College Algebra, Chapter 4, 4.3, Section 4.3, Problem 12

Two polynomials $P(x) = 6x^3 + x^2 - 12x + 5$ and $D(x) = 3x - 4$. Use either synthetic or long division to divide $P(x)$ by $D(x)$, and express the quotient $\displaystyle \frac{P(x)}{D(x)}$ in the form $\displaystyle \frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$.

Using Long Division







The process is complete at this point because $5$ is of lesser degree than the divisor $3x - 4$. We see that $Q(x) = 2x^2 + 3x$ and $R(x) = 5$, so


$\begin{equation} \begin{aligned} \frac{6x^3 + x^2 - 12x + 5}{3x - 4} =& 2x^2 + 3x + \frac{5}{3x - 4} \end{aligned} \end{equation}$