\begin{frame} \frametitle{Polynomials} \begin{block}{} A function $P$ is called \emph{polynomial} if \begin{talign} P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_2 x^2 + a_1 x + a_0 \end{talign} where \begin{itemize} \item $n$ is a non-negative integer, and \item $a_0,a_1,\ldots,a_n$ are constants, called \emph{coefficients}. \end{itemize} If $a_n \ne 0$ then $n$ is the \emph{degree} of the polynomial. \end{block} \pause\medskip % \begin{exampleblock}{} % $P(x) = 2x^6 - x^4 + \frac{3}{5}x + \sqrt{2}$ is a polynomial of degree $6$. % \end{exampleblock} % \pause\medskip \begin{minipage}{.3\textwidth} \scalebox{.6}{ \begin{tikzpicture}[default] \diagram{-2}{2}{-2}{3}{1} \diagramannotatez \diagramannotatex{-1,1} \diagramannotatey{-1,1} \draw[cblue,ultra thick] plot[smooth,domain=-1.67:1.53,samples=20] (\x,{pow(\x,3)-\x+1}); \end{tikzpicture} } \centerline{{\small $x^3-x+1$}} \end{minipage}\quad \begin{minipage}{.3\textwidth} \scalebox{.6}{ \begin{tikzpicture}[default,yscale=.5] {\def\diabordery{1cm} \diagram{-2}{2}{-4}{6}{1}} \diagramannotatez \diagramannotatex{-1,1} \diagramannotatey{-1,1} %,smooth,domain=-1.76:1.83,samples=20 \draw[cblue,ultra thick] plot[smooth,domain=-2:2] (\x,{pow(\x,4) - 3*pow(\x,2) + \x}); \end{tikzpicture} } \centerline{{\small $x^4 - 3x^2 + x$}} \end{minipage}\quad \begin{minipage}{.3\textwidth} \scalebox{.6}{ \begin{tikzpicture}[default,xscale=.75,yscale=.05] {\def\diaborderx{.5cm/.75} \def\diabordery{10cm} \diagram[10]{-3}{3}{-40}{60}{1}} \diagramannotatez \diagramannotatex{1} \diagramannotatey{-20,20,40} %,smooth,domain=-1.76:1.83,samples=20 \draw[cblue,ultra thick] plot[smooth,domain=-2.45:2.55] (\x,{3*pow(\x,5) - 25 *pow(\x,3) + 60*\x}); \end{tikzpicture} } \centerline{\ \ {\small $3x^5 -25 x^3 + 60x$}} \end{minipage} \vspace{10cm} \end{frame}