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229 lines
7.8 KiB
HTML
229 lines
7.8 KiB
HTML
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Simplify</h1>
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<p class="center"><i>Simplify: to make simpler!</i></p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/simplify.svg" alt="simplify 4x+2x to 6x" /></p>
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<p>One of the big jobs we do in Algebra is <b>simplification</b>.</p>
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<p>You will often be asked to put something "in simplest form"</p>
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<h2>What is the Simplest Form?</h2>
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<p>In general, it is simpler when it is <b>easier to use</b>.</p>
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<div class="example">
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<table style="border: 0;">
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<tr>
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<td align="right">Start with:</td>
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<td width="20"> </td>
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<td align="center" class="large">5x + x − 3</td>
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</tr>
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<tr>
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<td align="right"><a href="like-terms.html">Combine like terms</a>: </td>
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<td width="20"> </td>
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<td align="center" class="large">6x − 3</td>
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</tr>
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</table>
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<p>It is now a little easier to use.</p>
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</div>
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<p>And:</p>
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<div class="example">
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<table style="border: 0;">
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<tr>
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<td align="right">Start with:</td>
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<td width="20"> </td>
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<td align="center" class="large"><span class="large intbl">
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<em>3</em>
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<strong>6</strong>
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</span></td>
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</tr>
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<tr>
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<td align="right"><a href="../simplifying-fractions.html">Simplify</a> the fraction by dividing top and bottom by 3: </td>
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<td width="20"> </td>
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<td align="center" class="large"><span class="large intbl">
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<em>1</em>
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<strong>2</strong>
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</span></td>
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</tr>
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</table>
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<p>"Half" is definitely simpler than "three sixths", unless it is important to know that something was cut into sixths.</p>
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</div>
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<p>And:</p>
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<div class="example">
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<table style="border: 0;">
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<tr>
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<td align="right">Start with:</td>
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<td width="20"> </td>
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<td align="center" class="large">2w(5wy)</td>
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</tr>
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<tr>
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<td align="right"><a href="variables-exponents-multiply.html">Multiply the constants and variables</a>: </td>
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<td width="20"> </td>
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<td align="center" class="large">10w<sup>2</sup>y</td>
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</tr>
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</table>
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<p>It is now a little easier to use.</p>
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</div>
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<p>And:</p>
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<div class="example">
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<table style="border: 0;">
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<tr>
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<td align="right">Start with:</td>
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<td width="20"> </td>
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<td align="center" class="large"><span class="large intbl">
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<em>2x<sup>2</sup> − 6x + 2</em>
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<strong>x − 3</strong>
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</span></td>
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</tr>
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<tr>
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<td align="right"><a href="polynomials-division-long.html">Polynomial long division</a>: </td>
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<td width="20"> </td>
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<td align="center" class="large">2x + <span class="intbl">
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<em>2</em>
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<strong>x − 3</strong>
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</span></td>
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</tr>
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</table>
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</div>
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<p>And:</p>
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<div class="example">
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<table style="border: 0;">
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<tr>
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<td align="right">Start with:</td>
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<td width="20"> </td>
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<td align="center" class="large">x<sup>2</sup> − 2x − 3</td>
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</tr>
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<tr>
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<td align="right"><a href="factoring.html">Factoring</a>: </td>
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<td width="20"> </td>
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<td align="center" class="large">(x−3)(x+1)</td>
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</tr>
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</table>
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</div>
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<p>That last example can be argued with! Some people say to <b>remove</b> parentheses to make it "simpler", but <span class="large">(x−3)(x+1)</span> is usually a lot easier to use.</p>
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<p>The moral of the story:</p>
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<div class="def">
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<p>"Simplified" is sometimes obvious, but can also depend on what you want to do.</p>
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</div>
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<h2>How to Simplify</h2>
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<p>There are many ways to simplify!</p>
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<p>When we simplify we use similar skills to <a href="equations-solving.html">solving equations</a>, and that page has some good advice.</p>
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<p>Some of these things might help:</p>
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<ul>
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<li><a href="like-terms.html">Combine Like Terms</a></li>
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<li><a href="factoring.html">Factor</a></li>
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<li><a href="expanding.html">Expand</a> (the opposite of factoring)</li>
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<li>Clear out fractions by multiplying</li>
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<li>Find some pattern you have seen before, like the <a href="special-binomial-products.html">difference of squares</a>.</li>
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</ul>
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<h2>And Which Is Simpler Here?</h2>
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<p>Here is one more interesting case:</p>
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<div class="example">
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<table style="border: 0;">
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<tr>
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<td align="right">This:</td>
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<td width="20"> </td>
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<td align="center"><img src="images/latex/1-sqrt2.gif" alt="1/sqrt2" width="34" height="54" /></td>
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<td width="20" align="center"> </td>
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<td>seems simple enough</td>
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</tr>
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<tr>
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<td align="right">But this: </td>
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<td width="20"> </td>
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<td align="center"><img src="images/latex/sqrt2-2.gif" alt="sqrt2/2" width="35" height="51" /></td>
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<td width="20" align="center"> </td>
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<td>has a <a href="rationalize-denominator.html">rationalized denominator</a><br>
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(normally considered simpler, and preferred by teachers!)</td>
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</tr>
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</table>
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</div>
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<p>Which is simpler? You decide!</p>
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<p> </p>
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<div class="questions">
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<script>getQ(8709, 8710, 8711, 8712, 8713, 8714, 8715, 8716, 8717, 8718, 8719, 8720);</script>
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</div>
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<div class="related">
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<a href="index.html">Algebra Index</a>
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Copyright © 2020 MathsIsFun.com
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