document.write( "Question 191479: 35y^3-60y^2-20y\r
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Algebra.Com's Answer #143677 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"35y%5E3-60y%5E2-20y\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"5y%287y%5E2-12y-4%29\"$ Factor out the GCF $\"5y\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"7y%5E2-12y-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"7y%5E2-12y-4\"$ we can see that the first term is $\"7y%5E2\"$ and the last term is $\"-4\"$ where the coefficients are 7 and -4 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 7 and the last coefficient -4 to get -28. Now what two numbers multiply to -28 and add to the middle coefficient -12? Let's list all of the factors of -28:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -28:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,7,14,28\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-7,-14,-28 ...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 -28\r
\n" ); document.write( "\n" ); document.write( "(1)*(-28)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-14)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-7)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(28)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(14)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(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 -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12\r
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First Number	Second Number	Sum
1	-28	1+(-28)=-27
2	-14	2+(-14)=-12
4	-7	4+(-7)=-3
-1	28	-1+28=27
-2	14	-2+14=12
-4	7	-4+7=3

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 2 and -14 add up to -12 and multiply to -28\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"7y%5E2-12y-4\"$ , replace $\"-12y\"$ with $\"2y-14y\"$ (notice $\"2y-14y\"$ adds up to $\"-12y\"$ . So it is equivalent to $\"-12y\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"7y%5E2%2Bhighlight%282y-14y%29-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"7y%5E2%2B2y-14y-4\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%287y%5E2%2B2y%29%2B%28-14y-4%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"y%287y%2B2%29-2%287y%2B2%29\"$ Factor out the GCF of $\"y\"$ out of the first group. Factor out the GCF of $\"-2\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28y-2%29%287y%2B2%29\"$ Since we have a common term of $\"7y%2B2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"7y%5E2%2B2y-14y-4\"$ factors to $\"%28y-2%29%287y%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"7y%5E2-12y-4\"$ factors to $\"%28y-2%29%287y%2B2%29\"$ (since $\"7y%5E2-12y-4\"$ is equivalent to $\"7y%5E2%2B2y-14y-4\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"5y%287y%5E2-12y-4%29\"$ and factors further to $\"5y%28y-2%29%287y%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"35y%5E3-60y%5E2-20y\"$ completely factors to $\"5y%28y-2%29%287y%2B2%29\"$ \r
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