Write the partial fraction decomposition of the rational expression. $\frac { 3 x ^ { 2 } + 3 x + 18 } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
A) $\frac { 3 x + 3 } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
B) $\frac { 3 } { x ^ { 2 } + 6 } - \frac { 3 x } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
C) $\frac { 3 } { x ^ { 2 } + 6 } + \frac { 3 x } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
D) $\frac { 3 x + 18 } { x ^ { 2 } + 6 } + \frac { 3 x } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
E) $\frac { 3 } { x ^ { 2 } + 6 } + \frac { 3 } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$

Question

Write the partial fraction decomposition of the rational expression.​ $\frac { 3 x ^ { 2 } + 3 x + 18 } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$ ​
A) $\frac { 3 x + 3 } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
B) $\frac { 3 } { x ^ { 2 } + 6 } - \frac { 3 x } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
C) $\frac { 3 } { x ^ { 2 } + 6 } + \frac { 3 x } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
D) $\frac { 3 x + 18 } { x ^ { 2 } + 6 } + \frac { 3 x } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$
E) $\frac { 3 } { x ^ { 2 } + 6 } + \frac { 3 } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$

Quizplus · Accepted Answer

The partial fraction decomposition of the given rational expression is:

(5x + 7)/(x^2 - x -12) = A/(x-4) + B/(x+3)

Multiplying both sides by (x-4)(x+3), we get:

(5x + 7) = A(x+3) + B(x-4)

Setting x = -3, we get:

-8A = 2

A = -1/4

Setting x = 4, we get:

11B = 27

B = 27/11

Therefore, the partial fraction decomposition is:

(5x + 7)/(x^2 - x -12) = (-1/4)/(x-4) + (27/11)/(x+3)

Therefore, the correct choice is C.

Write the Partial Fraction Decomposition of the Rational Expression 3x2+3x+18(x2+6)2\frac { 3 x ^ { 2 } + 3 x + 18 } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }(x2+6)23x2+3x+18​

Write the Partial Fraction Decomposition of the Rational Expression $\frac { 3 x ^ { 2 } + 3 x + 18 } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }$