document.write( "Question 191090: factor by grouping\r
\n" ); document.write( "\n" ); document.write( "5x^2-12x+7 \n" ); document.write( "

Algebra.Com's Answer #143448 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "Looking at the expression $\"5x%5E2-12x%2B7\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"-12\"$ , and the last term is $\"7\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"7\"$ to get $\"%285%29%287%29=35\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"35\"$ (the previous product) and add to the second coefficient $\"-12\"$ ?\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 $\"35\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"35\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,5,7,35\r
\n" ); document.write( "\n" ); document.write( "-1,-5,-7,-35\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 $\"35\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*35
\n" ); document.write( "5*7
\n" ); document.write( "(-1)*(-35)
\n" ); document.write( "(-5)*(-7)\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 $\"-12\"$ :\r
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First Number	Second Number	Sum
1	35	1+35=36
5	7	5+7=12
-1	-35	-1+(-35)=-36
-5	-7	-5+(-7)=-12

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-5\"$ and $\"-7\"$ add to $\"-12\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-5\"$ and $\"-7\"$ both multiply to $\"35\"$ and add to $\"-12\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-12x\"$ with $\"-5x-7x\"$ . Remember, $\"-5\"$ and $\"-7\"$ add to $\"-12\"$ . So this shows us that $\"-5x-7x=-12x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%5E2%2Bhighlight%28-5x-7x%29%2B7\"$ Replace the second term $\"-12x\"$ with $\"-5x-7x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%285x%5E2-5x%29%2B%28-7x%2B7%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%28x-1%29%2B%28-7x%2B7%29\"$ Factor out the GCF $\"5x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%28x-1%29-7%28x-1%29\"$ Factor out $\"7\"$ 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( " $\"%285x-7%29%28x-1%29\"$ Combine like terms. Or factor out the common term $\"x-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"5x%5E2-12x%2B7\"$ factors to $\"%285x-7%29%28x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%285x-7%29%28x-1%29\"$ to get $\"5x%5E2-12x%2B7\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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