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\section{Polynomials}

\vspace{1pt}

\subsection{\ Sums, Differences, and Quotients of Polynomials\label%
{Adding polynomials}}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( 3x^{2}+3x\right) +\left( 8x^{2}+7\right) $

$\left( 3x^{2}+3x\right) -\left( 8x^{2}+7\right) $

$x^{2}+x+5-3x^{3}+5x^{2}+4x-13+2x^{2}$

$\left( 3x^{2}+3x\right) /\left( 8x^{2}+7\right) $
\end{quotation}

\strut

\subsection{\ Sums and Differences of Rational Expressions \label%
{Rational expressions}}

\begin{quote}
$\blacktriangleright $ \textsf{Simplify} (or \textsf{Factor})
\end{quote}

\begin{quotation}
$\dfrac{3x^{2}+3x}{8x^{2}+7}+\dfrac{5x^{2}+3}{2x^{2}+x+7}$

\ 

$\dfrac{\left( 3x^{2}+3x\right) ^{3}}{2x-5}-\dfrac{5x^{2}+3}{2x^{2}+x+7}$

\ 

$\dfrac{3x^{2}+4}{5x^{3}-3x}+\dfrac{4x+1}{x^{2}-1}$
\end{quotation}

\strut

\subsection{\ Products and Powers of Polynomials \label{Polynomial products}%
\label{Expand (2)}}

\begin{quote}
$\blacktriangleright $ \textsf{Expand}
\end{quote}

\begin{quotation}
$\left( 3x^{2}+3x\right) \left( 8x^{2}+7\right) $

$\left( 3x^{2}+3x\right) ^{3}$
\end{quotation}

\strut

\subsection{\ Division by Polynomials \label{Polynomials, divide}}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Divide}
\end{quote}

\begin{quotation}
$\left( 3x^{2}+3x\right) /\left( 8x^{2}+7\right) $

$\dfrac{3x^{5}+3x^{3}-4x^{2}+5}{8x^{2}+7}$

$\frac{128x^{6}+128x^{5}+128x^{4}+768x^{3}-128x^{2}+\allowbreak 128x-128}{%
16x^{4}+8x^{3}+70x^{2}+7x+49}$
\end{quotation}

\strut

\subsection{\ Collecting and Ordering Terms 
\index{Polynomials@Polynomials!sort@sort}\label{Polynomials, sort}}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Sort}
\end{quote}

\begin{quotation}
$5t^{2}+2t-16t^{5}+t^{3}-2t^{2}+9$

$x^{2}+3x+5-3x^{3}+5x^{2}+4x^{3}+13+2x^{4}$

$5t^{2}+2x-16t^{5}+y^{3}-2t^{2}+9$

$5t^{2}+2x-16t^{5}+y^{3}-2t^{2}+9$
\end{quotation}

\strut

\subsection{\ Factoring Polynomials \label{Polynomials, factor}}

\begin{quote}
$\blacktriangleright $ \textsf{Factor}
\end{quote}

\begin{quotation}
$%
\frac{1}{2}x^{2}+3x-\frac{20}{9}$

$x^{2}-y^{2}$

$x^{3}-y^{3}$

$x^{4}-y^{4}$

$x^{3}+y^{3}$

$x^{5}+y^{5}$

$x^{7}+y^{7}$
\end{quotation}

\strut

\subsection{\ Greatest Common Divisor of Two Polynomials \label%
{Polynomials, gcd}}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\gcd (5x^{2}-5x,10x-10)$

$\gcd \left( yx+3x-5y-15,\,xz-53x-5z+265\right) $

$\gcd \left( x^{2}+3x+yx+3y,\,x^{2}-4yx-5y^{2},3x^{2}+2yx-y^{2}\right) $
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Factor}
\end{quote}

\begin{quotation}
$5x^{2}-5x$

$10x-10$

$yx+3x-5y-15$

$xz-53x-5z+265$

$x^{2}+3x+yx+3y$

$x^{2}-4yx-5y^{2}$

$3x^{2}+2yx-y^{2}$
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate, } \textsf{Factor}
\end{quote}

\begin{quotation}
$\limfunc{lcm}(yx+3x-5y-15,xz-53x-5z+265)$
\end{quotation}

\strut

\subsection{\ Roots of Polynomials \label{Polynomials, roots}}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Roots}
\end{quote}

\begin{quotation}
$x^{3}-\frac{13}{5}ix^{2}-8x^{2}+\frac{29}{5}ix+\frac{81}{5}x+6i-\frac{18}{5}
$

$ax^{2}+bx+c$

$5x^{2}+2x-3$

$5x^{2}+x+3$,

$x^{3}+3x^{2}+3x+1$

$x^{4}+4x^{3}-12x^{2}-32x+64$

$x^{3}+cx-1$
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Factor}
\end{quote}

\begin{quotation}
$x^{3}+3x^{2}+3x+1$

$x^{4}+4x^{3}-12x^{2}-32x+64$

$x^{3}-\frac{3}{2}\sqrt[3]{2}x-1$
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Roots}
\end{quote}

\begin{quotation}
$x^{3}+3x+1$

$x^{3}+3x+1.0$
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( -.322185\right) ^{3}+3\left( -.322185\right) +1.0$

$\left( -.32218535462608559291\right) ^{3}+3\left(
-.32218535462608559291\right) +1$
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Roots}
\end{quote}

\begin{quotation}
$x^{4}-2x-3$

$x^{4}-2x-3.0$

$5x^{5}+5x^{4}-10x^{3}-10x^{2}+5x+5$

$\,x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$
\end{quotation}

\end{document}
