The remainder theorem states that if you want to find f(x) of any function, you can synthetically divide by whatever "x" is, get the remainder and you will have the corresponding "y" value. Lets go through an example: (I have to assume you know synthetic division)
Say you had the function #f(x) = 2x^2+3x+7# and wanted to find f(3), rather than plugging in 3, you could SYNTHETICALLY DIVIDE by 3 to find the answer.
To find f(3) you would set up synthetic division so that your "x" value (3 in this case) is in a box on the left and you write out all the coefficients of the function on the right! (Don't forget to add place holders if necessary!)
Just as a quick review for synthetic division, you bring the first term down, multiply by number on the left, write your answer in the next column, then add and so on!
After the synthetic division, you notice that the remainder is 34...
If I were to find f(3) by substitution I would get:
#f(3) = 2(3)^2 +3(3) +7#
#= 18 + 9 +7#
Hopefully you notice that the remainder is the same as the answer you get when using substitution! THIS WILL ALWAYS BE THE CASE IF YOU DO THE SYNTHETIC DIVISION CORRECTLY! Hopefully you've understood this! :)