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<!DOCTYPE html>
<html data-require="math math-format rational-expressions">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Dividing polynomials by binomials 2</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div>
<div class="vars">
<var id="X">randVar()</var>
<var id="Y" data-ensure="X !== Y">randVar()</var>
<var id="A">randRangeNonZero(-10, 10)</var>
<var id="B" data-ensure="abs(A) !== abs(B)">randRangeNonZero(-10, 10)</var>
<var id="C">randRangeExclude(-10, 10, [-1, 0, 1])</var>
<var id="CONSTANT">A * B</var>
<var id="LINEAR">-A - B</var>
<var id="SOLUTION">new RationalExpression([[C, X], C * -B])</var>
</div>
<p class="problem">Simplify the right-hand side of the equation below and state the condition under which the simplification is valid.</p>
<p class="question"><code><var>Y</var> =
\dfrac{<var>C</var><var>X</var>^2 + <var>C * LINEAR</var><var>X</var> + <var>C * CONSTANT</var>}{<var>X</var> - <var>A</var>}
</code></p>
<div class="solution" data-type="multiple">
<span><code><var>Y</var> = </code></span>
<div class="sol" data-type="expression">
<div class="set-sol" data-same-form=""><var>C</var><var>X</var> + <var>C * (-B)</var></div>
<div class="set-sol" data-same-form=""><var>C</var>(<var>X</var> + <var>(-B)</var>)</div>
</div>
<code><var>X</var> \neq </code> <span class="sol short50" data-type="number"><var>A</var></span>
</div>
<div class="hints">
<p>First factor the polynomial in the numerator.</p>
<div>
<p>We notice that all the terms in the numerator have a common factor of <code><var>C</var></code>, so we can rewrite the expression:</p>
<p><code>
<var>Y</var> =\dfrac{<var>C</var>(<var>X</var>^2 + <var>LINEAR</var><var>X</var> + <var>CONSTANT</var>)}{<var>X</var> - <var>A</var>}
</code></p>
</div>
<div>
<p>Then we factor the remaining polynomial:</p>
<p><code><var>X</var>^2
<var>LINEAR &gt; 0 ? "+" : ""</var> \green{<var>LINEAR</var>}<var>X</var>
<var>CONSTANT &gt; 0 ? "+" : ""</var> \blue{<var>CONSTANT</var>}
</code></p>
<p><code>\pink{<var>-A</var>} <var>B &lt; 0 ? "+" : ""</var> \pink{<var>-B</var>} = \green{<var>LINEAR</var>}</code></p>
<p><code>\pink{<var>-A</var>} \times \pink{<var>-B</var>} = \blue{<var>CONSTANT</var>}</code></p>
</div>
<p><code>
(<var>X</var> <var>A &lt; 0 ? "+" : ""</var> \pink{<var>-A</var>})
(<var>X</var> <var>B &lt; 0 ? "+" : ""</var> \pink{<var>-B</var>})
</code></p>
<div>
<p>This gives us a factored expression:</p>
<p><code><var>Y</var> =
\dfrac{<var>C</var>(<var>X</var> <var>A &lt; 0 ? "+" : ""</var> \pink{<var>-A</var>})
(<var>X</var> <var>B &lt; 0 ? "+" : ""</var> \pink{<var>-B</var>})}{<var>X</var> + <var>-A</var>}
</code></p>
</div>
<div>
<p>We can divide the numerator and denominator by <code>(<var>X</var> - <var>A</var>)</code>:</p>
<p><code><var>Y</var> =
\dfrac{<var>C</var>\cancel{(<var>X</var> <var>A &lt; 0 ? "+" : ""</var> \pink{<var>-A</var>})}
(<var>X</var> <var>B &lt; 0 ? "+" : ""</var> \pink{<var>-B</var>})}{\cancel{<var>X</var> - <var>A</var>}}
= <var>C</var>(<var>X</var> + <var>-B</var>)
</code></p>
</div>
<div>
<p>Because we divided by <code>(<var>X</var> - <var>A</var>)</code>,</p>
<p><code>\begin{align}
<var>X</var> - <var>A</var> &amp;\neq 0 \\
<var>X</var> &amp;\neq <var>A</var>
\end{align}
</code></p>
</div>
<div>
<p>Therefore,</p>
<p>
<code>\qquad <var>Y</var> = <var>C</var>(<var>X</var> - <var>B</var>)</code><br>
<code>\qquad <var>X</var> \neq <var>A</var></code>
</p>
</div>
</div>
</div>
</div>
</div>
</body>
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