# polynomialfunctions.tex

Nov 12th, 2011
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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*}
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}
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}