College Algebra, Chapter 4, 4.3, Section 4.3, Problem 14

Two polynomials $P(x) = x^5 + x^4 - 2x^3 + x + 1$ and $D(x) = x^2 + x - 1$. 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 $-x + 2$ is of lesser degree than the divisor $x^2 + x - 1$. We see that $Q(x) = x^3 - x + 1$ and $R(x) = -x + 2$, so


$
\begin{equation}
\begin{aligned}

\frac{x^5 + x^4 - 2x^3 + x + 1}{x^2 + x - 1} =& x^3 - x + 1 + \frac{2 - x}{x^2 + x - 1}

\end{aligned}
\end{equation}
$