Looking at we can see that the first term is and the last term is where the coefficients are 1 and 16 respectively.
Now multiply the first coefficient 1 and the last coefficient 16 to get 16. Now what two numbers multiply to 16 and add to the middle coefficient -8? Let's list all of the factors of 16:
Factors of 16:
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 16
1*16
2*8
4*4
(-1)*(-16)
(-2)*(-8)
(-4)*(-4)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -8? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -8
First Number
Second Number
Sum
1
16
1+16=17
2
8
2+8=10
4
4
4+4=8
-1
-16
-1+(-16)=-17
-2
-8
-2+(-8)=-10
-4
-4
-4+(-4)=-8
From this list we can see that -4 and -4 add up to -8 and multiply to 16
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 )
note: is equivalent to since the term occurs twice. So also factors to
(