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<h1>Polynomials</h1>
<p>Polynomials are a particular class of expressions that are simple enough to have many properties that can be analyzed. In particular, the key concepts of calculus: limits, continuity, derivatives, and integrals are all relatively trivial for polynomial functions. However, polynomials are flexible enough that they can be used to approximate a wide variety of functions. Indeed, though we don't pursue this, we mention that <code>Julia</code>'s <code>ApproxFun</code> package exploits this to great advantage.</p>
<p>Here we discuss some vocabulary and basic facts related to polynomials and show how the add-on <code>SymPy</code> package can be used to model polynomial expressions within <code>SymPy</code>.</p>
<p>For our purposes, a <em>monomial</em> is simply a non-negative integer power of <span class="math">$x$</span> (or some other indeterminate symbol) possibly multiplied by a scalar constant. For example, <span class="math">$5x^4$</span> is a monomial, as are constants, such as <span class="math">$-2=-2x^0$</span> and the symbol itself, as <span class="math">$x = x^1$</span>. In general, one may consider restrictions on where the constants can come from, and consider more than one symbol, but we won't pursue this here, restricting ourselves to the case of a single variable and real coefficients.</p>
<p>A <em>polynomial</em> is a sum of monomials. After combining terms with same powers, a non-zero polynomial may be written uniquely as:</p>
<p class="math">\[
a_n x^n + a_{n-1}x^{n-1} + \cdots a_1 x + a_0, \quad a_n \neq 0
\]</p>
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<figure>
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"/>
<figcaption><div class="markdown"><p>Polynomials of varying even degrees over [-1,1].</p>
</div></figcaption>
</figure>
</div>
<p>The numbers <span class="math">$a_0, a_1, \dots, a_n$</span> are the <strong>coefficients</strong> of the polynomial. With the convention that <span class="math">$x=x^1$</span> and <span class="math">$1 = x^0$</span>, the monomials above have their power match their coefficient's index, e.g., <span class="math">$a_ix^i$</span>. Outside of the coefficient <span class="math">$a_n$</span>, the other coefficients may be negative, positive, <em>or</em> <span class="math">$0$</span>. Except for the zero polynomial, the largest power <span class="math">$n$</span> is called the <a href="https://en.wikipedia.org/wiki/Degree_of_a_polynomial">degree</a>. The degree of the <a href="http://tinyurl.com/he6eg6s">zero</a> polynomial is typically not defined or defined to be <span class="math">$-1$</span>, so as to make certain statements easier to express. The term <span class="math">$a_n$</span> is called the <strong>leading coefficient</strong>. When the leading coefficient is <span class="math">$1$</span>, the polynomial is called a <strong>monic polynomial</strong>. The monomial <span class="math">$a_n x^n$</span> is the <strong>leading term</strong>.</p>
<p>For example, the polynomial <span class="math">$-16x^2 - 32x + 100$</span> has degree <span class="math">$2$</span>, leading coefficient <span class="math">$-16$</span> and leading term <span class="math">$-16x^2$</span>. It is not monic, as the leading coefficient is not 1.</p>
<p>Lower degree polynomials have special names: a degree <span class="math">$0$</span> polynomial (<span class="math">$a_0$</span>) is a non-zero constant, a degree 1 polynomial (<span class="math">$a_0+a_1x$</span>) is called linear, a degree <span class="math">$2$</span> polynomial is quadratic, and a degree <span class="math">$3$</span> polynomial is called cubic.</p>
<h2>Linear polynomials</h2>
<p>A special place is reserved for polynomials with degree 1. These are linear, as their graphs are straight lines. The general form,</p>
<p class="math">\[
a_1 x + a_0, \quad a_1 \neq 0,
\]</p>
<p>is often written as <span class="math">$mx + b$</span>, which is the <strong>slope-intercept</strong> form. The slope of a line determines how steeply it rises. The value of <span class="math">$m$</span> can be found from two points through the well-known formula:</p>
<p class="math">\[
m = \frac{y_1 - y_0}{x_1 - x_0} = \frac{\text{rise}}{\text{run}}
\]</p>
<div class="well well-sm">
<figure>
<img 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"/>
<figcaption><div class="markdown"><p>Graphs of y = mx for different values of m</p>
</div></figcaption>
</figure>
</div>
<p>The intercept, <span class="math">$b$</span>, comes from the fact that when <span class="math">$x=0$</span> the expression is <span class="math">$b$</span>. That is the graph of the function <span class="math">$f(x) = mx + b$</span> will have <span class="math">$(0,b)$</span> as a point on it.</p>
<p>More generally, we have the <strong>point-slope</strong> form of a line, written as a polynomial through</p>
<p class="math">\[
y_0 + m \cdot (x - x_0).
\]</p>
<p>The slope is <span class="math">$m$</span> and the point <span class="math">$(x_0, y_0)$</span>. Again, the line graphing this as a function of <span class="math">$x$</span> would have the point <span class="math">$(x_0,y_0)$</span> on it and have slope <span class="math">$m$</span>. This form is more useful in calculus, as the information we have convenient is more likely to be related to a specific value of <span class="math">$x$</span>, not the special value <span class="math">$x=0$</span>.</p>
<p>Thinking in terms of transformations, this looks like the function <span class="math">$f(x) = x$</span> (whose graph is a line with slope 1) stretched in the <span class="math">$y$</span> direction by a factor of <span class="math">$m$</span> then shifted right by <span class="math">$x_0$</span> units, and then shifted up by <span class="math">$y_0$</span> units. When <span class="math">$m>1$</span>, this means the line grows faster. When <span class="math">$m< 0$</span>, the line <span class="math">$f(x)=x$</span> is flipped through the <span class="math">$x$</span>-axis so would head downwards, not upwards like <span class="math">$f(x) = x$</span>.</p>
<h2>Symbolic math in Julia</h2>
<p>The indeterminate value <code>x</code> (or some other symbol) in a polynomial, is like a variable in a function and unlike a variable in <code>Julia</code>. Variables in <code>Julia</code> are identifiers, just a means to look up a specific, already determined, value. Rather, the symbol <code>x</code> is not yet determined, it is essentially a place holder for a future value. Although we have seen that <code>Julia</code> makes it very easy to work with mathematical functions, it is not the case that base <code>Julia</code> makes working with expressions of algebraic symbols easy. This makes sense, <code>Julia</code> is primarily designed for technical computing, where numeric approaches rule the day. However, symbolic math can be used from within <code>Julia</code> with an add-on package.</p>
<p>Symbolic math programs include well-known ones like the commercial programs Mathematica and Maple. Mathematica powers the popular <a href="www.wolframalpha.com">WolframAlpha</a> website, which turns "natural" language into the specifics of a programming language. The open-source Sage project is an alternative to these two commercial giants. It includes a wide-range of open-source math projects available within its umbrella framework. (<code>Julia</code> can even be run from within the free service <a href="https://cloud.sagemath.com/projects">cloud.sagemath.com</a>.) A more focused project for symbolic math, is the <a href="www.sympy.org">SymPy</a> Python library. SymPy is also used within Sage. However, SymPy provides a self-contained library that can be used standalone within a Python session. That is great for <code>Julia</code> users, as the <code>PyCall</code> package glues <code>Julia</code> to Python in a seamless manner. This allows the <code>Julia</code> package <code>SymPy</code> to provide functionality from SymPy within <code>Julia</code>.</p>
<div class="alert alert-info" role="alert">
<div class="markdown"><p><code>SymPy</code> is installed when the accompanying <code>CalculusWithJulia</code> package is installed. It could also be installed directly. The package relies on both Python being installed and SymPy being added to the installed Python. This is done automatically on installation, if needed, when the <code>PyCall</code> package is installed.</p>
</div>
</div>
<p>To use <code>SymPy</code>, we create symbolic objects to be our indeterminate symbols. The <code>symbols</code> function does this and is used like:</p>
<pre class='hljl'>
<span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>CalculusWithJulia</span><span class='hljl-t'> </span><span class='hljl-cs'># loads the `SymPy` package</span><span class='hljl-t'>
</span><span class='hljl-n'>a</span><span class='hljl-p'>,</span><span class='hljl-n'>b</span><span class='hljl-p'>,</span><span class='hljl-n'>c</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>symbols</span><span class='hljl-p'>(</span><span class='hljl-s'>"a,b,c"</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>symbols</span><span class='hljl-p'>(</span><span class='hljl-s'>"x"</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>real</span><span class='hljl-oB'>=</span><span class='hljl-kc'>true</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}x\end{equation*}</div>
<p>The first use, shows that multiple symbols can be defined at once. The second shows the extra keyword argument <code>real=true</code>, which instructs <code>SymPy</code> to assume the <code>x</code> is real, as otherwise it assumes it is possibly complex. There are many other <a href="http://docs.sympy.org/dev/modules/core.html#module-sympy.core.assumptions">assumptions</a> that can be made.</p>
<p>The <em>macro</em> <code>@vars</code> is like the second usage, only it does not need assignment, as the variable are created behind the scenes. This may be the easiest way to create symbolic values:</p>
<pre class='hljl'>
<span class='hljl-nd'>@vars</span><span class='hljl-t'> </span><span class='hljl-n'>h</span><span class='hljl-t'> </span><span class='hljl-n'>t</span>
</pre>
<pre class="output">
(h, t)
</pre>
<div class="alert alert-info" role="alert">
<div class="markdown"><p>Macros in <code>Julia</code> are just transformations of the syntax into other synax. The <code>@</code> indicates they behave differently than regular function calls. For the <code>@vars</code> macro, the arguments are <strong>not</strong> separated by commas, as a normal function cal would be.</p>
</div>
</div>
<p>The <code>SymPy</code> package does two basic things:</p>
<ul>
<li><p>It imports some of the functionality provided by <code>SymPy</code>, including the ability to create symbolic variables.</p>
</li>
<li><p>It overloads many <code>Julia</code> functions to work seamlessly with symbolic expressions. This makes working with polynomials quite natural.</p>
</li>
</ul>
<p>To illustrate, using the just defined <code>x</code>, here is how we can create the polynomial <span class="math">$-16x^2 + 100$</span>:</p>
<pre class='hljl'>
<span class='hljl-n'>p</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-ni'>16</span><span class='hljl-n'>x</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-ni'>100</span>
</pre>
<div class="well well-sm">\begin{equation*}- 16 x^{2} + 100\end{equation*}</div>
<p>That is, the expression is created just as you would create it within a function body. But here the result is still a symbolic object. We have assigned this expression to a variable <code>p</code>, and have not defined it as a function <code>p(x)</code>. Mentally keeping the distinction between expressions and functions is very important.</p>
<p>The <code>typeof</code> function shows that <code>p</code> is of a symbolic type (<code>Sym</code>):</p>
<pre class='hljl'>
<span class='hljl-nf'>typeof</span><span class='hljl-p'>(</span><span class='hljl-n'>p</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
SymPy.Sym
</pre>
<p>We can mix and match symbolic objects. This command creates an arbitrary quadratic polynomial:</p>
<pre class='hljl'>
<span class='hljl-n'>quad</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>a</span><span class='hljl-oB'>*</span><span class='hljl-n'>x</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>b</span><span class='hljl-oB'>*</span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>c</span>
</pre>
<div class="well well-sm">\begin{equation*}a x^{2} + b x + c\end{equation*}</div>
<p>Again, this is entered in a manner nearly identical to how we see such expressions typeset (<span class="math">$ax^2 + bx+c$</span>), though we must remember to explicitly place the multiplication operator, as the symbols are not numeric literals.</p>
<p>We can apply many of <code>Julia</code>'s mathematical functions and the result will still be symbolic:</p>
<pre class='hljl'>
<span class='hljl-nf'>sin</span><span class='hljl-p'>(</span><span class='hljl-n'>a</span><span class='hljl-oB'>*</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-n'>b</span><span class='hljl-oB'>*</span><span class='hljl-n'>pi</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>c</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}\sin{\left (a \left(- \pi b + x\right) + c \right )}\end{equation*}</div>
<p>Another example, might be the following combination:</p>
<pre class='hljl'>
<span class='hljl-n'>quad</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>quad</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-n'>quad</span><span class='hljl-oB'>^</span><span class='hljl-ni'>3</span>
</pre>
<div class="well well-sm">\begin{equation*}a x^{2} + b x + c - \left(a x^{2} + b x + c\right)^{3} + \left(a x^{2} + b x + c\right)^{2}\end{equation*}</div>
<h2>Substitution: subs, replace</h2>
<p>Algebraically working with symbolic expressions is straightforward. A different symbolic task is substitution. For example, replacing each instance of <code>x</code> in a polynomial, with, say, <code>(x-1)^2</code>. Substitution requires three things to be specified: an expression to work on, a variable to substitute, and a value to substitute in.</p>
<p>SymPy provides its <code>subs</code> function for this. This function is available in <code>Julia</code>, but it is easier to use notation reminiscent of function evaluation.</p>
<p>To illustrate, to do the task above for the polynomial <span class="math">$-16x^2 + 100$</span> we could have:</p>
<pre class='hljl'>
<span class='hljl-n'>p</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-ni'>16</span><span class='hljl-n'>x</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-ni'>100</span><span class='hljl-t'>
</span><span class='hljl-nf'>p</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-oB'>-</span><span class='hljl-ni'>1</span><span class='hljl-p'>)</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}- 16 \left(x - 1\right)^{4} + 100\end{equation*}</div>
<p>This "call" notation takes pairs (designated by <code>a=>b</code>) where the left-hand side is the variable to substitute for, and the right-hand side the new value. The value to substitute can depend on the variable, as illustrated; be a different variable; or be a numeric value, such as <span class="math">$2$</span>:</p>
<pre class='hljl'>
<span class='hljl-n'>y</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>p</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-oB'>=></span><span class='hljl-ni'>2</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}36\end{equation*}</div>
<p>The result will always be of a symbolic type, even if the answer is just a number:</p>
<pre class='hljl'>
<span class='hljl-nf'>typeof</span><span class='hljl-p'>(</span><span class='hljl-n'>y</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
SymPy.Sym
</pre>
<p>If there is just one free variable in an expression, the pair notation can be dropped:</p>
<pre class='hljl'>
<span class='hljl-nf'>p</span><span class='hljl-p'>(</span><span class='hljl-ni'>4</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-cs'># substitutes x=>4</span>
</pre>
<div class="well well-sm">\begin{equation*}-156\end{equation*}</div>
<h5>Example</h5>
<p>Suppose we have the polynomial <span class="math">$p = ax^2 + bx +c$</span>. What would it look like if we shifted right by <span class="math">$E$</span> units and up by <span class="math">$F$</span> units?</p>
<pre class='hljl'>
<span class='hljl-nd'>@vars</span><span class='hljl-t'> </span><span class='hljl-n'>a</span><span class='hljl-t'> </span><span class='hljl-n'>b</span><span class='hljl-t'> </span><span class='hljl-n'>c</span><span class='hljl-t'> </span><span class='hljl-n'>E</span><span class='hljl-t'> </span><span class='hljl-n'>F</span><span class='hljl-t'>
</span><span class='hljl-n'>p</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>a</span><span class='hljl-oB'>*</span><span class='hljl-n'>x</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>b</span><span class='hljl-oB'>*</span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>c</span><span class='hljl-t'>
</span><span class='hljl-nf'>p</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-n'>x</span><span class='hljl-oB'>-</span><span class='hljl-n'>E</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>F</span>
</pre>
<div class="well well-sm">\begin{equation*}F + a \left(- E + x\right)^{2} + b \left(- E + x\right) + c\end{equation*}</div>
<p>And expanded this becomes:</p>
<pre class='hljl'>
<span class='hljl-nf'>expand</span><span class='hljl-p'>(</span><span class='hljl-nf'>p</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-n'>x</span><span class='hljl-oB'>-</span><span class='hljl-n'>E</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>F</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}E^{2} a - 2 E a x - E b + F + a x^{2} + b x + c\end{equation*}</div>
<h3>Conversion of symbolic numbers to Julia numbers</h3>
<p>In the above, we substituted <code>2</code> in for <code>x</code> to get <code>y</code>:</p>
<pre class='hljl'>
<span class='hljl-n'>p</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-ni'>16</span><span class='hljl-n'>x</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-ni'>100</span><span class='hljl-t'>
</span><span class='hljl-n'>y</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>p</span><span class='hljl-p'>(</span><span class='hljl-ni'>2</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}36\end{equation*}</div>
<p>The value, <span class="math">$36$</span> is still symbolic, but clearly an integer. If we are just looking at the output, we can easily translate from the symbolic value to an integer, as they print similarly. However the conversion to an integer, or another type of number, does not happen automatically. If a number is needed to pass along to another <code>Julia</code> function, it may need to be converted. In general, conversions between different types are handled through various methods of <code>convert</code>. However, with <code>SymPy</code>, the <code>N</code> function will attempt to do the conversion for you:</p>
<pre class='hljl'>
<span class='hljl-nf'>N</span><span class='hljl-p'>(</span><span class='hljl-n'>y</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
36
</pre>
<p>Conversion by <code>N</code> also works for other types of data, such as <code>Rational</code> and <code>Float64</code>. For getting more digits of accuracy, a precision can be passed to <code>N</code>. The following command will take the symbolic value for <span class="math">$\pi$</span>, <code>PI</code>, and produce about 60 digits worth as a <code>BigFloat</code> value:</p>
<pre class='hljl'>
<span class='hljl-nf'>N</span><span class='hljl-p'>(</span><span class='hljl-n'>PI</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>60</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
3.141592653589793238462643383279502884197169399375105820974939
</pre>
<p>Conversion will fail if the value to be converted contains free symbols, as would be expected.</p>
<h2>Graphical properties of polynomials</h2>
<p>Consider the graph of the polynomial <code>x^5 - x + 1</code>:</p>
<pre class='hljl'>
<span class='hljl-nf'>plot</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-oB'>^</span><span class='hljl-ni'>5</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-ni'>1</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-ni'>3</span><span class='hljl-oB'>/</span><span class='hljl-ni'>2</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>3</span><span class='hljl-oB'>/</span><span class='hljl-ni'>2</span><span class='hljl-p'>)</span>
</pre>
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<p>(Plotting symbolic expressions is similar to plotting a function, in that the expression is passed in as the first argument. The expression must have only one free variable, as above, or an error will occur.)</p>
<p>This graph illustrates the key features of polynomial graphs:</p>
<ul>
<li><p>there may be values for <code>x</code> where the graph crosses the <span class="math">$x$</span> axis (real roots of the polynomial);</p>
</li>
<li><p>there may be peaks and valleys (local maxima and local minima)</p>
</li>
<li><p>except for constant polynomials, the ultimate behaviour for large values of <span class="math">$\lvert x\rvert$</span> is either both sides of the graph going to positive infinity, or negative infinity, or as in this graph one to the positive infinity and one to negative infinity. In particular, there is no <em>horizontal asymptote</em>.</p>
</li>
</ul>
<p>To investigate this last point, let's consider the case of the monomial <span class="math">$x^n$</span>. When <span class="math">$n$</span> is even, the following animation shows that larger values of <span class="math">$n$</span> have greater growth once outside of <span class="math">$[-1,1]$</span>:</p>
<div class="well well-sm">
<figure>
<img 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<figcaption><div class="markdown"><p>Demonstration that $x^{10}$ grows faster than $x^8$, ... and $x^2$ grows faster than $x^0$ (which is constant).</p>
</div></figcaption>
</figure>
</div>
<p>Of course, this is expected, as, for example, <span class="math">$2^2 < 2^4 < 2^6 < \cdots$</span>. The general shape of these terms is similar – <span class="math">$U$</span> shaped, and larger powers dominate the smaller powers as <span class="math">$\lvert x\rvert$</span> gets big.</p>
<p>For odd powers of <span class="math">$n$</span>, the graph of the monomial <span class="math">$x^n$</span> is no longer <span class="math">$U$</span> shaped, but rather constantly increasing. This graph of <span class="math">$x^5$</span> is typical:</p>
<pre class='hljl'>
<span class='hljl-nf'>plot</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-oB'>^</span><span class='hljl-ni'>5</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-ni'>2</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>2</span><span class='hljl-p'>)</span>
</pre>
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