document.write( "Question 316984: Factor:
\n" ); document.write( "3x^2-10xy+8y^2 \n" ); document.write( "

Algebra.Com's Answer #226912 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"3x%5E2-10xy%2B8y%5E2\"$ we can see that the first term is $\"3x%5E2\"$ and the last term is $\"8y%5E2\"$ where the coefficients are 3 and 8 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 3 and the last coefficient 8 to get 24. Now what two numbers multiply to 24 and add to the middle coefficient -10? 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 -10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -10\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 -4 and -6 add up to -10 and multiply to 24\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"3x%5E2-10xy%2B8y%5E2\"$ , replace $\"-10xy\"$ with $\"-4xy%2B-6xy\"$ (notice $\"-4xy%2B-6xy\"$ adds up to $\"-10xy\"$ . So it is equivalent to $\"-10xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%28-4xy%2B-6xy%29%2B8y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"3x%5E2-4xy-6xy%2B8y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2-4xy%29%2B%28-6xy%2B8y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x-4y%29-2y%283x-4y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"-2y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-2y%29%283x-4y%29\"$ Since we have a common term of $\"3x-4y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2-4xy-6xy%2B8y%5E2\"$ factors to $\"%28x-2y%29%283x-4y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"3x%5E2-10xy%2B8y%5E2\"$ factors to $\"%28x-2y%29%283x-4y%29\"$ (since $\"3x%5E2-10xy%2B8y%5E2\"$ is equivalent to $\"3x%5E2-4xy-6xy%2B8y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2-10xy%2B8y%5E2\"$ factors to $\"%28x-2y%29%283x-4y%29\"$
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