FACTOR THEOREM, REMAINDER THEOREM, AND RATIONAL ROOT THEOREM

FACTOR THEOREM
If f(x) is a polynomial and f(a) = 0, then (x–a) is a factor of f(x).

REMAINDER THEOREM
If f(x) is a polynomial and the remainder of dividing that polynomial by (x-a) is b, then f(a)=b.

RATIONAL ROOT THEOREM
Provides complete list of possible rational roots to eliminate the fuss of plugging and chugging through many numbers during synthetic division

     - It is all the factors of the polynomials last coefficient DIVIDED BY all the factors of the leading coefficient



In this case, it is all of the factors of c divided by all of the factors of a




APPLYING THESE THEOREMS...

Let's see what happens when we plug (x-1) into the equation below...

 






Notice when we plug (x-1) into the polynomial above, the remainder is zero.

In this case, the remainder theorem says that f(1)=0 and using the factor theorem we know that (x-1) must be a factor of the polynomial because f(1)=0.

HELPFUL VIDEO ON REMAINDER THEOREM AND FACTOR THEOREM