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<title>Dividing polynomials by binomials 1</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
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<body>
<div class="exercise">
<div class="problems">
<div id="perfect-square">
<div class="vars">
<var id="X">randVar()</var>
<var id="Y" data-ensure="X !== Y">randVar()</var>
<var id="A">randRangeExclude(-9, 9, [-1, 0, 1])</var>
<var id="CONSTANT">-A * A</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>X</var>^2 + <var>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" data-same-form="">
<var>X</var> - <var>A</var>
</div>
<span><code>\space <var>X</var> \neq \space </code>
<span class="sol" data-type="number"><var>-A</var></span>
</div>
<div class="hints" data-apply="appendContents">
<p>
The numerator is in the form <code>\pink{a^2} - \blue{b^2}</code>,
which is a difference of two squares so we can factor it as
<code>(\pink{a} + \blue{b})(\pink{a} - \blue{b})</code>.
</p>
<div>
<p><code>\qquad \pink{a = <var>X</var>}</code></p>
<p><code>\qquad \blue{b = \sqrt{<var>A * A</var>} = <var>abs(A)</var>}</code></p>
</div>
<div>
<p>So we can rewrite the expression as:</p>
<p><code><var>Y</var> =
<span data-if="A > 0">\dfrac{(\pink{<var>X</var>} + \blue{<var>A</var>})(\pink{<var>X</var>} \blue{<var>-A</var>})}</span>
<span data-else="">\dfrac{(\pink{<var>X</var>} \blue{<var>A</var>})(\pink{<var>X</var>} + \blue{<var>-A</var>})}</span>
{<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> =
<span data-if="A > 0">
\dfrac{\cancel{(\pink{<var>X</var>} + \blue{<var>A</var>})}(\pink{<var>X</var>} \blue{<var>-A</var>})}
</span><span data-else="">
\dfrac{\cancel{(\pink{<var>X</var>} \blue{<var>A</var>})}(\pink{<var>X</var>} + \blue{<var>-A</var>})}
</span>
{\cancel{<var>X</var> + <var>A</var>}} = <var>X</var> + <var>-A</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>X</var> - <var>A</var></code><br>
<code>\qquad <var>X</var> \neq <var>-A</var></code>
</p>
</div>
</div>
</div>
<div id="not-perfect-square">
<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="CONSTANT">A * B</var>
<var id="LINEAR">-A - 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>X</var>^2 + <var>plus(LINEAR + X)</var> + <var>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" data-simplify="">
<var>X</var> - <var>B</var>
</div>
<span><code>\space <var>X</var> \neq \space </code>
<span class="sol" data-type="number"><var>A</var></span>
</div>
<div class="hints" data-apply="appendContents">
<p><code>
<var>X</var>^2 + <var>plus(LINEAR + X)</var> + <var>CONSTANT</var> = (<var>X</var> - <var>A</var>)(<var>X</var> - <var>B</var>)
</code></p>
<div>
<p>So we can rewrite the expression as:</p>
<p><code><var>Y</var> =
\dfrac{(<var>X</var> + <var>-A</var>)(<var>X</var> + <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{\cancel{(<var>X</var> - <var>A</var>)}(<var>X</var> - <var>B</var>)}{\cancel{<var>X</var> - <var>A</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>X</var> - <var>B</var></code><br>
<code>\qquad <var>X</var> \neq <var>A</var></code>
</p>
</div>
</div>
</div>
</div>
<div class="hints">
<p>First factor the polynomial in the numerator.</p>
</div>
</div>
</body>
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