Prove the Factor Theorem which states For any polynomial fun

Prove the Factor Theorem which states: For any polynomial function f(x), (x - k) is a factor of the polynomial if and only if f(k) = 0.

Solution

As the Remainder Theorem points out, if youdivide a polynomial p(x) by a factor x – k of that polynomial, then you will get a zero remainder. Let\'s look again at that Division Algorithm expression of the polynomial:

p(x) = (x – k)q(x) + r(x)

If x – k is indeed a factor of p(x), then the remainder after division by x – a will be zero. That is:

p(x) = (x – k)q(x)

In terms of the Remainder Theorem, this means that, if x – k is a factor of p(x), then the remainder, when we do synthetic division by
x = k will be zero.

The point of the Factor Theorem is the reverse of the Remainder Theorem: If you synthetic-divide a polynomial by x = a and get a zero remainder, then, not only is x = a a zero of the polynomial (courtesy of the Remainder Theorem) i.e f(k) = 0 , but x – a is also a factor of the polynomial (courtesy of the Factor Theorem).

Just as with the Remainder Theorem, the point here is not to do the long division of a given polynomial by a given factor. This Theorem isn\'t repeating what you already know, but is instead trying to make your life simpler. When faced with a Factor Theorem exercise, you will apply synthetic division and then check for a zero remainder.