document.write( "Question 191403: please help factor out polynomial.
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\n" ); document.write( "14x^3y+21x^2y^2-35xy^3
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\n" ); document.write( "thanksPeacee \r
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\n" ); document.write( "--Kyle \n" ); document.write( "

Algebra.Com's Answer #143646 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"14x%5E3y%2B21x%5E2y%5E2-35xy%5E3\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"7xy%282x%5E2%2B3xy-5y%5E2%29\"$ Factor out the GCF $\"7xy\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"2x%5E2%2B3xy-5y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"2x%5E2%2B3xy-5y%5E2\"$ we can see that the first term is $\"2x%5E2\"$ and the last term is $\"-5y%5E2\"$ where the coefficients are 2 and -5 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 2 and the last coefficient -5 to get -10. Now what two numbers multiply to -10 and add to the middle coefficient 3? Let's list all of the factors of -10:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -10:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-5,-10 ...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 -10\r
\n" ); document.write( "\n" ); document.write( "(1)*(-10)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-5)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(10)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(5)\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 3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 3\r
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First Number	Second Number	Sum
1	-10	1+(-10)=-9
2	-5	2+(-5)=-3
-1	10	-1+10=9
-2	5	-2+5=3

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -2 and 5 add up to 3 and multiply to -10\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"2x%5E2%2B3xy-5y%5E2\"$ , replace $\"3xy\"$ with $\"-2xy%2B5xy\"$ (notice $\"-2xy%2B5xy\"$ adds up to $\"3xy\"$ . So it is equivalent to $\"3xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%28-2xy%2B5xy%29%2B-5y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"2x%5E2-2xy%2B5xy-5y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2-2xy%29%2B%285xy-5y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%28x-y%29%2B5y%28x-y%29\"$ Factor out the GCF of $\"2x\"$ out of the first group. Factor out the GCF of $\"5y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x%2B5y%29%28x-y%29\"$ Since we have a common term of $\"x-y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2-2xy%2B5xy-5y%5E2\"$ factors to $\"%282x%2B5y%29%28x-y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"2x%5E2%2B3xy-5y%5E2\"$ factors to $\"%282x%2B5y%29%28x-y%29\"$ (since $\"2x%5E2%2B3xy-5y%5E2\"$ is equivalent to $\"2x%5E2-2xy%2B5xy-5y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"7xy%282x%5E2%2B3xy-5y%5E2%29\"$ and factors further to $\"7xy%282x%2B5y%29%28x-y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"14x%5E3y%2B21x%5E2y%5E2-35xy%5E3\"$ completely factors to $\"7xy%282x%2B5y%29%28x-y%29\"$ \r
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