Find the partial fraction decomposition for the rational expression.
$\frac { 8 x ^ { 2 } - 97 x + 18 } { \left( x ^ { 2 } + 3 x - 6 \right) ( x - 8 ) }$
A) $\frac { 11 x } { x ^ { 2 } + 3 x - 6 } + \frac { - 3 } { x - 8 }$
B) $\frac { 11 x } { x ^ { 2 } + 3 x - 6 } + \frac { 3 } { x - 8 }$
C) $\frac { 11 x } { x - 8 } + \frac { - 3 } { x ^ { 2 } + 3 x - 6 }$
D) $\frac { 3 x } { x ^ { 2 } + 3 x - 6 } + \frac { - 11 } { x - 8 }$

Question

Quizplus · Accepted Answer

To find the partial fraction decomposition of the given rational expression, we first ensure that the numerator's degree is less than the denominator's degree, which is true in this case. Then, we set up the decomposition based on the factors of the denominator. The denominator has a quadratic term $x^2 + 3x - 6$ and a linear term $x - 8$, suggesting a decomposition into $\frac{Ax + B}{x^2 + 3x - 6} + \frac{C}{x - 8}$. By equating this decomposition to the original expression and solving for A, B, and C, we find that A = 11 and C = -3, fitting the form of choice A. Choice A correctly represents the partial fraction decomposition of the given rational expression.

Find the Partial Fraction Decomposition for the Rational Expression 8x2−97x+18(x2+3x−6)(x−8)\frac { 8 x ^ { 2 } - 97 x + 18 } { \left( x ^ { 2 } + 3 x - 6 \right) ( x - 8 ) }(x2+3x−6)(x−8)8x2−97x+18​

Find the Partial Fraction Decomposition for the Rational Expression $\frac { 8 x ^ { 2 } - 97 x + 18 } { \left( x ^ { 2 } + 3 x - 6 \right) ( x - 8 ) }$