document.write( "Question 182955: 4x²-4xy-120y \n" ); document.write( "

Algebra.Com's Answer #137352 by jim_thompson5910(35256) $\"\"$ $\"About$

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Are you sure that the expression is not $\"4x%5E2-4xy-120y%5E2\"$ ???\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2-4xy-120y%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"4%28x%5E2-xy-30y%5E2%29\"$ Factor out the GCF $\"4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"x%5E2-xy-30y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E2-xy-30y%5E2\"$ we can see that the first term is $\"1x%5E2\"$ and the last term is $\"-30y%5E2\"$ where the coefficients are 1 and -30 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -30 to get -30. Now what two numbers multiply to -30 and add to the middle coefficient -1? Let's list all of the factors of -30:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -30:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-30 ...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 -30\r
\n" ); document.write( "\n" ); document.write( "(1)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-10)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-6)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(30)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(15)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(10)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(6)\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 -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1\r
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First Number	Second Number	Sum
1	-30	1+(-30)=-29
2	-15	2+(-15)=-13
3	-10	3+(-10)=-7
5	-6	5+(-6)=-1
-1	30	-1+30=29
-2	15	-2+15=13
-3	10	-3+10=7
-5	6	-5+6=1

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 5 and -6 add up to -1 and multiply to -30\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"x%5E2-xy-30y%5E2\"$ , replace $\"-xy\"$ with $\"5xy-6xy\"$ (notice $\"5xy-6xy\"$ adds up to $\"-xy\"$ . So it is equivalent to $\"-xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%285xy-6xy%29-30y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2%2B5xy-6xy-30y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B5xy%29%2B%28-6xy-30y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B5y%29-6y%28x%2B5y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"-6y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-6y%29%28x%2B5y%29\"$ Since we have a common term of $\"x%2B5y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B5xy-6xy-30y%5E2\"$ factors to $\"%28x-6y%29%28x%2B5y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2-xy-30y%5E2\"$ factors to $\"%28x-6y%29%28x%2B5y%29\"$ (since $\"x%5E2-xy-30y%5E2\"$ is equivalent to $\"x%5E2%2B5xy-6xy-30y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"4%28x%5E2-xy-30y%5E2%29\"$ and factors further to $\"4%28x-6y%29%28x%2B5y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-4xy-120y%5E2\"$ completely factors to $\"4%28x-6y%29%28x%2B5y%29\"$ \r
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