e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
370 lines
14 KiB
HTML
370 lines
14 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/degree-expression.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:00:00 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Degree (of an Expression)</title>
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</script>
|
||
</head>
|
||
|
||
<body id="bodybg" class="adv">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo-adv.svg" alt="Math is Fun"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
<h1 class="center">Degree (of an Expression)</h1>
|
||
|
||
<p>"Degree" can mean several things in mathematics:</p>
|
||
<ul>
|
||
<li>In Geometry a degree (°) is a way of <a href="../geometry/degrees.html">measuring angles</a>,</li>
|
||
<li>But here we look at what degree means in <b>Algebra</b>.</li>
|
||
</ul>
|
||
<div class="words">
|
||
<p>In Algebra "Degree" is sometimes called "Order"</p>
|
||
</div>
|
||
<h2>Degree of a Polynomial (with one variable)</h2>
|
||
<p>A <a href="polynomials.html">polynomial</a> looks like this:</p>
|
||
<div class="beach">
|
||
<table align="center" cellpadding="5" border="0">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<td><img src="images/polynomial-1var-example.svg" alt="polynomial example"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>example of a polynomial<br>
|
||
this one has 3 terms</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
|
||
|
||
<div class="def">
|
||
<p>The <b>Degree</b> (for a polynomial with one variable, like <span class="times"><i>x</i></span>) is:</p>
|
||
<p class="center large">the <b><a href="../exponent.html">largest exponent</a></b> of that variable.</p>
|
||
<p class="center"><img src="images/degree-example-a.svg" alt="polynomial"></p>
|
||
</div>
|
||
<h3>More Examples:</h3>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td class=" large" align="center">4x</td>
|
||
<td align="center" width="20"> </td>
|
||
<td>The Degree is <b>1</b> (a variable without an<br>
|
||
exponent actually has an exponent of 1)</td>
|
||
</tr>
|
||
<tr>
|
||
<td class=" large" align="center"> </td>
|
||
<td style="text-align:center;"> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td class=" large" align="center">4x<sup>3</sup> − x + 3</td>
|
||
<td style="text-align:center;"> </td>
|
||
<td>The Degree is <b>3</b> (largest exponent of x)</td>
|
||
</tr>
|
||
<tr>
|
||
<td class=" large" align="center"> </td>
|
||
<td style="text-align:center;"> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td class=" large" align="center">x<sup>2</sup> + 2x<sup>5</sup> − x</td>
|
||
<td style="text-align:center;"> </td>
|
||
<td>The Degree is <b>5</b> (largest exponent of x)</td>
|
||
</tr>
|
||
<tr>
|
||
<td class=" large" align="center"> </td>
|
||
<td style="text-align:center;"> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td class=" large" align="center">z<sup>2</sup> − z + 3</td>
|
||
<td style="text-align:center;"> </td>
|
||
<td>The Degree is <b>2</b> (largest exponent of z)</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<h2>Names of Degrees</h2>
|
||
<p>When we know the degree we can also give it a name!</p>
|
||
<div class="beach">
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;" class=" large">
|
||
<th>Degree</th>
|
||
<th>Name</th>
|
||
<th>Example</th>
|
||
</tr>
|
||
<tr style="text-align:center;" class=" large">
|
||
<td style="width:40px;">0</td>
|
||
<td style="width:150px;">Constant</td>
|
||
<td style="width:150px;">7</td>
|
||
</tr>
|
||
<tr style="text-align:center;" class=" large">
|
||
<td style="width:40px;">1</td>
|
||
<td style="width:150px;"><a href="linear-equations.html">Linear</a></td>
|
||
<td style="width:150px;">x+3</td>
|
||
</tr>
|
||
<tr style="text-align:center;" class=" large">
|
||
<td style="width:40px;">2</td>
|
||
<td style="width:150px;"><a href="quadratic-equation.html">Quadratic</a></td>
|
||
<td style="width:150px;">x<sup>2</sup>−x+2</td>
|
||
</tr>
|
||
<tr style="text-align:center;" class=" large">
|
||
<td style="width:40px;">3</td>
|
||
<td style="width:150px;">Cubic </td>
|
||
<td style="width:150px;">x<sup>3</sup>−x<sup>2</sup>+5</td>
|
||
</tr>
|
||
<tr style="text-align:center;" class=" large">
|
||
<td style="width:40px;">4</td>
|
||
<td style="width:150px;">Quartic </td>
|
||
<td style="width:150px;">6x<sup>4</sup>−x<sup>3</sup>+x−2</td>
|
||
</tr>
|
||
<tr style="text-align:center;" class=" large">
|
||
<td style="width:40px;">5</td>
|
||
<td style="width:150px;">Quintic</td>
|
||
<td style="width:150px;">x<sup>5</sup>−3x<sup>3</sup>+x<sup>2</sup>+8</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div><br>
|
||
<div class="example">
|
||
<p>Example: <b>y = 2x + 7</b> has a degree of 1, so it is a <b>linear</b> equation</p>
|
||
</div>
|
||
<div class="example">
|
||
<p>Example: <b>5w<sup>2</sup> − 3</b> has a degree of 2, so it is <b>quadratic</b></p>
|
||
</div>
|
||
<p>Higher order equations are <b>usually</b> harder to solve:</p>
|
||
<ul>
|
||
<li>Linear equations are <b>easy</b> to solve</li>
|
||
<li>Quadratic equations are <b>a little harder</b> to solve</li>
|
||
<li>Cubic equations are harder again, but <b>there are formulas</b> to help</li>
|
||
<li>Quartic equations can also be solved, but the formulas are <b>very complicated</b></li>
|
||
<li>Quintic equations have no formulas, and <b>can sometimes be unsolvable</b>!</li>
|
||
</ul>
|
||
<h2>Degree of a Polynomial with More Than One Variable</h2>
|
||
<p>When a polynomial has more than one variable, we need to look at <b>each term</b>. Terms are separated by + or - signs:</p>
|
||
<div class="beach">
|
||
<table align="center" cellpadding="5" border="0">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<td><img src="images/polynomial-example.svg" alt="polynomial example"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>example of a polynomial<br>
|
||
with more than one variable</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>For <b>each term</b>:</p>
|
||
</div>
|
||
<ul>
|
||
<li>Find the degree by <b>adding the exponents of each variable</b> in it,</li>
|
||
</ul>
|
||
<p>The <b>largest</b> such degree is the degree of the polynomial.</p>
|
||
<div class="example">
|
||
<h3>Example: what is the degree of this polynomial:</h3>
|
||
<p class="center"><img src="images/degree-example.svg" alt="polynomial"></p>
|
||
<p>Checking each term:</p>
|
||
<ul>
|
||
<li><b>5xy<sup>2</sup></b> has a degree of <b>3</b> (x has an exponent of 1, y has 2, and 1+2=3)</li>
|
||
<li><b>3x</b> has a degree of <b>1</b> (x has an exponent of 1)</li>
|
||
<li><b>5y<sup>3</sup></b> has a degree of <b>3</b> (y has an exponent of 3)</li>
|
||
<li><b>3</b> has a degree of 0 (no variable)</li>
|
||
</ul>
|
||
<p>The largest degree of those is 3 (in fact two terms have a degree of 3), so the polynomial has a degree of <b>3</b></p>
|
||
</div>
|
||
<div class="example">
|
||
<h3>Example: what is the degree of this polynomial:</h3>
|
||
<p class="center large">4z<sup>3</sup> + 5y<sup>2</sup>z<sup>2</sup> + 2yz</p>
|
||
<p>Checking each term:</p>
|
||
<ul>
|
||
<li><b>4z<sup>3</sup></b> has a degree of <b>3</b> (z has an exponent of 3)</li>
|
||
<li><b>5y<sup>2</sup>z<sup>2</sup></b> has a degree of <b>4</b> (y has an exponent of 2, z has 2, and 2+2=4)</li>
|
||
<li><b>2yz</b> has a degree of <b>2</b> (y has an exponent of 1, z has 1, and 1+1=2)</li>
|
||
</ul>
|
||
<p>The largest degree of those is 4, so the polynomial has a degree of <b>4</b></p>
|
||
</div>
|
||
<h2>Writing it Down</h2>
|
||
<p>Instead of saying "<i>the degree of (whatever) is 3</i>" we write it like this:</p>
|
||
<p class="center"><img src="images/degree-notation.gif" alt="degree notation" height="30" width="175"></p>
|
||
<h2>When Expression is a Fraction</h2>
|
||
<p>We can work out the degree of a <a href="rational-expression.html">rational expression</a> (one that is in the form of a fraction) by taking the degree of the top (numerator) and subtracting the degree of the bottom (denominator).</p>
|
||
<p>Here are three examples:</p>
|
||
|
||
<div class="script">../algebra/images/degree-example.js?mode=x0</div>
|
||
<div class="script">../algebra/images/degree-example.js?mode=x1</div>
|
||
<div class="script">../algebra/images/degree-example.js?mode=xm1</div>
|
||
|
||
|
||
|
||
<h2>Calculating Other Types of Expressions</h2>
|
||
<p><b>Warning: Advanced Ideas Ahead!</b></p>
|
||
<p>We can sometimes work out the degree of an expression by dividing ...</p>
|
||
<ul>
|
||
<li>the logarithm of the function by</li>
|
||
<li>the logarithm of the variable</li>
|
||
</ul>
|
||
<p class="center">... then do that for larger and larger values, to see where the answer is "heading".</p>
|
||
<p><i>(More correctly we should work out the <a href="../calculus/limits-infinity.html">Limit to Infinity</a> of <span class="intbl"><em>ln(f(x))</em><strong>ln(x)</strong></span>, but I just want to keep this simple here).</i></p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td>
|
||
<p>Note: "<b>ln</b>" is the <a href="logarithms.html">natural logarithm</a> function.</p></td>
|
||
<td> </td>
|
||
<td><img src="../money/images/calculator-ln.gif" alt="calculator ln button" height="87" width="107"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>Here is an example:</p>
|
||
<div class="example">
|
||
<h3>Example: The degree of 3 + <span style="font-size:120%;">√</span><span class="overline">x</span> </h3>
|
||
<p>Let us try increasing values of x:</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<th align="right" width="120">x</th>
|
||
<th align="right" width="120">ln(3 + <span style="font-size:120%;">√</span><span class="overline">x</span>)</th>
|
||
<th align="right" width="120">ln(x)</th>
|
||
<th align="right" width="120"><span class="intbl"><em>ln(3 + <span style="font-size:120%;">√</span><span class="overline">x</span>)</em><strong>ln(x)</strong></span></th>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th width="120">2</th>
|
||
<td style="width:120px;">1.48483</td>
|
||
<td style="width:120px;">0.69315</td>
|
||
<td style="width:120px;">2.1422</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th width="120">4</th>
|
||
<td style="width:120px;">1.60944</td>
|
||
<td style="width:120px;">1.38629</td>
|
||
<td style="width:120px;">1.1610</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th width="120">10</th>
|
||
<td style="width:120px;">1.81845</td>
|
||
<td style="width:120px;">2.30259</td>
|
||
<td style="width:120px;">0.7897</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th width="120">100</th>
|
||
<td style="width:120px;">2.56495</td>
|
||
<td style="width:120px;">4.60517</td>
|
||
<td style="width:120px;">0.5570</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th width="120">1,000</th>
|
||
<td style="width:120px;">3.54451</td>
|
||
<td style="width:120px;">6.90776</td>
|
||
<td style="width:120px;">0.5131</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th width="120">10,000</th>
|
||
<td style="width:120px;">4.63473</td>
|
||
<td style="width:120px;">9.21034</td>
|
||
<td style="width:120px;">0.5032</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th width="120">100,000</th>
|
||
<td style="width:120px;">5.76590</td>
|
||
<td style="width:120px;">11.51293</td>
|
||
<td style="width:120px;">0.5008</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th width="120">1,000,000</th>
|
||
<td style="width:120px;">6.91075</td>
|
||
<td style="width:120px;">13.81551</td>
|
||
<td style="width:120px;">0.5002</td>
|
||
</tr>
|
||
</tbody></table>
|
||
|
||
<p>Looking at the table:</p>
|
||
<ul>
|
||
<li>as <b>x</b> gets larger then <b><span class="intbl"><em>ln(3 + <span style="font-size:120%;">√</span><span class="overline">x</span>)</em><strong>ln(x)</strong></span></b> gets closer and closer to <b>0.5</b></li>
|
||
</ul>
|
||
<p>So the Degree is 0.5 (in other words 1/2)</p>
|
||
<p><i>(Note: this agrees nicely with x<sup>½</sup> = square root of x, see <a href="exponent-fractional.html">Fractional Exponents</a>)</i></p>
|
||
</div>
|
||
<h2>Some Degree Values</h2>
|
||
<div class="beach">
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<th width="120">Expression</th>
|
||
<th width="120">Degree</th>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:120px;">log(x)</td>
|
||
<td style="width:120px;">0</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:120px;">e<sup>x</sup></td>
|
||
<td style="width:120px;"><span style="font-size: 20px; font-family: serif;">∞</span></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:120px;">1/x</td>
|
||
<td style="width:120px;">−1</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:120px;"><span style="font-size:120%;">√</span><span class="overline">x</span></td>
|
||
<td style="width:120px;">1/2</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">462, 4003, 2092, 4004,463, 1108, 2093, 4005, 1109, 4006</div>
|
||
|
||
<div class="related">
|
||
<a href="../geometry/degrees.html">Degree (Geometry)</a>
|
||
<a href="index.html">Algebra Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2021 MathsIsFun.com</div>
|
||
|
||
</div>
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/degree-expression.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:00:01 GMT -->
|
||
</html> |