-
Notifications
You must be signed in to change notification settings - Fork 1
/
af2d04f5-4652-41a6-bb28-502ea390e0ce.html
114 lines (114 loc) · 5.16 KB
/
af2d04f5-4652-41a6-bb28-502ea390e0ce.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
<h4>Factoring the GCF from Polynomials</h4>
<p>It is sometimes useful to represent a number as a product of factors, for example, 12 as \(2 \cdot 6\) or \(3 \cdot
4\). In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such
as \(3x^2+15x\), and end with its factors, \(3x(x + 5)\). To do this, we apply the Distributive Property “in
reverse.” We state the Distributive Property here just as you saw it in earlier chapters and “in
reverse.”</p>
<div class="os-raise-graybox">
<p class="os-raise-text-bold"> DISTRIBUTIVE PROPERTY </p>
<hr>
<p>If \(a\), \(b\), and \(c\) are real numbers, then</p>
<p>\(\;\;\;\;\;\;\;\;\;\;\;a(b + c) = ab + ac\) and \(ab + ac = a(b + c)\)</p>
<p>The form on the left is used to multiply. The form on the right is used to factor.</p>
</div>
<br>
<p>So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write
the polynomial as a product!</p>
<p class="os-raise-text-bold">Example 1</p>
<p>Factor: \(8m^3-12m^2n+20mn^2\).</p>
<p class="os-raise-text-bold">How to factor the greatest common factor from a polynomial:</p>
<p><strong>Step 1 - </strong>Find the GCF of all the terms of the polynomial. For this problem, find the GCF of \(8m^3\), \(12m^2n\), and
\(20mn^2\).</p>
<p><img height="135" src="https://k12.openstax.org/contents/raise/resources/2af0ae8a08f2a69dfa71f2cf930dc371ca3a0761"
width="300"></p>
<p><strong>Step 2 -</strong> Rewrite each term as a product using the GCF. In this case, rewrite \(8m^3\), \(12m^2n\), and \(20mn^2\) as products
of their GCF, \(4m\).</p>
<ul>
<li> \(8m^3 = {\style{color:red}4}
{\style{color:red}m} \cdot 2m^2\) </li>
<li> \(12m^2n = {\style{color:red}4}
{\style{color:red}m} \cdot 3mn\) </li>
<li> \(20mn^2 = {\style{color:red}4}
{\style{color:red}m} \cdot 5n^2\) </li>
</ul>
<p>\(8m^3-12m^2n+20mn^2\)<br>
\({\style{color:red}4}
{\style{color:red}m} \cdot 2m^2 - {\style{color:red}4}
{\style{color:red}m} \cdot 3mn + {\style{color:red}4}
{\style{color:red}m} \cdot 5n^2\)</p>
<p><strong>Step 3 - </strong>Use the “reverse” Distributive Property to factor the expression.</p>
<p>\(4m(2m^2-3mn+5n^2)\)</p>
<p><strong>Step 4 -</strong> Check by multiplying the factors.</p>
<p>\(4m(2m^2-3mn+5n^2)\)<br>
\(4m \cdot 2m^2 - 4m \cdot 3mn + 4m \cdot 5n^2\)<br>
\(8m^3-12m^2n+20mn^2\)</p>
<p>When the leading coefficient is negative, we factor the negative out as part of the GCF.</p>
<p class="os-raise-text-bold">Example 2</p>
<p>Factor: \(−4a^3 + 36a^2 − 8a\).</p>
<p>The leading coefficient is negative, so the GCF will be negative.</p>
<p><strong>Step 1 -</strong> Rewrite each term using the GCF, \(−4a\).</p>
<p> \(
{\style{color:red}-}
{\style{color:red}4}
{\style{color:red}a} \cdot a^2 −
{\style{color:red}(}
{\style{color:red}-}
{\style{color:red}4}
{\style{color:red}a}
{\style{color:red})} \cdot 9a +
{\style{color:red}(}{\style{color:red}-}
{\style{color:red}4}
{\style{color:red}a}
{\style{color:red})} \cdot 2\) </p>
<p><strong>Step 2 -</strong> Factor the GCF.</p>
<p> \(−4a(a^2 − 9a + 2)\)</p>
<p><strong>Step 3 -</strong> Check.</p>
<p> \(−4a(a^2 − 9a + 2)\)<br>
\(−4a \cdot a^2 − (−4a) \cdot 9a + (−4a) \cdot 2\)<br>
\(−4a^3 + 36a^2 − 8a\)</p>
<p>So far, our greatest common factors have been monomials. In the next example, the greatest common factor is a
binomial.</p>
<p class="os-raise-text-bold">Example 3</p>
<p>Factor: \(3y
{\style{color:red}(}
{\style{color:red}y} \;{\style{color:red}+}\; {\style{color:red}7}
{\style{color:red})}
− 4{\style{color:red}(}
{\style{color:red}y}\; {\style{color:red}+} \;{\style{color:red}7}
{\style{color:red})}\).</p>
<p>The GCF is the binomial \(y + 7\).</p>
<p>\(3y(y + 7) − 4(y + 7)\).</p>
<p><strong>Step 1 -</strong> Factor the GCF, \((y + 7)\).</p>
<p> \((y + 7)(3y − 4)\)</p>
<p><strong>Step 2 -</strong> Check on your own by multiplying.</p>
<br>
<h4>Try It: Factoring the GCF from Polynomials</h4>
<p>Factor.</p>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<li> \(9xy^2 + 6x^2y^2 + 21y^3\) </li>
</ol>
<ol class="os-raise-noindent" start="2">
<li> \(−4b^3 + 16b^2 − 8b\) </li>
</ol>
<ol class="os-raise-noindent" start="3">
<li> \(4m(m + 3) − 7(m + 3)\) </li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answers. Then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Here is how to factor the GCF from polynomials.</p>
<ol class="os-raise-noindent">
<li> \(3y^2(3x + 2x^2 + 7y)\) </li>
</ol>
<ol class="os-raise-noindent" start="2">
<li> \(−4b(b^2 − 4b + 2)\) </li>
</ol>
<ol class="os-raise-noindent" start="3">
<li> \((m + 3)(4m − 7)\) </li>
</ol>
</div>