Question

What is the difference between Remainder theorem and Factor theorem ? Give examples.

Answer 1

The remainder theorem tells us that for any polynomial f(x), if you divide it by the binomial x−a, the remainder is equal to the value of f(a).

The factor theorem tells us that if a is a zero of a polynomial f(x), then (x−a) is a factor of f(x), and vice-versa.

For example, let's consider the polynomial

f(x)=x2−2x+1

Using the remainder theorem

We can plug in 3 into f(x).

f(3)=32−2(3)+1
f(3)=9−6+1
f(3)=4

Therefore, by the remainder theorem, the remainder when you divide x2−2x+1 by x−3 is 4.

You can also apply this in reverse. Dividex2−2x+1 by x−3, and the remainder you get is the value of f(3).

Using the factor theorem

The quadratic polynomial f(x)=x2−2x+1equals 0 when x=1.
This tells us that (x−1) is a factor of x2−2x+1.

We can also apply the factor theorem in reverse:

We can factor x2−2x+1 into (x−1)2, therefore 1 is a zero of f(x).

Basically, the remainder theorem links the remainder of division by a binomial with the value of a function at a point, while the factor theorem links the factors of a polynomial to its zeros.