document.write( "Question 299375: reverse foil or grouping
\n" ); document.write( "2y^3+24y^2+72y \n" ); document.write( "

Algebra.Com's Answer #215135 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"2y%5E3%2B24y%5E2%2B72y\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"2y%28y%5E2%2B12y%2B36%29\"$ Factor out the GCF $\"2y\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"y%5E2%2B12y%2B36\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"y%5E2%2B12y%2B36\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"12\"$ , and the last term is $\"36\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"36\"$ to get $\"%281%29%2836%29=36\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"36\"$ (the previous product) and add to the second coefficient $\"12\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"36\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"36\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,12,18,36\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-36\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. 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 $\"36\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*36 = 36
\n" ); document.write( "2*18 = 36
\n" ); document.write( "3*12 = 36
\n" ); document.write( "4*9 = 36
\n" ); document.write( "6*6 = 36
\n" ); document.write( "(-1)*(-36) = 36
\n" ); document.write( "(-2)*(-18) = 36
\n" ); document.write( "(-3)*(-12) = 36
\n" ); document.write( "(-4)*(-9) = 36
\n" ); document.write( "(-6)*(-6) = 36\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"12\"$ :\r
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First Number	Second Number	Sum
1	36	1+36=37
2	18	2+18=20
3	12	3+12=15
4	9	4+9=13
6	6	6+6=12
-1	-36	-1+(-36)=-37
-2	-18	-2+(-18)=-20
-3	-12	-3+(-12)=-15
-4	-9	-4+(-9)=-13
-6	-6	-6+(-6)=-12

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"6\"$ and $\"6\"$ add to $\"12\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"6\"$ and $\"6\"$ both multiply to $\"36\"$ and add to $\"12\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"12y\"$ with $\"6y%2B6y\"$ . Remember, $\"6\"$ and $\"6\"$ add to $\"12\"$ . So this shows us that $\"6y%2B6y=12y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2%2Bhighlight%286y%2B6y%29%2B36\"$ Replace the second term $\"12y\"$ with $\"6y%2B6y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28y%5E2%2B6y%29%2B%286y%2B36%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y%2B6%29%2B%286y%2B36%29\"$ Factor out the GCF $\"y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y%2B6%29%2B6%28y%2B6%29\"$ Factor out $\"6\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28y%2B6%29%28y%2B6%29\"$ Combine like terms. Or factor out the common term $\"y%2B6\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28y%2B6%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "So $\"2y%28y%5E2%2B12y%2B36%29\"$ then factors further to $\"2y%28y%2B6%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"2y%5E3%2B24y%5E2%2B72y\"$ completely factors to $\"2y%28y%2B6%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"2y%5E3%2B24y%5E2%2B72y=2y%28y%2B6%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2y%28y%2B6%29%5E2\"$ to get $\"2y%5E3%2B24y%5E2%2B72y\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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