document.write( "Question 201040: Factor completely.\r
\n" ); document.write( "\n" ); document.write( "x^3y + 4x^2y^2 - 21xy^3 \n" ); document.write( "

Algebra.Com's Answer #151276 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"x%5E3y%2B4x%5E2y%5E2-21xy%5E3\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"xy%28x%5E2%2B4xy-21y%5E2%29\"$ Factor out the GCF $\"xy\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"x%5E2%2B4xy-21y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"1x%5E2%2B4xy-21y%5E2\"$ we can see that the first term is $\"1x%5E2\"$ and the last term is $\"-21y%5E2\"$ where the coefficients are 1 and -21 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -21 to get -21. Now what two numbers multiply to -21 and add to the middle coefficient 4? Let's list all of the factors of -21:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -21:\r
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\n" ); document.write( "\n" ); document.write( "-1,-3,-7,-21 ...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 -21\r
\n" ); document.write( "\n" ); document.write( "(1)*(-21)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-7)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(21)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(7)\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 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4\r
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First Number	Second Number	Sum
1	-21	1+(-21)=-20
3	-7	3+(-7)=-4
-1	21	-1+21=20
-3	7	-3+7=4

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -3 and 7 add up to 4 and multiply to -21\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"x%5E2%2B4xy-21y%5E2\"$ , replace $\"4xy\"$ with $\"-3xy%2B7xy\"$ (notice $\"-3xy%2B7xy\"$ adds up to $\"4xy\"$ . So it is equivalent to $\"4xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-3xy%2B7xy%29-21y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2-3xy%2B7xy-21y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-3xy%29%2B%287xy-21y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-3y%29%2B7y%28x-3y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"7y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B7y%29%28x-3y%29\"$ Since we have a common term of $\"x-3y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-3xy%2B7xy-21y%5E2\"$ factors to $\"%28x%2B7y%29%28x-3y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2%2B4xy-21y%5E2\"$ factors to $\"%28x%2B7y%29%28x-3y%29\"$ (since $\"x%5E2%2B4xy-21y%5E2\"$ is equivalent to $\"x%5E2-3xy%2B7xy-21y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"xy%28x%5E2%2B4xy-21y%5E2%29\"$ and factors further to $\"xy%28x%2B7y%29%28x-3y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E3y%2B4x%5E2y%5E2-21xy%5E3\"$ completely factors to $\"xy%28x%2B7y%29%28x-3y%29\"$ \r
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