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

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

Start with the given expression

Factor out the GCF

Now let's focus on the inner expression

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Looking at we can see that the first term is and the last term is 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 Number	Second Number	Sum
1	-96	1+(-96)=-95
2	-48	2+(-48)=-46
3	-32	3+(-32)=-29
4	-24	4+(-24)=-20
6	-16	6+(-16)=-10
8	-12	8+(-12)=-4
-1	96	-1+96=95
-2	48	-2+48=46
-3	32	-3+32=29
-4	24	-4+24=20
-6	16	-6+16=10
-8	12	-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 , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

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So our expression goes from and factors further to

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Answer:

So factors to

So the two binomial factors are and
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