document.write( "Question 148772: 6x2 - 28x + 16 (Factor Competely) \n" ); document.write( "

Algebra.Com's Answer #109113 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( " $\"6x%5E2-28x%2B16\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2%283x%5E2-14x%2B8%29\"$ Factor out the GCF $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"3x%5E2-14x%2B8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3x%5E2-14x%2B8\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"-14\"$ , and the last term is $\"8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"8\"$ to get $\"%283%29%288%29=24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"24\"$ (the previous product) and add to the second coefficient $\"-14\"$ ?\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 $\"24\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"24\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,24\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24\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 $\"24\"$ . For instance, $\"1%2A24=24\"$ , $\"2%2A12=24\"$ , etc.\r
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\n" ); document.write( "\n" ); document.write( "Since $\"24\"$ is positive, this means that either\r
\n" ); document.write( "\n" ); document.write( "a) both factors are positive, or...
\n" ); document.write( "b) both factors are negative.\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 $\"-14\"$ :\r
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First Number	Second Number	Sum
1	24	1+24=25
2	12	2+12=14
3	8	3+8=11
4	6	4+6=10
-1	-24	-1+(-24)=-25
-2	-12	-2+(-12)=-14
-3	-8	-3+(-8)=-11
-4	-6	-4+(-6)=-10

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"-12\"$ add to $\"-14\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"-12\"$ both multiply to $\"24\"$ and add to $\"-14\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-14x\"$ with $\"-2x-12x\"$ . Remember, $\"-2\"$ and $\"-12\"$ add to $\"-14\"$ . So this shows us that $\"-2x-12x=-14x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%28-2x-12x%29%2B8\"$ Replace the second term $\"-14x\"$ with $\"-2x-12x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2-2x%29%2B%28-12x%2B8%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x-2%29%2B%28-12x%2B8%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x-2%29-4%283x-2%29\"$ 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( " $\"%28x-4%29%283x-2%29\"$ Combine like terms. Or factor out the common term $\"3x-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2-14x%2B8\"$ factors to $\"%28x-4%29%283x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So this means that $\"2%283x%5E2-14x%2B8%29\"$ can be rewritten as $\"2%28x-4%29%283x-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E2-28x%2B16\"$ factors to $\"2%28x-4%29%283x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2%28x-4%29%283x-2%29\"$ to get $\"6x%5E2-28x%2B16\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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