document.write( "Question 201048: Factor completely.\r
\n" ); document.write( "\n" ); document.write( "x2 + 3xy - 208y2
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Algebra.Com's Answer #151298 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E2%2B3xy-208y%5E2\"$ we can see that the first term is $\"x%5E2\"$ and the last term is $\"-208y%5E2\"$ where the coefficients are 1 and -208 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -208 to get -208. Now what two numbers multiply to -208 and add to the middle coefficient 3? Let's list all of the factors of -208:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -208:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,8,13,16,26,52,104,208\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8,-13,-16,-26,-52,-104,-208 ...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 -208\r
\n" ); document.write( "\n" ); document.write( "(1)*(-208)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-104)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-52)\r
\n" ); document.write( "\n" ); document.write( "(8)*(-26)\r
\n" ); document.write( "\n" ); document.write( "(13)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(208)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(104)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(52)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(26)\r
\n" ); document.write( "\n" ); document.write( "(-13)*(16)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 3\r
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First Number	Second Number	Sum
1	-208	1+(-208)=-207
2	-104	2+(-104)=-102
4	-52	4+(-52)=-48
8	-26	8+(-26)=-18
13	-16	13+(-16)=-3
-1	208	-1+208=207
-2	104	-2+104=102
-4	52	-4+52=48
-8	26	-8+26=18
-13	16	-13+16=3

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -13 and 16 add up to 3 and multiply to -208\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"1x%5E2%2B3xy-208y%5E2\"$ , replace $\"3xy\"$ with $\"-13xy%2B16xy\"$ (notice $\"-13xy%2B16xy\"$ adds up to $\"3xy\"$ . So it is equivalent to $\"3xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-13xy%2B16xy%29-208y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2-13xy%2B16xy-208y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-13xy%29%2B%2816xy-208y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-13y%29%2B16y%28x-13y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"16y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B16y%29%28x-13y%29\"$ Since we have a common term of $\"x-13y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-13xy%2B16xy-208y%5E2\"$ factors to $\"%28x%2B16y%29%28x-13y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2%2B3xy-208y%5E2\"$ factors to $\"%28x%2B16y%29%28x-13y%29\"$ (since $\"x%5E2%2B3xy-208y%5E2\"$ is equivalent to $\"x%5E2-13xy%2B16xy-208y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B3xy-208y%5E2\"$ factors to $\"%28x%2B16y%29%28x-13y%29\"$
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