document.write( "Question 112854: I couldnt figure this one out(factoring)\r
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Algebra.Com's Answer #82258 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"8u%5E3-24u%5E2v%2B18uv%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2u%284u%5E2-12uv%2B9v%5E2%29\"$ Factor out the GCF $\"2u\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"4u%5E2-12uv%2B9v%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"4u%5E2-12uv%2B9v%5E2\"$ we can see that the first term is $\"4u%5E2\"$ and the last term is $\"9v%5E2\"$ where the coefficients are 4 and 9 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 4 and the last coefficient 9 to get 36. Now what two numbers multiply to 36 and add to the middle coefficient -12? Let's list all of the factors of 36:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 36:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18 ...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 36\r
\n" ); document.write( "\n" ); document.write( "1*36\r
\n" ); document.write( "\n" ); document.write( "2*18\r
\n" ); document.write( "\n" ); document.write( "3*12\r
\n" ); document.write( "\n" ); document.write( "4*9\r
\n" ); document.write( "\n" ); document.write( "6*6\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-9)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-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 -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12\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 this list we can see that -6 and -6 add up to -12 and multiply to 36\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"4u%5E2-12uv%2B9v%5E2\"$ , replace $\"-12uv\"$ with $\"-6uv%2B-6uv\"$ (notice $\"-6uv%2B-6uv\"$ adds up to $\"-12uv\"$ . So it is equivalent to $\"-12uv\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"4u%5E2%2Bhighlight%28-6uv%2B-6uv%29%2B9v%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"4u%5E2-6uv-6uv%2B9v%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%284u%5E2-6uv%29%2B%28-6uv%2B9v%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2u%282u-3v%29-3v%282u-3v%29\"$ Factor out the GCF of $\"2u\"$ out of the first group. Factor out the GCF of $\"-3v\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%282u-3v%29%282u-3v%29\"$ Since we have a common term of $\"2u-3v\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"4u%5E2-6uv-6uv%2B9v%5E2\"$ factors to $\"%282u-3v%29%282u-3v%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"4u%5E2-12uv%2B9v%5E2\"$ factors to $\"%282u-3v%29%282u-3v%29\"$ (since $\"4u%5E2-12uv%2B9v%5E2\"$ is equivalent to $\"4u%5E2-6uv-6uv%2B9v%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"2u%282u-3v%29%282u-3v%29\"$ Now replace the 2u\r
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\n" ); document.write( "\n" ); document.write( "So that means the original expression $\"8u%5E3-24u%5E2v%2B18uv%5E2\"$ factors to \r
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\n" ); document.write( "\n" ); document.write( " $\"2u%282u-3v%29%282u-3v%29\"$
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