document.write( "Question 914765: 3n^2+7n+4 \n" ); document.write( "

Algebra.Com's Answer #555274 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3n%5E2%2B7n%2B4\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"7\"$ , and the last term is $\"4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"4\"$ to get $\"%283%29%284%29=12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"12\"$ (the previous product) and add to the second coefficient $\"7\"$ ?\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 $\"12\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"12\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,12\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-12\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 $\"12\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*12 = 12
\n" ); document.write( "2*6 = 12
\n" ); document.write( "3*4 = 12
\n" ); document.write( "(-1)*(-12) = 12
\n" ); document.write( "(-2)*(-6) = 12
\n" ); document.write( "(-3)*(-4) = 12\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 $\"7\"$ :\r
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First Number	Second Number	Sum
1	12	1+12=13
2	6	2+6=8
3	4	3+4=7
-1	-12	-1+(-12)=-13
-2	-6	-2+(-6)=-8
-3	-4	-3+(-4)=-7

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"4\"$ add to $\"7\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"4\"$ both multiply to $\"12\"$ and add to $\"7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"7n\"$ with $\"3n%2B4n\"$ . Remember, $\"3\"$ and $\"4\"$ add to $\"7\"$ . So this shows us that $\"3n%2B4n=7n\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3n%5E2%2Bhighlight%283n%2B4n%29%2B4\"$ Replace the second term $\"7n\"$ with $\"3n%2B4n\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283n%5E2%2B3n%29%2B%284n%2B4%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3n%28n%2B1%29%2B%284n%2B4%29\"$ Factor out the GCF $\"3n\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3n%28n%2B1%29%2B4%28n%2B1%29\"$ Factor out $\"4\"$ 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( " $\"%283n%2B4%29%28n%2B1%29\"$ Combine like terms. Or factor out the common term $\"n%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3n%5E2%2B7n%2B4\"$ factors to $\"%283n%2B4%29%28n%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3n%5E2%2B7n%2B4=%283n%2B4%29%28n%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%283n%2B4%29%28n%2B1%29\"$ to get $\"3n%5E2%2B7n%2B4\"$ 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( "Let me know if you need more help or if you need me to explain a step in more detail.
\n" ); document.write( "Feel free to email me at jim_thompson5910@hotmail.com
\n" ); document.write( "or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html\r
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\n" ); document.write( "\n" ); document.write( "Thanks,\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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