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<!DOCTYPE html>
<html data-require="math">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Factor expressions by grouping</title>
<script src="../khan-exercise.js"></script>
<style type="text/css">
#answer_area input[type=text] {
width: 30px;
<div class="exercise">
<div class="problems">
<div id="factor-by-grouping">
<div class="vars">
<var id="a_start">randRange( 1, 10 )</var>
<var id="b_start">randRangeNonZero( -10, 10 )</var>
<var id="c_start">randRangeNonZero( -10, 10 )</var>
<var id="d_start">randRangeNonZero( -10, 10 )</var>
<var id="a">a_start / getGCD( a_start, b_start)</var>
<var id="b">b_start / getGCD( a_start, b_start)</var>
<var id="c">c_start / getGCD( c_start, d_start)</var>
<var id="d">d_start / getGCD( c_start, d_start)</var>
<var id="C11">a * c</var>
<var id="C10">a * d</var>
<var id="C01">b * c</var>
<var id="C00">b * d</var>
<p class="problem">Given the following polynomial, factor by grouping. In other words, if we put the polynomial below into the form <code>\color{#b22222}{(a r - b)}\color{#4169E1}{(c s - d)}</code>, what are the values of <code>\color{#b22222}{a}</code>, <code>\color{#b22222}{b}</code>, <code>\color{#4169E1}{c}</code>, and <code>\color{#4169E1}{d}</code>?</p>
<p><code><var>a * c</var> r s + <var>a * d</var> r + <var>b * c</var>s + <var>b * d</var></code> </p>
<div class="solution" data-type="set">
<div class="set-sol" data-type="multiple">
<span class="sol"><var>a</var></span>
<span class="sol"><var>b</var></span>
<span class="sol"><var>c</var></span>
<span class="sol"><var>d</var></span>
<div class="set-sol" data-type="multiple">
<span class="sol"><var>-a</var></span>
<span class="sol"><var>-b</var></span>
<span class="sol"><var>-c</var></span>
<span class="sol"><var>-d</var></span>
<div class="input-format">
<p class="entry" data-type="multiple">
<code>(</code><span class="sol"></span><code>\space r + \space</code><span class="sol"></span><code>)(</code><span class="sol"></span><code>\space s + \space</code><span class="sol"></span><code>)</code>
<p class="example">enter an integer for each coefficient</p>
<p class="example">pay attention to the sign of each number you enter to be sure the entire equation is correct</p>
<div class="hints">
<p>What is the largest factor we can pull out of both <code><var>a * c</var>rs</code> and <code><var>a * d</var>r</code>?</p>
<p>It's <code>\color{#b22222}{<var>a</var>r}</code>. What happens when we pull it out?</p>
<p>We get <code>\color{#b22222}{<var>a</var>r}\color{#4169E1}{(<var>c</var>s + <var>d</var>)} + <var>b * c</var>s + <var>b * d</var></code>.</p>
<p>Can we write <code><var>b * c</var>s + <var>b * d</var></code> as something times <code>\color{#4169E1}{(<var>c</var>s + <var>d</var>)}</code>?</p>
<p>It's <code>\color{#b22222}{<var>b</var>}</code> times <code>\color{#4169E1}{(<var>c</var>s + <var>d</var>)}</code>.</p>
<p>By the distributive property, <code>\color{#b22222}{<var>a</var>r}\color{#4169E1}{(<var>c</var>s + <var>d</var>)}+\color{#b22222}{<var>b</var>}\color{#4169E1}{(<var>c</var>s + <var>d</var>)}</code> = <code>\color{#b22222}{(<var>a</var>r + <var>b</var>)}\color{#4169E1}{(<var>c</var>s + <var>d</var>)}</code>.</p>
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