document.write( "Question 333713: Factor.\r
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Algebra.Com's Answer #239141 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"12x%5E2%2Bxy-6y%5E2\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"1\"$ , and the last coefficient is $\"-6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"12\"$ by the last coefficient $\"-6\"$ to get $\"%2812%29%28-6%29=-72\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-72\"$ (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 $\"-72\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-72\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,18,24,36,72\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72\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 $\"-72\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-72) = -72
\n" ); document.write( "2*(-36) = -72
\n" ); document.write( "3*(-24) = -72
\n" ); document.write( "4*(-18) = -72
\n" ); document.write( "6*(-12) = -72
\n" ); document.write( "8*(-9) = -72
\n" ); document.write( "(-1)*(72) = -72
\n" ); document.write( "(-2)*(36) = -72
\n" ); document.write( "(-3)*(24) = -72
\n" ); document.write( "(-4)*(18) = -72
\n" ); document.write( "(-6)*(12) = -72
\n" ); document.write( "(-8)*(9) = -72\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	-72	1+(-72)=-71
2	-36	2+(-36)=-34
3	-24	3+(-24)=-21
4	-18	4+(-18)=-14
6	-12	6+(-12)=-6
8	-9	8+(-9)=-1
-1	72	-1+72=71
-2	36	-2+36=34
-3	24	-3+24=21
-4	18	-4+18=14
-6	12	-6+12=6
-8	9	-8+9=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-8\"$ and $\"9\"$ add to $\"1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-8\"$ and $\"9\"$ both multiply to $\"-72\"$ and add to $\"1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"1xy\"$ with $\"-8xy%2B9xy\"$ . Remember, $\"-8\"$ and $\"9\"$ add to $\"1\"$ . So this shows us that $\"-8xy%2B9xy=1xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"12x%5E2%2Bhighlight%28-8xy%2B9xy%29-6y%5E2\"$ Replace the second term $\"1xy\"$ with $\"-8xy%2B9xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812x%5E2-8xy%29%2B%289xy-6y%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%283x-2y%29%2B%289xy-6y%5E2%29\"$ Factor out the GCF $\"4x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%283x-2y%29%2B3y%283x-2y%29\"$ Factor out $\"3y\"$ 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( " $\"%284x%2B3y%29%283x-2y%29\"$ Combine like terms. Or factor out the common term $\"3x-2y\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"12x%5E2%2Bxy-6y%5E2\"$ factors to $\"%284x%2B3y%29%283x-2y%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"12x%5E2%2Bxy-6y%5E2=%284x%2B3y%29%283x-2y%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%284x%2B3y%29%283x-2y%29\"$ to get $\"12x%5E2%2Bxy-6y%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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