Find the partial fraction decomposition for the rational expression.
-$\frac { - 31 x - 77 } { 2 x ^ { 2 } + 14 x + 20 }$
A) $\frac { - 5 } { 2 x + 4 } + \frac { - 13 } { x + 5 }$
B) $\frac { - 5 } { x + 4 } + \frac { - 13 } { 2 x + 5 }$
C) $\frac { 5 } { 2 x + 4 } + \frac { 13 } { x + 5 }$
D) $\frac { - 5 } { 2 x - 4 } + \frac { - 13 } { x - 5 }$

Question

Quizplus · Accepted Answer

First, factor the denominator: $2x^2 + 14x + 20 = 2(x^2 + 7x + 10) = 2(x + 2)(x + 5)$. Thus, the denominator becomes $2(x + 2)(x + 5)$, and the partial fraction decomposition must involve terms with denominators $x + 2$ and $x + 5$. After finding the coefficients for the numerators through algebraic methods, the correct decomposition is $\frac{-5}{2x + 4} + \frac{-13}{x + 5}$, which corresponds to choice A.

Find the Partial Fraction Decomposition for the Rational Expression −31x−772x2+14x+20\frac { - 31 x - 77 } { 2 x ^ { 2 } + 14 x + 20 }2x2+14x+20−31x−77​

Find the Partial Fraction Decomposition for the Rational Expression $\frac { - 31 x - 77 } { 2 x ^ { 2 } + 14 x + 20 }$