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Looking at we can see that the first term is and the last term is where the coefficients are 1 and 81 respectively.
Now multiply the first coefficient 1 and the last coefficient 81 to get 81. Now what two numbers multiply to 81 and add to the middle coefficient 18? Let's list all of the factors of 81:
Factors of 81:
1,3,9,27
-1,-3,-9,-27 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 81
1*81
3*27
9*9
(-1)*(-81)
(-3)*(-27)
(-9)*(-9)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 18? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 18
First Number
Second Number
Sum
1
81
1+81=82
3
27
3+27=30
9
9
9+9=18
-1
-81
-1+(-81)=-82
-3
-27
-3+(-27)=-30
-9
-9
-9+(-9)=-18
From this list we can see that 9 and 9 add up to 18 and multiply to 81
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
(