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HTML
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<title>Polynomials</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Polynomials</h1>
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<p class="larger">A polynomial looks like this:</p>
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<div class="beach">
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<table align="center" cellpadding="5" border="0">
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<tbody>
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<tr style="text-align:center;">
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<td><img src="images/polynomial-example.svg" alt="polynomial example" height="97" width="197"></td>
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</tr>
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<tr style="text-align:center;">
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<td>example of a polynomial<br>
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this one has 3 terms</td>
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</tr>
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</tbody></table>
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</div>
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<div class="words">
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<p><b>Polynomial</b> comes from <i>poly-</i> (meaning "many") and <i>-nomial</i> (in this case meaning "term") ... so it says "many terms"</p>
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</div>
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<p>Polynomials with one variable make nice smooth curves:</p>
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<p><img src="images/sage-walks-poly.svg" alt="algebra girl" height="226" width="635"></p>
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<p class="larger" style="margin-top:-40px;">A polynomial can have:</p>
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<table align="center" cellpadding="5" border="0">
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<tbody>
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<tr>
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<td><span class="larger"><b><a href="definitions.html">constants</a></b></span> (like <b>3</b>, <b>−20</b>, or <b>½</b>) </td>
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</tr>
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<tr>
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<td><span class="larger"><b><a href="definitions.html">variables</a></b></span> (like <i><b>x</b></i> and <i><b>y</b></i>)</td>
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</tr>
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<tr>
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<td><span class="larger"><b><a href="../exponent.html">exponents</a></b></span> (like the 2 in y<sup>2</sup>), but only <b> 0, 1, 2, 3, ...</b> etc are allowed</td>
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</tr>
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</tbody></table>
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<p>that can be combined using <b>addition, subtraction, multiplication and division</b> ...</p>
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<p class="center larger">... except ...</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><span class="larger">... <b>not</b> division by a variable</span> (so something like <b>2/x</b> is right out)</td>
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</tr>
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</tbody></table>
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<p>So:</p>
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<div class="def">
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<p class="center">A polynomial can have constants, variables and exponents,<br>
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but never division by a variable.</p>
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</div>
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<p>Also they can have one or more terms, but not an <a href="../numbers/infinity.html">infinite</a> number of terms.</p>
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<h2>Polynomial or Not?</h2>
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<p class="center"><img src="images/polynomial.svg" alt="polynomial" height="192" width="463"></p>
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<p>These <b>are</b> polynomials:</p>
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<ul>
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<div class="bigul">
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<li><b>3x</b></li>
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<li><b>x − 2</b></li>
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<li><b>−6y<sup>2</sup> − (<span class="intbl">
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<em>7</em>
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<strong>9</strong>
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</span>)x</b></li>
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<li><b>3xyz + 3xy<sup>2</sup>z − 0.1xz − 200y + 0.5</b></li>
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<li><b>512v<sup>5</sup></b> + <b>99w<sup>5</sup></b></li>
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<li><b>5</b></li>
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</div>
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</ul>
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<p>(Yes, "5" is a polynomial, <b>one term is allowed</b>, and it can be just a constant!)</p>
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<p> </p>
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<p>These are <b>not</b> polynomials</p>
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<ul>
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<li><b>3xy<sup>-2</sup></b> is not, because the exponent is "-2" (exponents can only be 0,1,2,...)</li>
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<li><b>2/(x+2)</b> is not, because dividing by a variable is not allowed</li>
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<li><b>1/x</b> is not either</li>
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<li><b>√x</b> is not, because the exponent is "½" (see <a href="exponent-fractional.html">fractional exponents</a>)</li>
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</ul>
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<p> </p>
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<p><b>But</b> these <b> are</b> allowed:</p>
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<ul>
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<li><b>x/2</b> <b>is allowed</b>, because you can divide by a constant</li>
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<li>also <b>3x/8</b> for the same reason</li>
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<li><b>√2</b> is allowed, because it is a constant (= 1.4142...etc)</li>
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</ul>
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<h2>Monomial, Binomial, Trinomial</h2>
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<p>There are special names for polynomials with 1, 2 or 3 terms:</p>
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<p class="center"><img src="images/polynomials.svg" alt="monomial, binomial, trinomial" height="59" width="529"></p>
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<p class="center"><i><b>How do you remember the names? Think cycles!<br>
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<br>
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</b></i> <img src="images/mono-bi-tri.svg" alt="monocycle bicycle tricycle" height="144" width="309"></p>
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<p class="center"><i>There is also quadrinomial (4 terms) and quintinomial (5 terms),<br>
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but those names are not often used.</i></p>
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<h2>Variables</h2>
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<p>Polynomials can have no variable at all</p>
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<div class="example">
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<p>Example: <span class="large">21</span> is a polynomial. It has just one term, which is a constant.</p>
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</div>
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<p>Or one variable</p>
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<div class="example">
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<p>Example: <span class="large">x<sup>4 </sup>− 2x<sup>2 </sup>+ x</span> has three terms, but only one variable (x)</p>
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</div>
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<p>Or two or more variables</p>
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<div class="example">
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<p>Example: <span class="large">xy<sup>4 </sup>− 5x<sup>2</sup>z</span> has two terms, and three variables (x, y and z)</p>
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</div>
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<h2>What is Special About Polynomials?</h2>
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<p>Because of the strict definition, polynomials are <b>easy to work with</b>.</p>
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<p>For example we know that:</p>
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<ul>
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<li>If you <a href="polynomials-adding-subtracting.html">add polynomials</a> you get a polynomial</li>
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<li>If you <a href="polynomials-multiplying.html">multiply polynomials</a> you get a polynomial</li>
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</ul>
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<p>So you can do lots of additions and multiplications, and still have a polynomial as the result.</p>
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<p>Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines.</p>
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<div class="example">
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<h3>Example: <span class="large">x<sup>4</sup>−2x<sup>2</sup>+x</span></h3>
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<div class="example2">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/polynomial-turning-points1.gif" alt="x^4-2x^2+x" height="212" width="216"></td>
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<td>
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<p>See how nice and<br>
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smooth the curve is?</p></td>
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</tr>
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</tbody></table>
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</div>
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</div>
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<p>You can also <a href="polynomials-dividing.html">divide polynomials</a> (but the result may not be a polynomial).</p>
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<h2>Degree</h2>
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<p>The <b>degree</b> of a polynomial with only one variable is the <b>largest exponent</b> of that variable.</p>
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<div class="example">
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<h3>Example:</h3>
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<div class="example2">
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<table align="center" width="80%" border="0">
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<tbody>
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<tr>
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<td class="larger" align="center" width="28%">4x<sup>3</sup> − x + 2</td>
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<td width="72%">The Degree is <b>3</b> (the largest exponent of <span class="times"><i>x</i></span>)</td>
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</tr>
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</tbody></table>
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</div>
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</div>
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<p>For more complicated cases, read <a href="degree-expression.html">Degree (of an Expression)</a>.</p>
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<h2>Standard Form</h2>
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<p>The <a href="standard-form.html">Standard Form</a> for writing a polynomial is to put the terms with the highest degree first.</p>
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<div class="example">
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<h3>Example: Put this in Standard Form: 3<b>x<sup>2</sup></b> − 7 + 4<b>x<sup>3</sup></b> + <b>x<sup>6</sup></b></h3>
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<p>The highest degree is 6, so that goes first, then 3, 2 and then the constant last:</p>
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<p class="center large"><b>x<sup>6</sup></b> + 4<b>x<sup>3</sup></b> + 3<b>x<sup>2</sup></b> − 7</p>
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</div>
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<p>You <b>don't have to</b> use Standard Form, but it helps.</p>
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<p> </p>
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<div class="questions">342, 343, 1095, 1096, 3182, 3183, 3184, 3185, 1097, 4002</div>
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<div class="related">
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<a href="polynomials-adding-subtracting.html">Adding Polynomials</a>
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<a href="introduction.html">Introduction to Algebra</a>
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<a href="polynomials-multiplying.html">Multiplying Polynomials</a>
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<a href="definitions.html">Algebra - Basic Definitions</a>
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<a href="index.html">Algebra Index</a>
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</div>
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