Factor theorem

Introduction to 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 expression of degree three or more using factor theorem and synthetic division. Let us see proof of Factor Theorem.
Proof of Factor Theorem

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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.

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