Linear Sentences in One Variable
Factoring Polynomials
Study this pattern for multiplying two binomials:
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Example 1: Factor 2 x2 − 5 x − 12.
Begin by writing two pairs of parentheses.
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For the first positions, find two factors whose product is 2 x2. For the last positions, find two factors whose product is −12. Below are the possibilities. With each possibility, the sum of outer and inner products is included.
Only possibility 11 will multiply out to produce the original polynomial. Therefore,
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Because many possibilities exist, some shortcuts are advisable:
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Shortcut 1: Be sure the GCF, if there is one, has been factored out.
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Shortcut 2: Try factors closest to one another first. For example, when considering factors of 12, try 3 and 4 before trying 6 and 2 and try 6 and 2 before trying 1 and 12.
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Shortcut 3: Avoid creating binomials that will have a GCF within them. This shortcut eliminates possibilities 1, 2, 5, 6, 7, 8, 9, and 10 (look at the underlined binomials; their terms each have some common factor), leaving only four possibilities to consider. Of the four remaining possibilities, 11 and 12 would be considered first using shortcut 2.
Example 2: Factor 8
x2 − 26
x + 20.
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For first factors, begin with 2
x and 2
x (closest factors). For last factors, begin with −5 and −2 (closest factors and the product is positive; and since the middle term is negative, both factors need to be negative).
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Shortcut 3 eliminates this possibility.
Now, try −1 and −10 for last factors.
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Shortcut 3 eliminates this possibility.
Now, try 1
x and 4
x for first factors and go back to −5 and −2 as last factors.
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Shortcut 3 eliminates this possibility. But because
x and 4
x are different factors, switching the −5 and −2 produces different results, as shown in the following:
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Therefore, 8 x2 − 26 x + 20 = 2( x − 2)(4 x − 5)
