Factor Theorem

Factor Theorem:

If p(x) is a polynomial x is divided by (x-a) and the remainder f (a) is equal to zero then (x-a) is an factor of p(x). We can factorize polynomial expressions of degree three or more using factor theorem and synthetic division. Let us see proof of Factor Theorem.Before we can prove the factor theorem I would like to mention about one of the important topics under this category that is trinomial factor calculator. us now move on to find the Proof of Factor Theorem

P(x) is divided by x-a,

Using remainder theorem,

R = p (a)

P(x) = (x-a).q(x) + p(a)

But p (a) = 0 is given.

Hence p(x) = (x-a).q(x)

(x-a) is the factor of p(x)

Conversely if x-a is a factor of p(x) then p(a)=0.

P(x) = (x-a).q(x) + R

If (x-a) is a factor then the remainder is zero (x-a divides p(x)

Exactly)

R=0

By remainder theorem, R = p (a)

Note:

1. If the sum of all coefficients in a polynomial including the constant term is zero, then x – 1 is a factor.

2. If the sum of the coefficients of the even powers together with the constant term is the same as the sum of the coefficients of odd powers, then x + 1 is a factor.Hope you like the above example of factor theorem.Please leave your comments, if you have any doubts.