+48x2+x−20\frac { x ^ { 3 } - 2 x ^ { 2 } - 14 x + 48 } { x ^ { 2 } + x - 20 }x2+x−20x3−2x2−14x+48​

Write the Partial Fraction Decomposition of the Improper Rational Expression $\frac { x ^ { 3 } - 2 x ^ { 2 } - 14 x + 48 } { x ^ { 2 } + x - 20 }$