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\problemgrouptitle{Advanced Factoring}

\begin{problemblock}[2]{Find all rational solutions of the following equations using the Rational Roots Test.}
\item $2x^3-3x^2-3x+2=0$
\answer{$x=2,1/2,-1$}
\item $x^4-x^3-4x^2-2x-12=0$
\answer{$x=-2,3$}
\end{problemblock}
\begin{problemblock}[2]{Use polynomial long division to evaluate the following.}
\item $\displaystyle (2x^3+2x^2-11x+3)\div (x+3)$
\answer{$\displaystyle 2x^2-4x+1\ \text{since}\\
  \polylongdiv{2x^3+2x^2-11x+3}{x+3}$}
\item $\displaystyle \frac{x^3+2x^2+3}{2x^2-1}$
\answer{$\displaystyle\frac{1}{2}x+1+\frac{\frac{1}{2}x+4}{2x^2-1}=\frac{1}{2}x+1+\frac{x+8}{4x^2-2}\ \text{since}\\
  \polylongdiv{x^3+2x^2+3}{2x^2-1}$}
\end{problemblock}
\begin{problemblock}{Solve the following equations by factoring by grouping.}
\item $\displaystyle x^3+4x^2-3x-12=0$
\answer{$\displaystyle x^3+4x^2-3x-12=(x+4)(x^2-3)\ \text{so}\ x=-4,\pm\sqrt{3}$}
\item $\displaystyle x^3-7x^2+6=0$ (Hint: Use a Rational Roots Test solution to inform breaking up $-7x^2$.)
\answer{$\displaystyle x^3-7x^2+6=(x-1)(x^2-6x-6)\ \text{so}\ x=1,3\pm\sqrt{15}$}
\item $\displaystyle 3x^2+10x=8$
\answer{$\displaystyle 3x^2+10x-8=(x+4)(3x-2)\ \text{so}\ x=-4,2/3$}
\end{problemblock}

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