document.write( "Question 146141This question is from textbook
\n" ); document.write( ": Factoring: 12y^2 - 11y + 2 \n" ); document.write( "

Algebra.Com's Answer #106691 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"12y%5E2-11y%2B2\"$ we can see that the first term is $\"12y%5E2\"$ and the last term is $\"2\"$ where the coefficients are 12 and 2 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 12 and the last coefficient 2 to get 24. Now what two numbers multiply to 24 and add to the middle coefficient -11? Let's list all of the factors of 24:\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
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. 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\r
\n" ); document.write( "\n" ); document.write( "2*12\r
\n" ); document.write( "\n" ); document.write( "3*8\r
\n" ); document.write( "\n" ); document.write( "4*6\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-8)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-6)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -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 this list we can see that -3 and -8 add up to -11 and multiply to 24\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"12y%5E2-11y%2B2\"$ , replace $\"-11y\"$ with $\"-3y%2B-8y\"$ (notice $\"-3y%2B-8y\"$ adds up to $\"-11y\"$ . So it is equivalent to $\"-11y\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"12y%5E2%2Bhighlight%28-3y%2B-8y%29%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"12y%5E2-3y-8y%2B2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812y%5E2-3y%29%2B%28-8y%2B2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3y%284y-1%29-2%284y-1%29\"$ Factor out the GCF of $\"3y\"$ out of the first group. Factor out the GCF of $\"-2\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283y-2%29%284y-1%29\"$ Since we have a common term of $\"4y-1\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"12y%5E2-3y-8y%2B2\"$ factors to $\"%283y-2%29%284y-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"12y%5E2-11y%2B2\"$ factors to $\"%283y-2%29%284y-1%29\"$ (since $\"12y%5E2-11y%2B2\"$ is equivalent to $\"12y%5E2-3y-8y%2B2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"12y%5E2-11y%2B2\"$ factors to $\"%283y-2%29%284y-1%29\"$
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