Write the partial fraction decomposition of the rational expression.
-$$\frac { 8 x + 1 } { ( x - 1 ) \left( x ^ { 2 } + x + 1 \right) }$$
A) $\frac { 3 } { x - 1 } + \frac { - 3 x + 2 } { x ^ { 2 } + x + 1 }$
B) $\frac { 3 } { x - 1 } + \frac { - 3 } { x + 1 } + \frac { 2 } { x - 1 }$
C) $\frac { - 3 } { x - 1 } + \frac { 3 x + 2 } { x ^ { 2 } + x + 1 }$
D) $\frac { 3 } { x - 1 } + \frac { 2 x - 3 } { x ^ { 2 } + x + 1 }$

Question

Quizplus · Accepted Answer

The correct partial fraction decomposition of the given rational expression is $\frac { 3 } { x - 1 } + \frac { - 3 x + 2 } { x ^ { 2 } + x + 1 }$. This is because when decomposing a rational expression into partial fractions, the coefficients are determined by solving the equation obtained by equating the original expression to the sum of its partial fractions. In this case, option A correctly represents the decomposition, where the linear term $x-1$ and the quadratic term $x^2+x+1$ are the denominators of the partial fractions, and their respective numerators are determined accordingly.

Write the Partial Fraction Decomposition of the Rational Expression 8x+1(x−1)(x2+x+1)\frac { 8 x + 1 } { ( x - 1 ) \left( x ^ { 2 } + x + 1 \right) }(x−1)(x2+x+1)8x+1​

Write the Partial Fraction Decomposition of the Rational Expression $\frac { 8 x + 1 } { ( x - 1 ) \left( x ^ { 2 } + x + 1 \right) }$