document.write( "Question 537148: I'm having trouble with this one in particular\r
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\n" ); document.write( "\n" ); document.write( "I would like to know which factoring technique to use here. Thx, Brett \n" ); document.write( "

Algebra.Com's Answer #352740 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"21r%5E2%2B4r-32\"$ , we can see that the first coefficient is $\"21\"$ , the second coefficient is $\"4\"$ , and the last term is $\"-32\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"21\"$ by the last term $\"-32\"$ to get $\"%2821%29%28-32%29=-672\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-672\"$ (the previous product) and add to the second coefficient $\"4\"$ ?\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 $\"-672\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-672\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,7,8,12,14,16,21,24,28,32,42,48,56,84,96,112,168,224,336,672\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-7,-8,-12,-14,-16,-21,-24,-28,-32,-42,-48,-56,-84,-96,-112,-168,-224,-336,-672\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 $\"-672\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-672) = -672
\n" ); document.write( "2*(-336) = -672
\n" ); document.write( "3*(-224) = -672
\n" ); document.write( "4*(-168) = -672
\n" ); document.write( "6*(-112) = -672
\n" ); document.write( "7*(-96) = -672
\n" ); document.write( "8*(-84) = -672
\n" ); document.write( "12*(-56) = -672
\n" ); document.write( "14*(-48) = -672
\n" ); document.write( "16*(-42) = -672
\n" ); document.write( "21*(-32) = -672
\n" ); document.write( "24*(-28) = -672
\n" ); document.write( "(-1)*(672) = -672
\n" ); document.write( "(-2)*(336) = -672
\n" ); document.write( "(-3)*(224) = -672
\n" ); document.write( "(-4)*(168) = -672
\n" ); document.write( "(-6)*(112) = -672
\n" ); document.write( "(-7)*(96) = -672
\n" ); document.write( "(-8)*(84) = -672
\n" ); document.write( "(-12)*(56) = -672
\n" ); document.write( "(-14)*(48) = -672
\n" ); document.write( "(-16)*(42) = -672
\n" ); document.write( "(-21)*(32) = -672
\n" ); document.write( "(-24)*(28) = -672\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 $\"4\"$ :\r
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First Number	Second Number	Sum
1	-672	1+(-672)=-671
2	-336	2+(-336)=-334
3	-224	3+(-224)=-221
4	-168	4+(-168)=-164
6	-112	6+(-112)=-106
7	-96	7+(-96)=-89
8	-84	8+(-84)=-76
12	-56	12+(-56)=-44
14	-48	14+(-48)=-34
16	-42	16+(-42)=-26
21	-32	21+(-32)=-11
24	-28	24+(-28)=-4
-1	672	-1+672=671
-2	336	-2+336=334
-3	224	-3+224=221
-4	168	-4+168=164
-6	112	-6+112=106
-7	96	-7+96=89
-8	84	-8+84=76
-12	56	-12+56=44
-14	48	-14+48=34
-16	42	-16+42=26
-21	32	-21+32=11
-24	28	-24+28=4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-24\"$ and $\"28\"$ add to $\"4\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-24\"$ and $\"28\"$ both multiply to $\"-672\"$ and add to $\"4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"4r\"$ with $\"-24r%2B28r\"$ . Remember, $\"-24\"$ and $\"28\"$ add to $\"4\"$ . So this shows us that $\"-24r%2B28r=4r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"21r%5E2%2Bhighlight%28-24r%2B28r%29-32\"$ Replace the second term $\"4r\"$ with $\"-24r%2B28r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2821r%5E2-24r%29%2B%2828r-32%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3r%287r-8%29%2B%2828r-32%29\"$ Factor out the GCF $\"3r\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3r%287r-8%29%2B4%287r-8%29\"$ 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( " $\"%283r%2B4%29%287r-8%29\"$ Combine like terms. Or factor out the common term $\"7r-8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"21r%5E2%2B4r-32\"$ factors to $\"%283r%2B4%29%287r-8%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"21r%5E2%2B4r-32=%283r%2B4%29%287r-8%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%283r%2B4%29%287r-8%29\"$ to get $\"21r%5E2%2B4r-32\"$ 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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