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<!DOCTYPE html>
<html data-require="math math-format">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Factoring polynomials with two variables</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">
<div data-ensure="A !== B && A !== -B">
<var id="A">randRangeNonZero( -10, 10 )</var>
<var id="B">randRangeNonZero( -10, 10 )</var>
</div>
<var id="CONSTANT">A * B</var>
<var id="CONSTANT_DISP">CONSTANT === 1 ? "" : CONSTANT === -1 ? "-" : CONSTANT</var>
<var id="LINEAR">-A - B</var>
<var id="LINEAR_DISP">LINEAR === 1 ? "" : LINEAR === -1 ? "-" : LINEAR</var>
</div>
<p class="problem">Factor the following expression:</p>
<p class="question"><code>
x^2 + <var>LINEAR_DISP</var>xy + <var>CONSTANT_DISP</var>y^2
</code></p>
<p class="solution" data-type="expression" data-same-form>(x-<var>A</var>y)(x-<var>B</var>y)</p>
<div class="hints">
<div>
<p>When we factor a polynomial of this form, we are basically reversing this process of multiplying linear expressions together:</p>
<p><code>\qquad
\begin{eqnarray}
(x + ay)(x + by)&amp;=&amp;xx &amp;+&amp; xby + ayx &amp;+&amp; ayby \\ \\
&amp;=&amp; x^2 &amp;+&amp; \green{(a+b)}xy &amp;+&amp; \blue{ab}y^2 \\
&amp;\hphantom{=}&amp; \hphantom{x^2} &amp;\hphantom{+}&amp; \hphantom{\green{<var>LINEAR</var>}xy} &amp;\hphantom{+}&amp; \hphantom{\blue{<var>CONSTANT</var>}y^2}
\end{eqnarray}
</code></p>
</div>
<div>
<p><code>\qquad
\begin{eqnarray}
\hphantom{(x + ay)(x + by)}&amp;\hphantom{=}&amp;\hphantom{xx} &amp;\hphantom{+}&amp; \hphantom{xby + ayx} &amp;\hphantom{+}&amp;\hphantom{ayby} \\
&amp;\hphantom{=}&amp; \hphantom{x^2} &amp;\hphantom{+}&amp;\hphantom{\green{(a+b)}xy}&amp;\hphantom{+}&amp;\hphantom{\blue{ab}y^2} \\
&amp;=&amp; x^2 &amp;+&amp; \green{<var>LINEAR</var>}xy &amp;+&amp; \blue{<var>CONSTANT</var>}y^2
\end{eqnarray}
</code></p>
<p>
The coefficient on the <code>xy</code> term is <code class="hint_green"><var>LINEAR</var></code>
and the coefficient on the <code>y^2</code> term is <code class="hint_blue"><var>CONSTANT</var></code>, so to reverse the steps above, we need to find two numbers
that <span class="hint_green">add up to <code><var>LINEAR</var></code></span> and <span class="hint_blue">multiply to
<code><var>CONSTANT</var></code></span>.
</p>
</div>
<div>
<p>You can start by trying to guess which factors of <code class="hint_blue"><var>CONSTANT</var></code> add up to
<span class="hint_green"><code><var>LINEAR</var></code></span>. In other words, you need to find the values for <code class="hint_pink">a</code> and
<code class="hint_pink">b</code> that meet the following conditions:</p>
<p><code>\qquad \pink{a} + \pink{b} = \green{<var>LINEAR</var>}</code></p>
<p><code>\qquad \pink{a} \times \pink{b} = \blue{<var>CONSTANT</var>}</code></p>
<p>If you're stuck, try listing out every single factor of <code class="hint_blue"><var>CONSTANT</var></code> and its opposite as
<code class="hint_pink">a</code> in these equations, and see if it gives a value for <code class="hint_pink">b</code>
that validates both conditions. For example, since <code><var>abs(A)</var></code> is a factor of <code class="hint_blue"><var>CONSTANT</var></code>,
try substituting <code><var>abs(A)</var></code> for <code class="hint_pink">a</code> as well as <code><var>-abs(A)</var></code>.</p>
</div>
<div>
<p>The two numbers <code class="hint_pink"><var>-A</var></code> and <code class="hint_pink"><var>-B</var></code> satisfy both conditions:</p>
<p><code>
\qquad \pink{<var>-A</var>} + \pink{<var>-B</var>} = \green{<var>LINEAR</var>}
</code></p>
<p><code>
\qquad \pink{<var>-A</var>} \times \pink{<var>-B</var>} = \blue{<var>CONSTANT</var>}
</code></p>
</div>
<p>
So we can factor the polynomial as <code>(<var>plus("x", -A + "y")</var>)(<var>plus("x", -B + "y")</var>)</code>.
</p>
</div>
</div>
</div>
</div>
</body>
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