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  1. \chapter{Polynomial functions}
  2. \minitoc
  3.  
  4. \section{Polynomial functions}
  5. \listspecialproblems
  6. %===================================
  7. %   Author: Hughes
  8. %   Date:   May 2011
  9. %===================================
  10. \begin{pccdefinition}[Polynomial functions]
  11. Polynomial functions have the form
  12. \[
  13.    p(x)=a_nx^n+a_{n+1}x^{n+1}+\ldots+a_1x+a_0
  14. \]
  15. where $a_n$, $a_{n+1}$, $a_{n+2}$, \ldots, $a_0$ are real numbers.
  16. \begin{itemize}
  17.    \item We call $n$ the degree of the polynomial;
  18.    \item $a_n$, $a_{n+1}$, $a_{n+2}$, \ldots, $a_0$ are called the coefficients;
  19.    \item We typically write polynomial functions in descending powers of $x$.
  20. \end{itemize}
  21. In particular, we call $a_n$ the {\em leading} coefficient.
  22. \end{pccdefinition}
  23.  
  24. %===================================
  25. %   Author: Hughes
  26. %   Date:   May 2011
  27. %===================================
  28. \begin{figure}[!h]
  29. \centering
  30. \setlength{\figurewidth}{\textwidth/6}
  31. \setwindow{-10}{-10}{10}{10}{\figurewidth}
  32. \begin{subfigure}{\figurewidth}
  33. \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  34.      \pccpsframe(\xmin,\ymin)(\xmax,\ymax)
  35.      \psaxes[dx=10,Dx=10,dy=100,Dy=100]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  36.      \pccpsplot{-10}{8}{(x+2)}
  37.    \end{pspicture}
  38. \caption{$a_1>0$}
  39. \end{subfigure}
  40. \hfill
  41. \begin{subfigure}{\figurewidth}
  42. \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  43.      \pccpsframe(\xmin,\ymin)(\xmax,\ymax)
  44.      \psaxes[dx=10,Dx=10,dy=100,Dy=100]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  45.      \pccpsplot{-4}{4}{(x^2-6)}
  46.    \end{pspicture}
  47. \caption{$a_2>0$}
  48. \end{subfigure}
  49. \hfill
  50. \begin{subfigure}{\figurewidth}
  51. \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  52.      \pccpsframe(\xmin,\ymin)(\xmax,\ymax)
  53.      \psaxes[dx=10,Dx=10,dy=100,Dy=100]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  54.      \pccpsplot{-7.5}{7.5}{0.05*(x+6)*x*(x-6)}
  55.    \end{pspicture}
  56. \caption{$a_3>0$}
  57. \end{subfigure}
  58. \hfill
  59. \begin{subfigure}{\figurewidth}
  60. \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  61.      \pccpsframe(\xmin,\ymin)(\xmax,\ymax)
  62.      \psaxes[dx=10,Dx=10,dy=100,Dy=100]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  63.      \pccpsplot[plotpoints=1000]{-2.35}{5.35}{0.2*(x-5)*x*(x-3)*(x+2)}
  64.    \end{pspicture}
  65. \caption{$a_4>0$}
  66. \end{subfigure}
  67. \hfill
  68. \begin{subfigure}{\figurewidth}
  69. \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  70.      \pccpsframe(\xmin,\ymin)(\xmax,\ymax)
  71.      \psaxes[dx=10,Dx=10,dy=100,Dy=100]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  72.      \pccpsplot[plotpoints=1000]{-5.5}{6.3}{0.01*(x+2)*x*(x-3)*(x+5)*(x-6)}
  73.    \end{pspicture}
  74. \caption{$a_5>0$}
  75. \end{subfigure}
  76. \caption{Graphs to illustrate typical shapes of polynomials.}
  77. \end{figure}
  78.  
  79. %===================================
  80. %   Author: Hughes
  81. %   Date:   May 2011
  82. %===================================
  83. \begin{problem}[Polynomial or not?][special]
  84. Identify whether each of the following functions is a polynomial or not.
  85. If the function is a polynomial, state its degree.
  86. \begin{multicols}{3}
  87. \begin{subproblem}[special]
  88. $p(x)=2x+1$
  89.    \begin{shortsolution}
  90.        $p$ is a polynomial (you might also describe $p$ as linear). The degree of $p$ is 1.
  91.    \end{shortsolution}
  92. \end{subproblem}
  93. \begin{subproblem}
  94. $p(x)=7x^2+4x$
  95.    \begin{shortsolution}
  96.        $p$ is a polynomial (you might also describe $p$ as quadratic). The degree of $p$ is 2.
  97.    \end{shortsolution}
  98. \end{subproblem}
  99. \begin{subproblem}
  100. $p(x)=\sqrt{x}+2x+1$
  101.    \begin{shortsolution}
  102.        $p$ is not a polynomial; we require the powers of $x$ to be integer values.
  103.    \end{shortsolution}
  104. \end{subproblem}
  105. \begin{subproblem}
  106. $p(x)=2^x-45$
  107.    \begin{shortsolution}
  108.        $p$ is not a polynomial; the $2^x$ term is exponential.
  109.    \end{shortsolution}
  110. \end{subproblem}
  111. \begin{subproblem}
  112. $p(x)=6x^4-5x^3+9$
  113.    \begin{shortsolution}
  114.        $p$ is a polynomial- the degree of $p$ is $6$.
  115.    \end{shortsolution}
  116. \end{subproblem}
  117. \begin{subproblem}[special]
  118. $p(x)=-5x^{17}+9x+2$
  119.    \begin{shortsolution}
  120.        $p$ is a polynomial- the degree of $p$ is 17.
  121.    \end{shortsolution}
  122. \end{subproblem}
  123. \end{multicols}
  124. \end{problem}
  125.  
  126. %===================================
  127. %   Author: Hughes
  128. %   Date:   May 2011
  129. %===================================
  130. \begin{problem}[Polynomial graphs]
  131. Three polynomial functions are shown in  \crefrange{poly:fig:functionp}{poly:fig:functionn}.
  132. The functions $p$, $m$, $n$ have the following formulas
  133. \begin{align*}
  134.    p(x)&= (x-1)(x+2)(x-3)\\
  135.    m(x)&= -(x-1)(x+2)(x-3)\\
  136.    n(x)&= (x-1)(x+2)(x-3)(x+1)(x+4)
  137. \end{align*}
  138. Note that for our present purposes we are not concerned with the vertical scale of the graphs.
  139. \end{problem}
  140. \begin{subproblem}
  141. Identify both on the graph {\em and} algebraically, the zeros of the polynomial.
  142.    \begin{shortsolution}
  143. \setlength{\figurewidth}{\solutionfigurewidth}
  144. \begin{figure}[!h]
  145. \begin{subfigure}{\figurewidth}
  146.        \setwindow{-5}{-10}{5}{10}{\figurewidth}
  147.        \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  148.            \psaxes[dx=1,Dx=1,dy=0,Dy=10]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  149.            \pccpsplot{-2.5}{3.5}{(x-1)*(x+2)*(x-3)}
  150.            \pccpsSolDot(-2,0)(1,0)(3,0)
  151.        \end{pspicture}
  152. \caption{$y=p(x)$}
  153.   \end{subfigure}
  154. \begin{subfigure}{\figurewidth}
  155.        \setwindow{-5}{-10}{5}{10}{\figurewidth}
  156.            \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  157.                \psaxes[dx=1,Dx=1,dy=0,Dy=10]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  158.                \pccpsplot{-2.5}{3.5}{-1*(x-1)*(x+2)*(x-3)}
  159.                \pccpsSolDot(-2,0)(1,0)(3,0)
  160.            \end{pspicture}
  161. \caption{$y=m(x)$}
  162.   \end{subfigure}
  163.    \begin{subfigure}{\figurewidth}
  164.        \setwindow{-5}{-90}{5}{70}{\figurewidth}
  165.            \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  166.                \psaxes[dx=1,Dx=1,dy=0,Dy=100]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  167.                \pccpsplot{-4.15}{3.15}{(x-1)*(x+2)*(x-3)*(x+1)*(x+4)}
  168.                \pccpsSolDot(-4,0)(-2,0)(-1,0)(1,0)(3,0)
  169.            \end{pspicture}
  170. \caption{$y=n(x)$}
  171.   \end{subfigure}
  172. \end{figure}
  173.  
  174. The zeros of $p$ are $x=-2,1,3$; the zeros of $m$ are $x=-2,1,3$; the zeros of $n$ are $x=-4,-2,-1$, and $3$.
  175.    \end{shortsolution}
  176. \end{subproblem}
  177. \begin{subproblem}[special]
  178. Write down the degree, how many times the graph `turns around', and how many zeros it has
  179.    \begin{shortsolution}
  180.    The degree of $p$ is 3, and it turns around twice. The degree of $q$ is also 3, and it turns around twice. The degree
  181.    of $n$ is $5$, and it turns around 4 times.
  182.    \end{shortsolution}
  183. \end{subproblem}
  184.  
  185. \setlength{\figurewidth}{0.25\textwidth}
  186. \begin{figure}[!h]
  187. \begin{subfigure}{\figurewidth}
  188.        \setwindow{-5}{-10}{5}{10}{\figurewidth}
  189.            \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  190.                \psaxes[dx=1,Dx=1,dy=0,Dy=10]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  191.                \pccpsplot{-2.5}{3.5}{(x-1)*(x+2)*(x-3)}
  192.            \end{pspicture}
  193. \caption{$y=p(x)$}
  194.        \label{poly:fig:functionp}
  195.       \end{subfigure}
  196.       \hfill
  197. \begin{subfigure}{\figurewidth}
  198.        \setwindow{-5}{-10}{5}{10}{\figurewidth}
  199.            \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  200.                \psaxes[dx=1,Dx=1,dy=0,Dy=10]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  201.                \pccpsplot{-2.5}{3.5}{-1*(x-1)*(x+2)*(x-3)}
  202.            \end{pspicture}
  203. \caption{$y=m(x)$}
  204.        \label{poly:fig:functionm}
  205.       \end{subfigure}
  206.       \hfill
  207. \begin{subfigure}{\figurewidth}
  208.        \setwindow{-5}{-90}{5}{70}{\figurewidth}
  209.            \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  210.                \psaxes[dx=1,Dx=1,dy=0,Dy=100]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  211.                \pccpsplot{-4.15}{3.15}{(x-1)*(x+2)*(x-3)*(x+1)*(x+4)}
  212.            \end{pspicture}
  213. \caption{$y=n(x)$}
  214.        \label{poly:fig:functionn}
  215.       \end{subfigure}
  216.    \caption{}
  217. \end{figure}
  218.  
  219.  
  220. %===================================
  221. %   Author: Hughes
  222. %   Date:   May 2011
  223. %===================================
  224. \begin{problem}[Horizontal intercepts][special]\label{poly:prob:matchpolys}%
  225. State the horizontal intercepts (as ordered pairs) of the following polynomials.
  226. \end{problem}
  227.    \begin{subproblem}\label{poly:prob:degree5}
  228.        $p(x)=(x-1)(x+2)(x-3)(x+1)(x+4)$
  229.        \begin{shortsolution}
  230.            $(-4,0)$, $(-2,0)$, $(-1,0)$, $(1,0)$, $(3,0)$
  231.        \end{shortsolution}
  232.    \end{subproblem}
  233.    \begin{subproblem}
  234.        $q(x)=-(x-1)(x+2)(x-3)$
  235.        \begin{shortsolution}
  236.            $(-2,0)$, $(1,0)$, $(3,0)$
  237.        \end{shortsolution}
  238.    \end{subproblem}
  239.    \begin{subproblem}
  240.        $r(x)=(x-1)(x+2)(x-3)$
  241.        \begin{shortsolution}
  242.            $(-2,0)$, $(1,0)$, $(3,0)$
  243.        \end{shortsolution}
  244.    \end{subproblem}
  245.    \begin{subproblem}\label{poly:prob:degree2}
  246.        $s(x)=(x-2)(x+2)$
  247.        \begin{shortsolution}
  248.            $(-2,0)$, $(2,0)$
  249.        \end{shortsolution}
  250.    \end{subproblem}
  251.  
  252.  
  253.  
  254. \begin{pccdefinition}[Linear factors of a polynomial]
  255. If a polynomial $p$ can be written in factored form
  256. \[
  257.    p(x)=(x-x_1)(x-x_2)\ldots(x-x_n)
  258. \]
  259. then we call each of the factors $(x-x_i)$ the linear factors of $p$.
  260. \end{pccdefinition}
  261.  
  262. %===================================
  263. %   Author: Hughes
  264. %   Date:   May 2011
  265. %===================================
  266. \begin{problem}[Multiple zeros]
  267. Consider the polynomial
  268. \[
  269. p(x) = (x+3)^2(x-1).
  270. \]
  271. \end{problem}
  272. \begin{marginfigure}
  273. \setlength{\figurewidth}{\marginparwidth}
  274. \setwindow{-5}{-20}{2}{10}{\figurewidth}
  275.    \centering
  276.        \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  277.            \psaxes[dx=1,Dx=1,dy=0,Dy=100]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  278.            \pccpsplot{-4.75}{1.5}{(x-1)*(x+3)^2}
  279.        \end{pspicture}
  280.    \caption{$p(x)=(x+3)^2(x-1)$}
  281.    \label{poly:fig:multiplezeros}
  282. \end{marginfigure}
  283. \begin{subproblem}
  284. How is this different to the polynomials we have seen so far?
  285.    \begin{shortsolution}
  286.    $p$ has a repeated linear factor.
  287.    \end{shortsolution}
  288. \end{subproblem}
  289. \begin{subproblem}
  290. How many linear factors does $p$ have?
  291.    \begin{shortsolution}
  292.    $p$ has 3 linear factors.
  293.    \end{shortsolution}
  294. \end{subproblem}
  295. \begin{subproblem}
  296. What is the degree of this polynomial?
  297.    \begin{shortsolution}
  298.    The degree of $p$ is $3$.
  299.    \end{shortsolution}
  300. \end{subproblem}
  301. Note that this polynomial can be written as
  302. \begin{align*}
  303. p(x) &=(x+3)^2(x-1)\\
  304.      &=(x+3)(x+3)(x-1).
  305. \end{align*}
  306. Does this change your answers?
  307. \begin{subproblem}
  308. The graph of this polynomial is shown in \cref{poly:fig:multiplezeros}. Notice in
  309. particular the behavior of $p$ at $(-3,0)$. Does $p$ cut the horizontal axis, or bounce off it?
  310.    \begin{shortsolution}
  311.    $p$ bounces off the horizontal axis at $(-3,0)$.
  312.    \end{shortsolution}
  313. \end{subproblem}
  314. \begin{subproblem}
  315. Now consider the polynomial functions in \cref{poly:fig:moremultiple}.
  316. The formulas for $p$, $q$, and $r$ are as follows
  317. \begin{align*}
  318.  p(x)&=(x-3)^2(x+4)^2\\
  319.  q(x)&=x(x+2)^2(x-1)^2(x-3)\\
  320.  r(x)&=x(x-3)^3(x+1)^2
  321. \end{align*}
  322. Find the degree of $p$, $q$, and $r$, and decide if the functions bounce or cut at
  323. each of their zeros.
  324.    \begin{shortsolution}
  325.    \begin{itemize}
  326.        \item The degree of $p$ is 4. It bounces at both zeros, $x=3$ and $x=4$.
  327.        \item The degree of $q$ is 6. It bounces at $x=-2$ and $x=1$, and cuts at $x=0,3$.
  328.        \item The degree of $r$ is 6. It bounces at $x=-1$, and cuts at $x=0$. It also
  329.        cuts at $x=3$, although is flattened immediately to the left and right of $x=3$.
  330.    \end{itemize}
  331.    \end{shortsolution}
  332. \end{subproblem}
  333.  
  334. \setlength{\figurewidth}{0.25\textwidth}
  335. \begin{figure}[!h]
  336. \centering
  337. \begin{subfigure}{\figurewidth}
  338.        \setwindow{-5}{-30}{5}{200}{\figurewidth}
  339.            \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  340.                \psaxes[dx=2,Dx=2,dy=0,Dy=200]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  341.                \pccpsplot{-5}{4.25}{(x-3)^2*(x+4)^2}
  342.            \end{pspicture}
  343. \caption{$y=p(x)$}
  344.   \end{subfigure}
  345.   \hfill
  346. \begin{subfigure}{\figurewidth}
  347.        \setwindow{-3}{-60}{4}{40}{\figurewidth}
  348.            \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  349.                \psaxes[dx=1,Dx=1,dy=0,Dy=200]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  350.                \pccpsplot{-2.45}{3.05}{x*(x+2)^2*(x-1)^2*(x-3)}
  351.            \end{pspicture}
  352. \caption{$y=q(x)$}
  353.   \end{subfigure}
  354.   \hfill
  355. \begin{subfigure}{\figurewidth}
  356.        \setwindow{-2}{-40}{4}{40}{\figurewidth}
  357.            \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)
  358.                \psaxes[dx=1,Dx=1,dy=0,Dy=200]{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)[$x$,-90][$y$,180]
  359.                \pccpsplot{-1.45}{3.75}{x*(x-3)^3*(x+1)^2}
  360.            \end{pspicture}
  361. \caption{$y=r(x)$}
  362.   \end{subfigure}
  363.    \caption{}
  364.    \label{poly:fig:moremultiple}
  365. \end{figure}
  366.  
  367. \begin{pccdefinition}[Multiple zeros]
  368. Let $p$ be a polynomial that has a repeated linear factor $(x-k)^n$. Then we say
  369. that $p$ has a multiple zero at $x=k$ and
  370. \begin{itemize}
  371. \item if the factor $(x-k)$ is repeated an even number of times, the graph of $y=p(x)$ does not
  372.  cross the $x$ axis at $x=k$, but `bounces' off the $x$ axis at $x=k$.
  373. \item if the factor $(x-k)$ is repeated an odd number of times, the graph of $y=p(x)$ crosses the
  374.  $x$ axis at $x=k$, but it looks `flattened' there
  375. \end{itemize}
  376. \end{pccdefinition}
  377.  
  378. %===================================
  379. %   Author: Hughes
  380. %   Date:   May 2011
  381. %===================================
  382. \begin{margintable}
  383. \centering
  384.        \caption{$p$ and $q$}
  385.        \begin{tabular}{rrr}
  386.            \beforeheading
  387.            \heading{$x$}     &   \heading{$p(x)$}  & \heading{$q(x)$} \\ \afterheading
  388.            $-4$    &   $-56$   &  $-16$      \\\normalline
  389.            $-3$    &   $-18$   &  $-3$      \\ \normalline
  390.            $-2$    &   $0$     &  $0$      \\  \normalline
  391.            $-1$    &   $4$     &  $-1$      \\ \normalline
  392.            $0$     &   $0$     &  $0$      \\  \normalline
  393.            $1$     &   $-6$    &  $9$      \\  \normalline
  394.            $2$     &   $-8$    &  $32$      \\ \normalline
  395.            $3$     &   $0$     &  $75$      \\ \normalline
  396.            $4$     &   $24$    &  $144$     \\\lastline
  397.        \end{tabular}
  398.        \label{poly:tab:findformula}
  399. \end{margintable}
  400. \begin{problem}[Find a formula from a table]
  401. \Cref{poly:tab:findformula} shows two polynomial functions, $p$ and $q$.
  402. \end{problem}
  403. \begin{subproblem}
  404. Assuming that all of the zeros of $p$ are shown, how many zeros does $p$ have?
  405.    \begin{shortsolution}
  406.    $p$ has 3 zeros.
  407.    \end{shortsolution}
  408. \end{subproblem}
  409. \begin{subproblem}
  410. What is the degree of $p$?
  411.    \begin{shortsolution}
  412.    $p$ is degree 3.
  413.    \end{shortsolution}
  414. \end{subproblem}
  415. \begin{subproblem}
  416. Write a formula for $p(x)$.
  417.    \begin{shortsolution}
  418.    $p(x)=x(x+2)(x-3)$
  419.    \end{shortsolution}
  420. \end{subproblem}
  421. \begin{subproblem}
  422. Assuming that all of the zeros of $q$ are shown, how many zeros does $q$ have?
  423.    \begin{shortsolution}
  424.    $q$ has 2 zeros.
  425.    \end{shortsolution}
  426. \end{subproblem}
  427. \begin{subproblem}
  428. Describe the difference in behavior of $p$ and $q$ at $x=-2$.
  429.    \begin{shortsolution}
  430.    $p$ changes sign at $x=-2$, and $q$ does not change sign at $x=-2$.
  431.    \end{shortsolution}
  432. \end{subproblem}
  433. \begin{subproblem}[special]
  434. Given that $q$ is a degree 3 polynomial, write a formula for $q(x)$.
  435.    \begin{shortsolution}
  436.    $q(x)=x(x+2)^2$
  437.    \end{shortsolution}
  438. \end{subproblem}
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