write diffrence between reminnder theorom and factor theorom with example?
Answers
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) = x^2 - 2x + 1
Using the remainder theorem
We can plug in 3 into f(x).
f(3) = 3^2 - 2(3) + 1
f(3) = 9 - 6 + 1
f(3) = 4
Therefore, by the remainder theorem, the remainder when you divide x^2 - 2x + 1by x-3 is 4.
You can also apply this in reverse. Divide x^2 - 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) = x^2 - 2x + 1 equals 0 when x=1.
This tells us that (x-1) is a factor of x^2 - 2x + 1.
We can also apply the factor theorem in reverse:
We can factor x^2 - 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.