SOLUTION: What are the two binomial factors of 6s^2+40s-64?

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Question 174855: What are the two binomial factors of 6s^2+40s-64?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

6s%5E2%2B40s-64 Start with the given expression


2%283s%5E2%2B20s-32%29 Factor out the GCF 2


Now let's focus on the inner expression 3s%5E2%2B20s-32




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Looking at 3s%5E2%2B20s-32 we can see that the first term is 3s%5E2 and the last term is -32 where the coefficients are 3 and -32 respectively.

Now multiply the first coefficient 3 and the last coefficient -32 to get -96. Now what two numbers multiply to -96 and add to the middle coefficient 20? Let's list all of the factors of -96:



Factors of -96:
1,2,3,4,6,8,12,16,24,32,48,96

-1,-2,-3,-4,-6,-8,-12,-16,-24,-32,-48,-96 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -96
(1)*(-96)
(2)*(-48)
(3)*(-32)
(4)*(-24)
(6)*(-16)
(8)*(-12)
(-1)*(96)
(-2)*(48)
(-3)*(32)
(-4)*(24)
(-6)*(16)
(-8)*(12)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20

First NumberSecond NumberSum
1-961+(-96)=-95
2-482+(-48)=-46
3-323+(-32)=-29
4-244+(-24)=-20
6-166+(-16)=-10
8-128+(-12)=-4
-196-1+96=95
-248-2+48=46
-332-3+32=29
-424-4+24=20
-616-6+16=10
-812-8+12=4



From this list we can see that -4 and 24 add up to 20 and multiply to -96


Now looking at the expression 3s%5E2%2B20s-32, replace 20s with -4s%2B24s (notice -4s%2B24s adds up to 20s. So it is equivalent to 20s)

3s%5E2%2Bhighlight%28-4s%2B24s%29%2B-32


Now let's factor 3s%5E2-4s%2B24s-32 by grouping:


%283s%5E2-4s%29%2B%2824s-32%29 Group like terms


s%283s-4%29%2B8%283s-4%29 Factor out the GCF of s out of the first group. Factor out the GCF of 8 out of the second group


%28s%2B8%29%283s-4%29 Since we have a common term of 3s-4, we can combine like terms

So 3s%5E2-4s%2B24s-32 factors to %28s%2B8%29%283s-4%29


So this also means that 3s%5E2%2B20s-32 factors to %28s%2B8%29%283s-4%29 (since 3s%5E2%2B20s-32 is equivalent to 3s%5E2-4s%2B24s-32)



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So our expression goes from 2%283s%5E2%2B20s-32%29 and factors further to 2%28s%2B8%29%283s-4%29


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Answer:

So 6s%5E2%2B40s-64 factors to 2%28s%2B8%29%283s-4%29

So the two binomial factors are s%2B8 and 3s-4