Looking at we can see that the first term is and the last term is where the coefficients are 8 and -1 respectively.
Now multiply the first coefficient 8 and the last coefficient -1 to get -8. Now what two numbers multiply to -8 and add to the middle coefficient -7? Let's list all of the factors of -8:
Factors of -8:
1,2,4,8
-1,-2,-4,-8 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -8
(1)*(-8)
(2)*(-4)
(-1)*(8)
(-2)*(4)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to -7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -7
First Number
Second Number
Sum
1
-8
1+(-8)=-7
2
-4
2+(-4)=-2
-1
8
-1+8=7
-2
4
-2+4=2
From this list we can see that 1 and -8 add up to -7 and multiply to -8
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
Now factor to get (use the difference of cubes formula) and factor to get (use the sum of cubes formula)
(