document.write( "Question 703574: please help me factor the trinomial $\"21y%5E3-36y%5E2-12y\"$ \n" ); document.write( "

Algebra.Com's Answer #433557 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"21y%5E3-36y%5E2-12y\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"3y%287y%5E2-12y-4%29\"$ Factor out the GCF $\"3y\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"7y%5E2-12y-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"7y%5E2-12y-4\"$ , we can see that the first coefficient is $\"7\"$ , the second coefficient is $\"-12\"$ , and the last term is $\"-4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"7\"$ by the last term $\"-4\"$ to get $\"%287%29%28-4%29=-28\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-28\"$ (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 $\"-28\"$ (the previous product).\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
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-7,-14,-28\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 $\"-28\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-28) = -28
\n" ); document.write( "2*(-14) = -28
\n" ); document.write( "4*(-7) = -28
\n" ); document.write( "(-1)*(28) = -28
\n" ); document.write( "(-2)*(14) = -28
\n" ); document.write( "(-4)*(7) = -28\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	-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 the table, we can see that the two numbers $\"2\"$ and $\"-14\"$ add to $\"-12\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"-14\"$ both multiply to $\"-28\"$ and add to $\"-12\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-12y\"$ with $\"2y-14y\"$ . Remember, $\"2\"$ and $\"-14\"$ add to $\"-12\"$ . So this shows us that $\"2y-14y=-12y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"7y%5E2%2Bhighlight%282y-14y%29-4\"$ Replace the second term $\"-12y\"$ with $\"2y-14y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%287y%5E2%2B2y%29%2B%28-14y-4%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%287y%2B2%29%2B%28-14y-4%29\"$ Factor out the GCF $\"y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%287y%2B2%29-2%287y%2B2%29\"$ Factor out $\"2\"$ 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( " $\"%28y-2%29%287y%2B2%29\"$ Combine like terms. Or factor out the common term $\"7y%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3y%287y%5E2-12y-4%29\"$ then factors further to $\"3y%28y-2%29%287y%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"21y%5E3-36y%5E2-12y\"$ completely factors to $\"3y%28y-2%29%287y%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"21y%5E3-36y%5E2-12y=3y%28y-2%29%287y%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"3y%28y-2%29%287y%2B2%29\"$ to get $\"21y%5E3-36y%5E2-12y\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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