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<!DOCTYPE html>
<html data-require="math expressions">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Partial fraction expansion</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
<style type="text/css">
#solutionarea .soln-bot input[type="text"] {
margin-left: 3px;
}
#solutionarea td {
text-align: center;
vertical-align: middle;
padding: 0px;
}
#solutionarea .soln-top {
padding-bottom: 1px;
}
#solutionarea .soln-middle {
padding: 0px 5px;
}
#solutionarea .soln-bot {
padding-top: 1px;
border-top: 1px solid black;
}
</style>
</head>
<body>
<div class="exercise">
<div class="problems">
<div>
<div class="vars">
<div data-ensure="A + B !== 0 &amp;&amp;
A*D + B*C !== 0 &amp;&amp;
C !== D">
<var id="A">randRangeNonZero(-5, 5)</var>
<var id="B">randRangeNonZero(-5, 5)</var>
<var id="C">randRangeNonZero(-5, 5)</var>
<var id="D">randRangeNonZero(-5, 5)</var>
</div>
<var id="E">A + B</var>
<var id="F">-A * D - B * C</var>
<var id="NUMER">expr(["+", ["*", E, "x"], F])</var>
<var id="DENOM">expr(C === -D ?
["+", "x^2", C*D] :
["+", "x^2", ["*", -C - D, "x"], C*D]
)</var>
<var id="ADENOM">expr(["+", "x", -C])</var>
<var id="BDENOM">expr(["+", "x", -D])</var>
</div>
<p class="question">
Expand
<code>\dfrac{<var>NUMER</var>}{<var>DENOM</var>}</code>
using partial fractions.
</p>
<div class="solution" data-type="set">
<div class="set-sol" data-type="multiple">
<span class="sol" data-forms="integer"><var>A</var></span>
<span class="sol" data-forms="integer"><var>B</var></span>
<span class="sol" data-forms="integer"><var>C</var></span>
<span class="sol" data-forms="integer"><var>D</var></span>
</div>
<div class="set-sol" data-type="multiple">
<span class="sol" data-forms="integer"><var>B</var></span>
<span class="sol" data-forms="integer"><var>A</var></span>
<span class="sol" data-forms="integer"><var>D</var></span>
<span class="sol" data-forms="integer"><var>C</var></span>
</div>
<div class="input-format">
<div class="entry" data-type="multiple">
<table>
<tbody><tr>
<td class="soln-top">
<span class="sol short35">a</span>
</td>
<td class="soln-middle" rowspan="2">
<code>+</code>
</td>
<td class="soln-top">
<span class="sol short35">a</span>
</td>
</tr>
<tr>
<td class="soln-bot">
<code>x-</code><span class="sol short35">a</span>
</td>
<td class="soln-bot">
<code>x-</code><span class="sol short35">a</span>
</td>
</tr>
</tbody></table>
</div>
</div>
</div>
<div class="hints">
<p>
First, factor the denominator to find the denominators of
the two fractions we will split our fraction into.
</p>
<p>
<code>
<var>DENOM</var> =
(<var>ADENOM</var>)(<var>BDENOM</var>)
</code>
</p>
<p>
Because the original denominator can be factored into these
two parts, we can write out our original fraction as the
sum of two fractions whose denominators are the two factors
we just found.
</p>
<p>
<code>
\dfrac{<var>NUMER</var>}{
(<var>ADENOM</var>)(<var>BDENOM</var>)
} =
\dfrac{?}{<var>ADENOM</var>} +
\dfrac{?}{<var>BDENOM</var>}
</code>
</p>
<div>
<p>
Now, we replace the numerators with polynomials of a
degree one less than the degree of the polynomial in
the denominator.
</p>
<p>
In our case, both of the denominators have a degree of
<code>1</code>, so we replace our numerators with
polynomials of degree <code>0</code>, or constants. We
will use the constants <code>A</code> and
<code>B</code>.
</p>
</div>
<p>
<code>
\dfrac{<var>NUMER</var>}{
(<var>ADENOM</var>)(<var>BDENOM</var>)
} =
\dfrac{A}{<var>ADENOM</var>} +
\dfrac{B}{<var>BDENOM</var>}
</code>
</p>
<p>
Now, to get rid of the fractions, we multiply by the common
denominator,
<code>(<var>ADENOM</var>)(<var>BDENOM</var>)</code>.
</p>
<p>
<code>
<var>NUMER</var> =
A(<var>BDENOM</var>) + B(<var>ADENOM</var>)
</code>
</p>
<p>
Now we can solve for <code>A</code> and <code>B</code>. An
easy way to do this is to try to choose values for
<code>x</code> that will get one of <code>A</code> or
<code>B</code> to cancel out, and then solve for the other
one.
</p>
<div>
<p>
Let's try to cancel out <code>B</code>. We see that if
we plug in <code><var>C</var></code> for
<code>x</code>, the term with <code>B</code> cancels
out, and we are left with:
</p>
<p>
<code>
<var>expr(["+", E * C, F])</var> =
A(<var>expr(["+", C, -D])</var>)
</code>
</p>
</div>
<p>
<code>
<var>E * C + F</var> =
<var>expr(["*", C - D, "A"])</var>
</code>
</p>
<p>
<code>A=<var>A</var></code>
</p>
<div>
<p>
We can do the same thing to solve for <code>B</code>,
but instead plugging in <code><var>D</var></code> for
<code>x</code>:
</p>
<p>
<code>
<var>expr(["+", E * D, F])</var> =
B(<var>expr(["+", D, -C])</var>)
</code>
</p>
</div>
<p>
<code>
<var>E * D + F</var> =
<var>expr(["*", D - C, "B"])</var>
</code>
</p>
<p>
<code>B=<var>B</var></code>
</p>
<div>
<p>
Now, we plug back in to our fractions, and get:
</p>
<p>
<code>
\dfrac{<var>NUMER</var>}{<var>DENOM</var>} =
\dfrac{<var>A</var>}{<var>ADENOM</var>} +
\dfrac{<var>B</var>}{<var>BDENOM</var>}
</code>
</p>
</div>
</div>
</div>
</div>
</div>
</body>
</html>
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