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<title>Remainder Theorem and Factor Theorem</title>
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<h1 align="center">Remainder Theorem<br />
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and Factor Theorem</h1>
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<p align="center"><i>Or: how to avoid Polynomial Long Division when finding factors</i></p>
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<p>Do you remember doing division in Arithmetic?</p>
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<p align="center" class="larger"><img src="../numbers/images/remainder-7-2.svg" alt="7/2=3 remainder 1" /></p>
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<p align="center" class="larger"><i> "7 divided by 2 equals <b>3</b> with a <b>remainder of 1</b>"</i></p>
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<p>Each part of the division has names:</p>
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<p align="center"><span class="larger"><img src="../numbers/images/division-7d2.svg" alt="dividend/divisor=quotient with remainder" /></span></p>
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<p>Which can be <b>rewritten</b> as a sum like this:</p>
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<p align="center"><img src="../numbers/images/division-multiply.svg" alt="7 = 2 times 3 + 1" /></p>
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<h2>Polynomials</h2>
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<p>Well, we can also <a href="polynomials-division-long.html">divide polynomials</a>.</p>
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<p align="center"><span class="large">f(x) ÷ d(x) = q(x) with a remainder of r(x)</span></p>
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<p>But it is better to write it as a sum like this: </p>
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<p align="center" class="large"><img src="images/polynomial-division-names.svg" alt="f(x) = d(x) times q(x) + r(x)" /></p>
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<p>Like in this example using <a href="polynomials-division-long.html">Polynomial Long Division</a>:</p>
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<div class="example">
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<h3>Example: 2x<sup>2</sup>−5x−1 divided by x−3</h3>
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<ul>
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<li>f(x) is 2x<sup>2</sup>−5x−1</li>
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<li>d(x) is x−3</li>
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</ul>
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<p align="center"><img src="images/polynomial-long-division2.gif" alt="polynomial long division 2x^/2-5x-1 / x-3 = 2x+1 R 2" width="274" height="160" /></p>
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<p>After dividing we get the answer <span class="large">2x+1</span>, but there is a remainder of <span class="large">2</span>.</p>
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<ul>
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<li>q(x) is 2x+1</li>
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<li>r(x) is 2</li>
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</ul>
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<p>In the style <span class="large">f(x) = d(x)·q(x) + r(x)</span> we can write:</p>
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<p align="center" class="large">2x<sup>2</sup>−5x−1 = (x−3)(2x+1) + 2</p>
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</div>
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<p>But you need to know one more thing:</p>
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<div class="def">
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<p>The <a href="degree-expression.html">degree</a> of r(x) is always less than d(x)</p>
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</div>
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<p>Say we divide by a polynomial of <b>degree 1</b> (such as "x−3") the remainder will have <b>degree 0</b> (in other words a constant, like "4").</p>
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<p>We will use that idea in the
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"Remainder Theorem":</p>
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<h2>The Remainder Theorem</h2>
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<p>When we divide <span class="large">f(x)</span> by the simple polynomial <span class="large">x−c</span> we get:</p>
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<p align="center"> <span class="large">f(x) = (x−c)·q(x) + r(x)</span></p>
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<p><span class="large">x−c</span> is <b>degree 1</b>, so <span class="large">r(x)</span> must have <b>degree 0</b>, so it is just some constant <span class="large">r</span> <i>:</i></p>
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<p align="center"><span class="large">f(x) = (x−c)·q(x) + <span class="hi">r</span></span></p>
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<p>Now see what happens when we have <span class="large">x equal to c</span>:</p>
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<div class="tbl">
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<div class="row"><span class="left">f(c) =</span><span class="right">(c−c)·q(c) + r</span></div>
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<div class="row"><span class="left">f(c) =</span><span class="right">(0)·q(c) + r</span></div>
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<div class="row"><span class="left">f(c) =</span><span class="right">r</span></div>
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</div>
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<p>So we get this:</p>
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<div class="def">
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<p><b>The Remainder Theorem:</b></p>
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<p align="center">When we divide a polynomial <span class="large">f(x)</span> by <span class="large">x−c</span> the remainder is <span class="large">f(c)</span></p>
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</div>
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<p>So to find the remainder after dividing by <span class="large">x-c</span> we don't need to do any division: </p>
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<p align="center" class="larger">Just calculate <span class="large">f(c)</span>.</p>
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<p>Let us see that in practice:</p>
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<div class="example">
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<h3>Example: The remainder after 2x<sup>2</sup>−5x−1 is divided by x−3</h3>
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<p>(Our example from above)</p>
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<p>We don't need to divide by <b>(x−3)</b> ... just calculate <b>f(3)</b>:</p>
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<p align="center" class="larger">2(3)<sup>2</sup>−5(3)−1 = 2x9−5x3−1 <br>
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= 18−15−1 <br>
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= <b>2</b></p>
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<p>And that is the remainder we got from our calculations above.</p>
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<p>We didn't need to do Long Division at all!</p>
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</div>
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<div class="example">
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<h3>Example: The remainder after 2x<sup>2</sup>−5x−1 is divided by x−5</h3>
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<p>Same example as above but this time we divide by "x−5"</p>
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<p>"c" is 5, so let us check f(5):</p>
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<p align="center" class="larger">2(5)<sup>2</sup>−5(5)−1 = 2x25−5x5−1 <br>
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= 50−25−1 <br>
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= <b>24</b></p>
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<p>The remainder is <b>24</b></p>
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<p>Once again ... We didn't need to do Long Division to find that. </p>
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</div>
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<h2>The Factor Theorem</h2>
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<p>Now ...</p>
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<div class="center80">
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<p>What if we calculate <b>f(c)</b> and it is <b>0</b>?</p>
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<p align="center">... that means the <b>remainder is 0</b>, and ...</p>
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<p align="right">... <b>(x−c) must be a factor</b> of the polynomial!</p>
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</div>
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<p>We see this when dividing whole numbers. For example 60 ÷ 20 = 3 with no remainder. So 20 must be a factor of 60.
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</p>
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<div class="example">
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<h3>Example: x<sup>2</sup>−3x−4</h3>
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<p align="center" class="larger">f(4) = (4)<sup>2</sup>−3(4)−4 = 16−12−4 = 0</p>
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<p>so (x−4) must be a factor of x<sup>2</sup>−3x−4</p>
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</div>
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<p>And so we have:</p>
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<div class="def">
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<p><b>The Factor Theorem:</b></p>
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<p align="center">When <span class="large">f(c)=0</span> then <span class="large">x−c</span> is a factor of <span class="large">f(x)</span></p>
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<p><i>And the other way around, too:</i></p>
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<p align="center">When <span class="large">x−c</span> is a factor of <span class="large">f(x)</span> then <span class="large">f(c)=0</span></p>
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</div>
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<h2>Why Is This Useful?</h2>
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<p>Knowing that <span class="large">x−c</span> is a factor is the same as knowing that <span class="large">c </span>is a root (and vice versa).</p>
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<div class="center80">
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<p>The <b>factor "x−c"</b> and the <b>root "c"</b> are the same thing</p>
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<p>Know one and we know the other</p>
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</div>
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<p>For one thing, it means that we can quickly check if (x−c) is a factor of the polynomial.</p>
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<div class="example">
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<h3>Example: Find the factors of 2x<sup>3</sup>−x<sup>2</sup>−7x+2</h3>
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<p>The polynomial is degree 3, and could be difficult to solve. So let us plot it first:</p>
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<p align="center"><img src="images/graph-2x3mx2m7xp2.gif" alt="graph of 2x^3-x^2-7x+2" width="248" height="134" /></p>
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<p>The curve crosses the x-axis at three points, and one of them <b>might be at 2</b>. We can check easily:</p>
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<p align="center"><b>f(2)</b> = 2(2)<sup>3</sup>−(2)<sup>2</sup>−7(2)+2 <br>
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= 16−4−14+2 <br>
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= <b>0</b></p>
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<p>Yes! <b>f(2)=0</b>, so we have found a root <b>and</b> a factor.</p>
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<div class="center80">
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<p align="center" class="large">So (x−2) must be a factor of 2x<sup>3</sup>−x<sup>2</sup>−7x+2</p>
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</div>
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<p> </p>
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<p>How about where it crosses near <b>−1.8</b>?</p>
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<p align="center"><b>f(−1.8)</b> = 2(−1.8)<sup>3</sup>−(−1.8)<sup>2</sup>−7(−1.8)+2 <br>
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= −11.664−3.24+12.6+2 <br>
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= <b>−0.304</b></p>
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<p>No, (x+1.8) is not a factor. We could try some other values near by and maybe get lucky. </p>
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<p>But at least we know <span class="large">(x−2)</span> is a factor, so let's use <a href="polynomials-division-long.html">Polynomial Long Division</a>:</p>
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<div class="mono">
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<span style="border-bottom: 1px solid black;">2x<sup>2</sup>+3x−1 </span><br>
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x−2)2x<sup>3</sup>− x<sup>2</sup>−7x+2<br>
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<span style="border-bottom: 1px solid black;">2x<sup>3</sup>−4x<sup>2</sup></span><br>
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3x<sup>2</sup>−7x<br>
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<span style="border-bottom: 1px solid black;">3x<sup>2</sup>−6x</span><br>
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−x+2<br>
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<span style="border-bottom: 1px solid black;">−x+2</span><br>
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0
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</div>
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<p>As expected the remainder is zero. </p>
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<p>Better still, we are left with the <a href="quadratic-equation.html">quadratic equation</a> <b>2x<sup>2</sup>+3x−1</b> which is easy to <a href="../quadratic-equation-solver.html">solve</a>.</p>
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<p>It's roots are −1.78... and 0.28..., so the final result is:</p>
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<p class="center large">2x<sup>3</sup>−x<sup>2</sup>−7x+2 = (x−2)(x+1.78...)(x−0.28...)</p>
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<p>We were able to solve a difficult polynomial.</p>
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</div>
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<h2>Summary</h2>
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<div class="dotpoint">
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<p><b>The Remainder Theorem:</b> </p>
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<ul>
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<li>When we divide a polynomial <span class="large">f(x)</span> by <span class="large">x−c</span> the remainder is <span class="large">f(c)</span></li>
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</ul>
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</div>
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<div class="dotpoint">
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<p><b>The Factor Theorem:</b> </p>
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<ul>
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<li>When <span class="large">f(c)=0</span> then <span class="large">x−c</span> is a factor of <span class="large">f(x)</span></li>
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<li>When <span class="large">x−c</span> is a factor of <span class="large">f(x)</span> then <span class="large">f(c)=0</span></li>
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</ul>
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</div><p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(482, 483, 4014, 4015, 484, 485, 4016, 1124, 1125, 4017);</script>
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<br />
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Challenging Questions:
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<a href="javascript:doQ(96)">1</a>
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<a href="javascript:doQ(227)">2</a>
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<a href="javascript:doQ(228)">3</a>
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<a href="javascript:doQ(229)">4</a>
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<a href="javascript:doQ(230)">5</a>
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<a href="javascript:doQ(231)">6</a>
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</div>
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<div class="related"><a href="polynomials-division-long.html">Polynomial Long Division</a> <a href="index.html">Algebra Index</a></div>
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