document.write( "Question 473453: Factor 2x^2 + 11x + 12 \n" ); document.write( "

Algebra.Com's Answer #324670 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2x%5E2%2B11x%2B12\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"11\"$ , and the last term is $\"12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"12\"$ to get $\"%282%29%2812%29=24\"$ .\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 $\"11\"$ ?\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 $\"24\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*24 = 24
\n" ); document.write( "2*12 = 24
\n" ); document.write( "3*8 = 24
\n" ); document.write( "4*6 = 24
\n" ); document.write( "(-1)*(-24) = 24
\n" ); document.write( "(-2)*(-12) = 24
\n" ); document.write( "(-3)*(-8) = 24
\n" ); document.write( "(-4)*(-6) = 24\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 $\"11\"$ :\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 $\"3\"$ and $\"8\"$ add to $\"11\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"8\"$ both multiply to $\"24\"$ and add to $\"11\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"11x\"$ with $\"3x%2B8x\"$ . Remember, $\"3\"$ and $\"8\"$ add to $\"11\"$ . So this shows us that $\"3x%2B8x=11x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%283x%2B8x%29%2B12\"$ Replace the second term $\"11x\"$ with $\"3x%2B8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2%2B3x%29%2B%288x%2B12%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%282x%2B3%29%2B%288x%2B12%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%282x%2B3%29%2B4%282x%2B3%29\"$ 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( " $\"%28x%2B4%29%282x%2B3%29\"$ Combine like terms. Or factor out the common term $\"2x%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2%2B11x%2B12\"$ factors to $\"%28x%2B4%29%282x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"2x%5E2%2B11x%2B12=%28x%2B4%29%282x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B4%29%282x%2B3%29\"$ to get $\"2x%5E2%2B11x%2B12\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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