SOLUTION: Factor. 3w^2 - 2w - 16

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Question 384021: Factor.
3w^2 - 2w - 16
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at the expression 3w%5E2-2w-16, we can see that the first coefficient is 3, the second coefficient is -2, and the last term is -16.


Now multiply the first coefficient 3 by the last term -16 to get %283%29%28-16%29=-48.


Now the question is: what two whole numbers multiply to -48 (the previous product) and add to the second coefficient -2?


To find these two numbers, we need to list all of the factors of -48 (the previous product).


Factors of -48:
1,2,3,4,6,8,12,16,24,48
-1,-2,-3,-4,-6,-8,-12,-16,-24,-48


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


These factors pair up and multiply to -48.
1*(-48) = -48
2*(-24) = -48
3*(-16) = -48
4*(-12) = -48
6*(-8) = -48
(-1)*(48) = -48
(-2)*(24) = -48
(-3)*(16) = -48
(-4)*(12) = -48
(-6)*(8) = -48

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -2:


First NumberSecond NumberSum
1-481+(-48)=-47
2-242+(-24)=-22
3-163+(-16)=-13
4-124+(-12)=-8
6-86+(-8)=-2
-148-1+48=47
-224-2+24=22
-316-3+16=13
-412-4+12=8
-68-6+8=2



From the table, we can see that the two numbers 6 and -8 add to -2 (the middle coefficient).


So the two numbers 6 and -8 both multiply to -48 and add to -2


Now replace the middle term -2w with 6w-8w. Remember, 6 and -8 add to -2. So this shows us that 6w-8w=-2w.


3w%5E2%2Bhighlight%286w-8w%29-16 Replace the second term -2w with 6w-8w.


%283w%5E2%2B6w%29%2B%28-8w-16%29 Group the terms into two pairs.


3w%28w%2B2%29%2B%28-8w-16%29 Factor out the GCF 3w from the first group.


3w%28w%2B2%29-8%28w%2B2%29 Factor out 8 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.


%283w-8%29%28w%2B2%29 Combine like terms. Or factor out the common term w%2B2


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Answer:


So 3w%5E2-2w-16 factors to %283w-8%29%28w%2B2%29.


In other words, 3w%5E2-2w-16=%283w-8%29%28w%2B2%29.


Note: you can check the answer by expanding %283w-8%29%28w%2B2%29 to get 3w%5E2-2w-16 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, feel free to check out my tutoring website

Jim