document.write( "Question 152716: Factor 3x^2-13x+12 \n" ); document.write( "

Algebra.Com's Answer #112226 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3x%5E2-13x%2B12\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"-13\"$ , and the last term is $\"12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"12\"$ to get $\"%283%29%2812%29=36\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"36\"$ (the previous product) and add to the second coefficient $\"-13\"$ ?\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 $\"36\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"36\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,12,18,36\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-36\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 $\"36\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*36
\n" ); document.write( "2*18
\n" ); document.write( "3*12
\n" ); document.write( "4*9
\n" ); document.write( "6*6
\n" ); document.write( "(-1)*(-36)
\n" ); document.write( "(-2)*(-18)
\n" ); document.write( "(-3)*(-12)
\n" ); document.write( "(-4)*(-9)
\n" ); document.write( "(-6)*(-6)\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 $\"-13\"$ :\r
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First Number	Second Number	Sum
1	36	1+36=37
2	18	2+18=20
3	12	3+12=15
4	9	4+9=13
6	6	6+6=12
-1	-36	-1+(-36)=-37
-2	-18	-2+(-18)=-20
-3	-12	-3+(-12)=-15
-4	-9	-4+(-9)=-13
-6	-6	-6+(-6)=-12

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"-9\"$ add to $\"-13\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"-9\"$ both multiply to $\"36\"$ and add to $\"-13\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-13x\"$ with $\"-4x-9x\"$ . Remember, $\"-4\"$ and $\"-9\"$ add to $\"-13\"$ . So this shows us that $\"-4x-9x=-13x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%28-4x-9x%29%2B12\"$ Replace the second term $\"-13x\"$ with $\"-4x-9x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2-4x%29%2B%28-9x%2B12%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x-4%29%2B%28-9x%2B12%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x-4%29-3%283x-4%29\"$ Factor out $\"3\"$ 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( " $\"%28x-3%29%283x-4%29\"$ Combine like terms. Or factor out the common term $\"3x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2-13x%2B12\"$ factors to $\"%28x-3%29%283x-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28x-3%29%283x-4%29\"$ to get $\"3x%5E2-13x%2B12\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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