document.write( "Question 614323: factor completely: -36m^2 + 6m + 12 \n" ); document.write( "

Algebra.Com's Answer #386502 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"-36m%5E2%2B6m%2B12\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"-6%286m%5E2-m-2%29\"$ Factor out the GCF $\"-6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"6m%5E2-m-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6m%5E2-m-2\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"-1\"$ , and the last term is $\"-2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"-2\"$ to get $\"%286%29%28-2%29=-12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-12\"$ (the previous product) and add to the second coefficient $\"-1\"$ ?\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 $\"-12\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-12\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,12\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-12\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 $\"-12\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-12) = -12
\n" ); document.write( "2*(-6) = -12
\n" ); document.write( "3*(-4) = -12
\n" ); document.write( "(-1)*(12) = -12
\n" ); document.write( "(-2)*(6) = -12
\n" ); document.write( "(-3)*(4) = -12\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 $\"-1\"$ :\r
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First Number	Second Number	Sum
1	-12	1+(-12)=-11
2	-6	2+(-6)=-4
3	-4	3+(-4)=-1
-1	12	-1+12=11
-2	6	-2+6=4
-3	4	-3+4=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"-4\"$ add to $\"-1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"-4\"$ both multiply to $\"-12\"$ and add to $\"-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-1m\"$ with $\"3m-4m\"$ . Remember, $\"3\"$ and $\"-4\"$ add to $\"-1\"$ . So this shows us that $\"3m-4m=-1m\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6m%5E2%2Bhighlight%283m-4m%29-2\"$ Replace the second term $\"-1m\"$ with $\"3m-4m\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286m%5E2%2B3m%29%2B%28-4m-2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3m%282m%2B1%29%2B%28-4m-2%29\"$ Factor out the GCF $\"3m\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3m%282m%2B1%29-2%282m%2B1%29\"$ Factor out $\"2\"$ 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( " $\"%283m-2%29%282m%2B1%29\"$ Combine like terms. Or factor out the common term $\"2m%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"-6%286m%5E2-m-2%29\"$ then factors further to $\"-6%283m-2%29%282m%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"-36m%5E2%2B6m%2B12\"$ completely factors to $\"-6%283m-2%29%282m%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"-36m%5E2%2B6m%2B12=-6%283m-2%29%282m%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"-6%283m-2%29%282m%2B1%29\"$ to get $\"-36m%5E2%2B6m%2B12\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you\r
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