document.write( "Question 146160: Factor completely\r
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\n" ); document.write( "\n" ); document.write( "x^10y^3 – 4x^9y^2 – 21x^8y
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Algebra.Com's Answer #106709 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"x%5E10y%5E3-4x%5E9y%5E2-21x%5E8y\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E8y%28x%5E2y%5E2-4xy-21%29\"$ Factor out the GCF $\"x%5E8y\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"x%5E2y%5E2-4xy-21\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"1x%5E2y%5E2-4xy-21\"$ we can see that the first term is $\"1x%5E2y%5E2\"$ and the last term is $\"-21\"$ 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
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\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 $\"1x%5E2y%5E2-4xy-21\"$ , replace $\"-4xy\"$ with $\"3xy%2B-7xy\"$ (notice $\"3xy%2B-7xy\"$ adds up to $\"-4xy\"$ . So it is equivalent to $\"-4xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"1x%5E2y%5E2%2Bhighlight%283xy%2B-7xy%29%2B-21\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"1x%5E2y%5E2%2B3xy-7xy-21\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%281x%5E2y%5E2%2B3xy%29%2B%28-7xy-21%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"xy%28xy%2B3%29-7%28xy%2B3%29\"$ Factor out the GCF of $\"xy\"$ out of the first group. Factor out the GCF of $\"-7\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28xy-7%29%28xy%2B3%29\"$ Since we have a common term of $\"xy%2B3\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"1x%5E2y%5E2%2B3xy-7xy-21\"$ factors to $\"%28xy-7%29%28xy%2B3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"1x%5E2y%5E2-4xy-21\"$ factors to $\"%28xy-7%29%28xy%2B3%29\"$ (since $\"1x%5E2y%5E2-4xy-21\"$ is equivalent to $\"1x%5E2y%5E2%2B3xy-7xy-21\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"x%5E8y%28x%5E2y%5E2-4xy-21%29\"$ and factors further to $\"x%5E8y%28xy-7%29%28xy%2B3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E10y%5E3-4x%5E9y%5E2-21x%5E8y\"$ factors to $\"x%5E8y%28xy-7%29%28xy%2B3%29\"$ \r
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