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Proof of the Polynomial Remainder Theorem - Revision history
2024-03-29T04:34:20Z
Revision history for this page on the wiki
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https://artofproblemsolving.com/wiki/index.php?title=Proof_of_the_Polynomial_Remainder_Theorem&diff=105547&oldid=prev
Zhang2018 at 22:15, 30 April 2019
2019-04-30T22:15:23Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:15, 30 April 2019</td>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Synopsis: Written below is a brief description of the polynomial remainder theorem. The theorem has a wide range of applications spanning from Algebra to Number Theory. This depicts how important the polynomial remainder theorem truly is, and why it must be taught in all courses and is a great tool.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The remainder theorem states when a polynomial denoted as <math>f(x)</math> is divided by <math>x-a</math> for some value of <math>x</math>, whether real or unreal, the remainder of <math>\frac{f(x)}{x-a}=f(a)</math> Written below is the proof of the polynomial remainder theorem.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The remainder theorem states when a polynomial denoted as <math>f(x)</math> is divided by <math>x-a</math> for some value of <math>x</math>, whether real or unreal, the remainder of <math>\frac{f(x)}{x-a}=f(a)</math> Written below is the proof of the polynomial remainder theorem.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>All polynomials can be written in the form <math>f(x)=d(x)\cdot<del class="diffchange diffchange-inline">\</del>q(x)+r(x)</math>, where <math>d(x)</math> is the divisor of the function/polynomial <math>f(x)</math>, <math>q(x)</math> is the quotient. amd <math>r(x)</math> is the remainder. Because the <math>deg r=0</math> or <math>deg r<del class="diffchange diffchange-inline">\less\</del>deg <del class="diffchange diffchange-inline">d</del></math> <del class="diffchange diffchange-inline">and the </del><math><del class="diffchange diffchange-inline">deg </del>d<del class="diffchange diffchange-inline">=1</del></math><del class="diffchange diffchange-inline">, </del>degrees must be whole numbers, <del class="diffchange diffchange-inline">and so </del><math>deg r=0</math><del class="diffchange diffchange-inline">. So </del>to speak, <math>r(x)</math> is a constant<del class="diffchange diffchange-inline">. We </del>denote <del class="diffchange diffchange-inline">this constant </del><math>b</math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>All polynomials can be written in the form <math>f(x)=d(x)\cdot<ins class="diffchange diffchange-inline">{</ins>q(x)<ins class="diffchange diffchange-inline">}</ins>+r(x)</math>, where <math>d(x)</math> is the divisor of the function/polynomial <math>f(x)</math>, <math>q(x)</math> is the quotient. amd <math>r(x)</math> is the remainder. Because the <math>deg<ins class="diffchange diffchange-inline"></math> <math></ins>r=0</math> or <ins class="diffchange diffchange-inline">the </ins><math>deg<ins class="diffchange diffchange-inline"></math> <math></ins>r<ins class="diffchange diffchange-inline"><</ins>deg</math> <math>d</math> <ins class="diffchange diffchange-inline">and the fact that </ins>degrees must be whole numbers<ins class="diffchange diffchange-inline">(<math>0</math> and the positive numbers)</ins>, <ins class="diffchange diffchange-inline">the </ins><math>deg<ins class="diffchange diffchange-inline"></math> <math></ins>r=0</math><ins class="diffchange diffchange-inline">, and so </ins>to speak, <math>r(x)</math> is a constant<ins class="diffchange diffchange-inline">, which we will </ins>denote <ins class="diffchange diffchange-inline">as </ins><math>b</math>.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Knowing this, we can write</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Knowing this, we can write</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>f(x)=d(x)\cdot<del class="diffchange diffchange-inline">\</del>q(x)+b</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>f(x)=d(x)\cdot<ins class="diffchange diffchange-inline">{</ins>q(x)<ins class="diffchange diffchange-inline">}</ins>+b</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>f(x)=(x-a)\cdot<del class="diffchange diffchange-inline">\</del>q(x)+b</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>f(x)=(x-a)\cdot<ins class="diffchange diffchange-inline">{</ins>q(x)<ins class="diffchange diffchange-inline">}</ins>+b</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>f(a)=(a-a)\cdot<del class="diffchange diffchange-inline">\</del>q(a)+b</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>f(a)=(a-a)\cdot<ins class="diffchange diffchange-inline">{</ins>q(a)<ins class="diffchange diffchange-inline">}</ins>+b</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>f(a)=b</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>f(a)=b</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>We have hereby proven when the quantity <math>x-a</math> is divided into a polynomial <math>f(x)</math> of any degree, the value of <math>f(a)=b</math>, where b is the remainder. The remainder must be a constant because <math>deg r=0</math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>We have hereby proven when the quantity <math>x-a</math> is divided into a polynomial <math>f(x)</math> of any degree, the value of <math>f(a)=b</math>, where b is the remainder. The remainder must be a constant because <math>deg<ins class="diffchange diffchange-inline"></math> <math></ins>r=0</math>.</div></td></tr>
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Zhang2018
https://artofproblemsolving.com/wiki/index.php?title=Proof_of_the_Polynomial_Remainder_Theorem&diff=105546&oldid=prev
Zhang2018: Synopsis: Written below is a brief description of the polynomial remainder theorem. The theorem has a wide range of application in subjects spanning from Algebra to
2019-04-30T21:59:46Z
<p>Synopsis: Written below is a brief description of the polynomial remainder theorem. The theorem has a wide range of application in subjects spanning from Algebra to</p>
<p><b>New page</b></p><div>The remainder theorem states when a polynomial denoted as <math>f(x)</math> is divided by <math>x-a</math> for some value of <math>x</math>, whether real or unreal, the remainder of <math>\frac{f(x)}{x-a}=f(a)</math> Written below is the proof of the polynomial remainder theorem.<br />
<br />
All polynomials can be written in the form <math>f(x)=d(x)\cdot\q(x)+r(x)</math>, where <math>d(x)</math> is the divisor of the function/polynomial <math>f(x)</math>, <math>q(x)</math> is the quotient. amd <math>r(x)</math> is the remainder. Because the <math>deg r=0</math> or <math>deg r\less\deg d</math> and the <math>deg d=1</math>, degrees must be whole numbers, and so <math>deg r=0</math>. So to speak, <math>r(x)</math> is a constant. We denote this constant <math>b</math>.<br />
<br />
Knowing this, we can write<br />
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<math>f(x)=d(x)\cdot\q(x)+b</math><br />
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<math>f(x)=(x-a)\cdot\q(x)+b</math><br />
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<math>f(a)=(a-a)\cdot\q(a)+b</math><br />
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<math>f(a)=b</math><br />
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We have hereby proven when the quantity <math>x-a</math> is divided into a polynomial <math>f(x)</math> of any degree, the value of <math>f(a)=b</math>, where b is the remainder. The remainder must be a constant because <math>deg r=0</math>.</div>
Zhang2018