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factoring_polynomials_by_grouping_1.html
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factoring_polynomials_by_grouping_1.html
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<!DOCTYPE html>
<html data-require="math math-format graphie">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Factoring polynomials by grouping 1</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div>
<div class="vars" data-ensure="B !== 0 && getGCD(A, E) === 1">
<var id="A" data-ensure="abs(A) !== 1">randRangeNonZero(-9, 9)</var>
<var id="E">randRangeNonZero(-9, 9)</var>
<var id="F">randRangeNonZero(-5, 5)</var>
<var id="B">E + A * F</var>
<var id="C">E * F</var>
<var id="AX_PLUS_E">"\\(\\s*" + (A < 0 ? "[-\u2212]" : "") + abs(A) + "[xX]\\s*" + (E < 0 ? "[-\u2212]" : "\\+") + "\\s*" + abs(E) + "\\s*\\)"</var>
<var id="E_PLUS_AX">"\\(\\s*" + (E < 0 ? "[-\u2212]" : "") + "\\s*" + abs(E) + (A < 0 ? "[-\u2212]" : "") + abs(A) + "[xX]\\s*" + "\\s*\\)"</var>
<var id="X_PLUS_F">"\\(\\s*[xX]\\s*" + (F < 0 ? "[-\u2212]" : "\\+") + "\\s*" + abs(F) + "\\s*\\)"</var>
<var id="F_PLUS_X">"\\(\\s*" + (F < 0 ? "[-\u2212]" : "") + "\\s*" + abs(F) + "\\s*\\+\\s*[xX]\\s*\\)"</var>
<var id="NEG_AX_MINUS_E">"\\(\\s*" + (A > 0 ? "[-\u2212]" : "") + abs(A) + "[xX]\\s*" + (E > 0 ? "[-\u2212]" : "\\+") + "\\s*" + abs(E) + "\\s*\\)"</var>
<var id="NEG_E_MINUS_AX">"\\(\\s*" + (E > 0 ? "[-\u2212]" : "") + "\\s*" + abs(E) + "\\s*" + (A > 0 ? "[-\u2212]" : "\\+") + abs(A) + "\\s*[xX]\\s*" + "\\s*\\)"</var>
<var id="NEG_X_MINUS_F">"\\(\\s*[-\u2212]\\s*[xX]\\s*" + (F > 0 ? "[-\u2212]" : "\\+") + "\\s*" + abs(F) + "\\s*\\)"</var>
<var id="NEG_F_MINUS_X">"\\(\\s*" + (F > 0 ? "[-\u2212]" : "") + "\\s*" + abs(F) + "\\s*[-\u2212]\\s*[xX]\\s*\\)"</var>
</div>
<p class="problem">Factor the following expression:</p>
<p class="question">
<span id="question-a"><code><var>A</var></code></span><code>x^2<span data-if="B > 0">+</span></code>
<span id="question-b"><code><var>B</var></code></span><code>x<span data-if="C > 0">+</span></code>
<span id="question-c"><code><var>C</var></code></span>
</p>
<div class="solution" data-type="set">
<!-- TODO(eater): Add a magical helper thing that assembles all these permutations automatically -->
<div class="set-sol" data-type="regex">^\s*<var>AX_PLUS_E</var>\s*<var>X_PLUS_F</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>AX_PLUS_E</var>\s*<var>F_PLUS_X</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>E_PLUS_AX</var>\s*<var>X_PLUS_F</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>E_PLUS_AX</var>\s*<var>F_PLUS_X</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>X_PLUS_F</var>\s*<var>AX_PLUS_E</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>F_PLUS_X</var>\s*<var>AX_PLUS_E</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>X_PLUS_F</var>\s*<var>E_PLUS_AX</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>F_PLUS_X</var>\s*<var>E_PLUS_AX</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>NEG_AX_MINUS_E</var>\s*<var>NEG_X_MINUS_F</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>NEG_AX_MINUS_E</var>\s*<var>NEG_F_MINUS_X</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>NEG_E_MINUS_AX</var>\s*<var>NEG_X_MINUS_F</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>NEG_E_MINUS_AX</var>\s*<var>NEG_F_MINUS_X</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>NEG_X_MINUS_F</var>\s*<var>NEG_AX_MINUS_E</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>NEG_F_MINUS_X</var>\s*<var>NEG_AX_MINUS_E</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>NEG_X_MINUS_F</var>\s*<var>NEG_E_MINUS_AX</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>NEG_F_MINUS_X</var>\s*<var>NEG_E_MINUS_AX</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>NEG_AX_MINUS_E</var>\s*<var>X_PLUS_F</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>NEG_AX_MINUS_E</var>\s*<var>F_PLUS_X</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>NEG_E_MINUS_AX</var>\s*<var>X_PLUS_F</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>NEG_E_MINUS_AX</var>\s*<var>F_PLUS_X</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>NEG_X_MINUS_F</var>\s*<var>AX_PLUS_E</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>NEG_F_MINUS_X</var>\s*<var>AX_PLUS_E</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>NEG_X_MINUS_F</var>\s*<var>E_PLUS_AX</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>NEG_F_MINUS_X</var>\s*<var>E_PLUS_AX</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>X_PLUS_F</var>\s*<var>NEG_AX_MINUS_E</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>F_PLUS_X</var>\s*<var>NEG_AX_MINUS_E</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>X_PLUS_F</var>\s*<var>NEG_E_MINUS_AX</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>F_PLUS_X</var>\s*<var>NEG_E_MINUS_AX</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>AX_PLUS_E</var>\s*<var>NEG_X_MINUS_F</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>AX_PLUS_E</var>\s*<var>NEG_F_MINUS_X</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>E_PLUS_AX</var>\s*<var>NEG_X_MINUS_F</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*[-\u2212]\s*<var>E_PLUS_AX</var>\s*<var>NEG_F_MINUS_X</var>\s*$</div>
<div class="input-format"><span class="entry"></span></div>
<div class="example">a factored expression, like <b>(2x+1)(x+2)</b></div>
</div>
<div class="hints">
<div>
<p>
This expression is in the form <code>\blue{A}x^2 + \green{B}x + \pink{C}</code>.
You can factor it by grouping.
</p>
<div class="graphie" style="outline: none;">
$("#question-a").addClass("hint_blue");
$("#question-b").addClass("hint_green");
$("#question-c").addClass("hint_pink");
</div>
</div>
<div>
<p>
First, find two values, <code class="hint_purple">a</code> and
<code class="hint_purple">b</code>, so:
</p>
<code>
\qquad \begin{eqnarray}
\purple{ab} &=& \blue{A}\pink{C} \\
\purple{a} + \purple{b} &=& \green{B}
\end{eqnarray}
</code>
</div>
<div>
<p>In this case:</p>
<code>
\qquad \begin{eqnarray}
\purple{ab} &=&
\blue{(<var>A</var>)}\pink{(<var>C</var>)} &=& <var>A * C</var> \\
\purple{a} + \purple{b} &=& &=&
\green{<var>B</var>}
\end{eqnarray}
</code>
</div>
<p>
In order to find <code>\purple{a}</code> and <code>\purple{b}</code>, list out the factors of
<code><var>A * C</var></code> and add them together.
<span data-if="A * C < 0">
Remember, since <code><var>A * C</var></code> is negative, one of the factors must be
negative.
</span>
The factors that add up to <code>\green{<var>B</var>}</code> will be your
<code>\purple{a}</code> and <code>\purple{b}</code>.
</p>
<div>
<p>
When <code>\purple{a}</code> is <code>\purple{<var>E</var>}</code> and
<code>\purple{b}</code> is <code>\purple{<var>A * F</var>}</code>:
</p>
<code>
\qquad \begin{eqnarray}
\purple{ab} &=& (\purple{<var>E</var>})(\purple{<var>A * F</var>})
&=& <var>E * A * F</var> \\
\purple{a} + \purple{b} &=& \purple{<var>E</var>} + \purple{<var>A * F</var>}
&=& <var>E + A * F</var>
\end{eqnarray}
</code>
</div>
<div>
<p>
Next, rewrite the expression as <code>\blue{A}x^2 + \purple{a}x + \purple{b}x + \pink{C}</code>:
</p>
<code>
\qquad \blue{<var>A</var>}x^2
<span data-if="E > 0">+</span>\purple{<var>E</var>}x
<span data-if="A * F > 0">+</span>\purple{<var>A * F</var>}x
<span data-if="C > 0">+</span>\pink{<var>C</var>}
</code>
</div>
<div>
<p>
Group the terms so that there is a common factor in each group:
</p>
<code>
\qquad (\blue{<var>A</var>}x^2 <span data-if="E > 0">+</span>\purple{<var>E</var>}x)
+ (\purple{<var>A * F</var>}x <span data-if="C > 0">+</span>\pink{<var>C</var>})
</code>
</div>
<div>
<p>
Factor out the common factors:
</p>
<code>
\qquad x(<var>A</var>x + <var>E</var>) + <var>F</var>(<var>A</var>x + <var>E</var>)
</code>
</div>
<p>
Notice how <code>(<var>A</var>x + <var>E</var>)</code> has become a common factor.
Factor this out to find the answer.
</p>
<p><code>(<var>A</var>x + <var>E</var>)(x + <var>F</var>)</code></p>
</div>
</div>
</div>
</div>
</body>
</html>