Find the partial fraction decomposition for the rational expression.
$\frac { - 31 x - 113 } { 2 x ^ { 2 } + 20 x + 42 }$
A) $\frac { - 5 } { 2 x - 6 } + \frac { - 13 } { x - 7 }$
B) $\frac { 5 } { 2 x + 6 } + \frac { 13 } { x + 7 }$
C) $\frac { - 5 } { x + 6 } + \frac { - 13 } { 2 x + 7 }$
D) $\frac { - 5 } { 2 x + 6 } + \frac { - 13 } { x + 7 }$

Question

Quizplus · Accepted Answer

First, factor the denominator: $2x^2 + 20x + 42 = 2(x^2 + 10x + 21) = 2(x + 3)(x + 7)$. Thus, the denominator can be rewritten as $2(x + 3)(x + 7)$, and the partial fraction decomposition should have the form $\frac{A}{x + 3} + \frac{B}{x + 7}$. Multiplying both sides by the common denominator to clear the fractions, we can solve for A and B. The correct coefficients that satisfy the equation are A = -5/2 and B = -13, leading to the decomposition $\frac{-5}{2(x + 3)} + \frac{-13}{x + 7}$, which simplifies to choice D when considering the factor of 2 in the first term's denominator.

Find the Partial Fraction Decomposition for the Rational Expression −31x−1132x2+20x+42\frac { - 31 x - 113 } { 2 x ^ { 2 } + 20 x + 42 }2x2+20x+42−31x−113​

Find the Partial Fraction Decomposition for the Rational Expression $\frac { - 31 x - 113 } { 2 x ^ { 2 } + 20 x + 42 }$