document.write( "Question 299482: reverse foil or trial factoring\r
\n" ); document.write( "\n" ); document.write( "2x^2-12x+16 \n" ); document.write( "

Algebra.Com's Answer #215164 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"2x%5E2-12x%2B16\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%28x%5E2-6x%2B8%29\"$ Factor out the GCF $\"2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"x%5E2-6x%2B8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-6x%2B8\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-6\"$ , and the last term is $\"8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"8\"$ to get $\"%281%29%288%29=8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"8\"$ (the previous product) and add to the second coefficient $\"-6\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"8\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"8\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"8\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*8 = 8
\n" ); document.write( "2*4 = 8
\n" ); document.write( "(-1)*(-8) = 8
\n" ); document.write( "(-2)*(-4) = 8\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-6\"$ :\r
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First Number	Second Number	Sum
1	8	1+8=9
2	4	2+4=6
-1	-8	-1+(-8)=-9
-2	-4	-2+(-4)=-6

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"-4\"$ add to $\"-6\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"-4\"$ both multiply to $\"8\"$ and add to $\"-6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-6x\"$ with $\"-2x-4x\"$ . Remember, $\"-2\"$ and $\"-4\"$ add to $\"-6\"$ . So this shows us that $\"-2x-4x=-6x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-2x-4x%29%2B8\"$ Replace the second term $\"-6x\"$ with $\"-2x-4x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-2x%29%2B%28-4x%2B8%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-2%29%2B%28-4x%2B8%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-2%29-4%28x-2%29\"$ Factor out $\"4\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-4%29%28x-2%29\"$ Combine like terms. Or factor out the common term $\"x-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"2%28x%5E2-6x%2B8%29\"$ then factors further to $\"2%28x-4%29%28x-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2-12x%2B16\"$ completely factors to $\"2%28x-4%29%28x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"2x%5E2-12x%2B16=2%28x-4%29%28x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2%28x-4%29%28x-2%29\"$ to get $\"2x%5E2-12x%2B16\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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