document.write( "Question 582002: Factor x^2-xy-56y^2 \n" ); document.write( "

Algebra.Com's Answer #371912 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-xy-56y%5E2\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-1\"$ , and the last coefficient is $\"-56\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last coefficient $\"-56\"$ to get $\"%281%29%28-56%29=-56\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-56\"$ (the previous product) and add to the second coefficient $\"-1\"$ ?\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 $\"-56\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-56\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,7,8,14,28,56\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-7,-8,-14,-28,-56\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 $\"-56\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-56) = -56
\n" ); document.write( "2*(-28) = -56
\n" ); document.write( "4*(-14) = -56
\n" ); document.write( "7*(-8) = -56
\n" ); document.write( "(-1)*(56) = -56
\n" ); document.write( "(-2)*(28) = -56
\n" ); document.write( "(-4)*(14) = -56
\n" ); document.write( "(-7)*(8) = -56\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 $\"-1\"$ :\r
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First Number	Second Number	Sum
1	-56	1+(-56)=-55
2	-28	2+(-28)=-26
4	-14	4+(-14)=-10
7	-8	7+(-8)=-1
-1	56	-1+56=55
-2	28	-2+28=26
-4	14	-4+14=10
-7	8	-7+8=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"7\"$ and $\"-8\"$ add to $\"-1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"7\"$ and $\"-8\"$ both multiply to $\"-56\"$ and add to $\"-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-1xy\"$ with $\"7xy-8xy\"$ . Remember, $\"7\"$ and $\"-8\"$ add to $\"-1\"$ . So this shows us that $\"7xy-8xy=-1xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%287xy-8xy%29-56y%5E2\"$ Replace the second term $\"-1xy\"$ with $\"7xy-8xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B7xy%29%2B%28-8xy-56y%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B7y%29%2B%28-8xy-56y%5E2%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B7y%29-8y%28x%2B7y%29\"$ Factor out $\"-8y\"$ from the second group.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-8y%29%28x%2B7y%29\"$ Factor out $\"x%2B7y\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-xy-56y%5E2\"$ completely factors to $\"%28x-8y%29%28x%2B7y%29\"$ \n" ); document.write( "

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