Answer
$8x^2-2x+6+\dfrac{2}{x-\frac{1}{2}}$
Work Step by Step
Equating the divisor to zero of the given expression, $
(8x^3-1+7x-6x^2)\div\left(x-\dfrac{1}{2}\right)
$ which is equivalent to $
(8x^3-6x^2+7x-1)\div\left(x-\dfrac{1}{2}\right)
,$ and then solving for the variable, then \begin{align*}
x-\dfrac{1}{2}&=0
\\
x&=\dfrac{1}{2}
.\end{align*} Using $
\dfrac{1}{2}
$ in manipulating the coefficients of the dividend results to the picture shown below. The last row of numbers, $\{
8,-2,6,2
\},$ represent the coefficients of the quotient and the remainder. Hence, $(8x^3-1+7x-6x^2)\div\left(x-\dfrac{1}{2}\right)$ is equal to \begin{align*}
8x^2-2x+6+\dfrac{2}{x-\frac{1}{2}}
.\end{align*}
