document.write( "Question 539461: 6y^3+25y^2+14y \n" ); document.write( "

Algebra.Com's Answer #353428 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"6y%5E3%2B25y%5E2%2B14y\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%286y%5E2%2B25y%2B14%29\"$ Factor out the GCF $\"y\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"6y%5E2%2B25y%2B14\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6y%5E2%2B25y%2B14\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"25\"$ , and the last term is $\"14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"14\"$ to get $\"%286%29%2814%29=84\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"84\"$ (the previous product) and add to the second coefficient $\"25\"$ ?\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 $\"84\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"84\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,7,12,14,21,28,42,84\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-7,-12,-14,-21,-28,-42,-84\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 $\"84\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*84 = 84
\n" ); document.write( "2*42 = 84
\n" ); document.write( "3*28 = 84
\n" ); document.write( "4*21 = 84
\n" ); document.write( "6*14 = 84
\n" ); document.write( "7*12 = 84
\n" ); document.write( "(-1)*(-84) = 84
\n" ); document.write( "(-2)*(-42) = 84
\n" ); document.write( "(-3)*(-28) = 84
\n" ); document.write( "(-4)*(-21) = 84
\n" ); document.write( "(-6)*(-14) = 84
\n" ); document.write( "(-7)*(-12) = 84\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 $\"25\"$ :\r
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First Number	Second Number	Sum
1	84	1+84=85
2	42	2+42=44
3	28	3+28=31
4	21	4+21=25
6	14	6+14=20
7	12	7+12=19
-1	-84	-1+(-84)=-85
-2	-42	-2+(-42)=-44
-3	-28	-3+(-28)=-31
-4	-21	-4+(-21)=-25
-6	-14	-6+(-14)=-20
-7	-12	-7+(-12)=-19

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"4\"$ and $\"21\"$ add to $\"25\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"4\"$ and $\"21\"$ both multiply to $\"84\"$ and add to $\"25\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"25y\"$ with $\"4y%2B21y\"$ . Remember, $\"4\"$ and $\"21\"$ add to $\"25\"$ . So this shows us that $\"4y%2B21y=25y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6y%5E2%2Bhighlight%284y%2B21y%29%2B14\"$ Replace the second term $\"25y\"$ with $\"4y%2B21y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286y%5E2%2B4y%29%2B%2821y%2B14%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2y%283y%2B2%29%2B%2821y%2B14%29\"$ Factor out the GCF $\"2y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2y%283y%2B2%29%2B7%283y%2B2%29\"$ Factor out $\"7\"$ 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( " $\"%282y%2B7%29%283y%2B2%29\"$ Combine like terms. Or factor out the common term $\"3y%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"y%286y%5E2%2B25y%2B14%29\"$ then factors further to $\"y%282y%2B7%29%283y%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6y%5E3%2B25y%5E2%2B14y\"$ completely factors to $\"y%282y%2B7%29%283y%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"6y%5E3%2B25y%5E2%2B14y=y%282y%2B7%29%283y%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"y%282y%2B7%29%283y%2B2%29\"$ to get $\"6y%5E3%2B25y%5E2%2B14y\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you\r
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