document.write( "Question 143604: I need your help in Factoring this problem. It must be factored completely.\r
\n" ); document.write( "\n" ); document.write( "x^5y^3 - 9x^4y^2 + 20x^3y\r
\n" ); document.write( "\n" ); document.write( "I have tried, but I am coming up with an incorrect answer.\r
\n" ); document.write( "\n" ); document.write( "My answer: x^3y(x -x5)(x - 4)
\n" ); document.write( " = x^3y [(x0(x) - (x)(-4) - (-5)(x) + (-5)(-4)]
\n" ); document.write( " = x^3y(x^2) - (4x^2) - (5x^2) + 20
\n" ); document.write( " = x^5y - 9x^2 + 20\r
\n" ); document.write( "\n" ); document.write( "I am unsure of how to determine the Greatest common factor on this type of problem. \n" ); document.write( "

Algebra.Com's Answer #104500 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"x%5E5y%5E3-9x%5E4y%5E2%2B20x%5E3y\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E3y%28x%5E2y%5E2-9xy%2B20%29\"$ Factor out the GCF $\"x%5E3y\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"x%5E2y%5E2-9xy%2B20\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"1x%5E2y%5E2-9xy%2B20\"$ we can see that the first term is $\"1x%5E2y%5E2\"$ and the last term is $\"20\"$ where the coefficients are 1 and 20 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 20 to get 20. Now what two numbers multiply to 20 and add to the middle coefficient -9? Let's list all of the factors of 20:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 20:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,10,20\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-20 ...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 20\r
\n" ); document.write( "\n" ); document.write( "1*20\r
\n" ); document.write( "\n" ); document.write( "2*10\r
\n" ); document.write( "\n" ); document.write( "4*5\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-10)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-5)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9\r
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First Number	Second Number	Sum
1	20	1+20=21
2	10	2+10=12
4	5	4+5=9
-1	-20	-1+(-20)=-21
-2	-10	-2+(-10)=-12
-4	-5	-4+(-5)=-9

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -4 and -5 add up to -9 and multiply to 20\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"1x%5E2y%5E2-9xy%2B20\"$ , replace $\"-9xy\"$ with $\"-4xy%2B-5xy\"$ (notice $\"-4xy%2B-5xy\"$ adds up to $\"-9xy\"$ . So it is equivalent to $\"-9xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"1x%5E2y%5E2%2Bhighlight%28-4xy%2B-5xy%29%2B20\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"1x%5E2y%5E2-4xy-5xy%2B20\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%281x%5E2y%5E2-4xy%29%2B%28-5xy%2B20%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"xy%28xy-4%29-5%28xy-4%29\"$ Factor out the GCF of $\"xy\"$ out of the first group. Factor out the GCF of $\"-5\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28xy-5%29%28xy-4%29\"$ Since we have a common term of $\"xy-4\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"1x%5E2y%5E2-4xy-5xy%2B20\"$ factors to $\"%28xy-5%29%28xy-4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"1x%5E2y%5E2-9xy%2B20\"$ factors to $\"%28xy-5%29%28xy-4%29\"$ (since $\"1x%5E2y%5E2-9xy%2B20\"$ is equivalent to $\"1x%5E2y%5E2-4xy-5xy%2B20\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"x%5E3y%28x%5E2y%5E2-9xy%2B20%29\"$ and factors further to $\"x%5E3y%28xy-5%29%28xy-4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E5y%5E3-9x%5E4y%5E2%2B20x%5E3y\"$ factors to $\"x%5E3y%28xy-5%29%28xy-4%29\"$ \r
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