SOLUTION: I'm having trouble with this one in particular FACTOR COMPLETELY 21r^2+4r-32 I would like to know which factoring technique to use here. Thx, Brett

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I'm having trouble with this one in particular FACTOR COMPLETELY 21r^2+4r-32 I would like to know which factoring technique to use here. Thx, Brett      Log On


   

Question 537148: I'm having trouble with this one in particular
FACTOR COMPLETELY
21r^2+4r-32
I would like to know which factoring technique to use here. Thx, Brett
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

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.


Now multiply the first coefficient 21 by the last term -32 to get %2821%29%28-32%29=-672.


Now the question is: what two whole numbers multiply to -672 (the previous product) and add to the second coefficient 4?


To find these two numbers, we need to list all of the factors of -672 (the previous product).


Factors of -672:
1,2,3,4,6,7,8,12,14,16,21,24,28,32,42,48,56,84,96,112,168,224,336,672
-1,-2,-3,-4,-6,-7,-8,-12,-14,-16,-21,-24,-28,-32,-42,-48,-56,-84,-96,-112,-168,-224,-336,-672


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -672.
1*(-672) = -672
2*(-336) = -672
3*(-224) = -672
4*(-168) = -672
6*(-112) = -672
7*(-96) = -672
8*(-84) = -672
12*(-56) = -672
14*(-48) = -672
16*(-42) = -672
21*(-32) = -672
24*(-28) = -672
(-1)*(672) = -672
(-2)*(336) = -672
(-3)*(224) = -672
(-4)*(168) = -672
(-6)*(112) = -672
(-7)*(96) = -672
(-8)*(84) = -672
(-12)*(56) = -672
(-14)*(48) = -672
(-16)*(42) = -672
(-21)*(32) = -672
(-24)*(28) = -672

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 4:


First NumberSecond NumberSum
1-6721+(-672)=-671
2-3362+(-336)=-334
3-2243+(-224)=-221
4-1684+(-168)=-164
6-1126+(-112)=-106
7-967+(-96)=-89
8-848+(-84)=-76
12-5612+(-56)=-44
14-4814+(-48)=-34
16-4216+(-42)=-26
21-3221+(-32)=-11
24-2824+(-28)=-4
-1672-1+672=671
-2336-2+336=334
-3224-3+224=221
-4168-4+168=164
-6112-6+112=106
-796-7+96=89
-884-8+84=76
-1256-12+56=44
-1448-14+48=34
-1642-16+42=26
-2132-21+32=11
-2428-24+28=4



From the table, we can see that the two numbers -24 and 28 add to 4 (the middle coefficient).


So the two numbers -24 and 28 both multiply to -672 and add to 4


Now replace the middle term 4r with -24r%2B28r. Remember, -24 and 28 add to 4. So this shows us that -24r%2B28r=4r.


21r%5E2%2Bhighlight%28-24r%2B28r%29-32 Replace the second term 4r with -24r%2B28r.


%2821r%5E2-24r%29%2B%2828r-32%29 Group the terms into two pairs.


3r%287r-8%29%2B%2828r-32%29 Factor out the GCF 3r from the first group.


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.


%283r%2B4%29%287r-8%29 Combine like terms. Or factor out the common term 7r-8


===============================================================


Answer:


So 21r%5E2%2B4r-32 factors to %283r%2B4%29%287r-8%29.


In other words, 21r%5E2%2B4r-32=%283r%2B4%29%287r-8%29.


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).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim