Factor Theorem

For the polynomial P ( x ) ,

1. If value of r is a zero of P ( x ) then x - r will be a factor of P ( x ) .

2. If x - r is a factor of P ( x ) then r will be a zero of P ( x ) .

The factor theorem leads to the below fact.

Fact 1

If P ( x ) is a polynomial of degree n & r is a zero of P (x ) then P ( x ) can be written in the given form.

                                     P ( x ) = ( x - r ) Q (x )

Where Q (x) refer to a polynomial with degree n -1 .Q (x) can be found by dividing P (x) by x - r .

There is one more fact that we have to get out of the way.

Fact 2

If P ( x ) = ( x - r ) Q ( x )& x = t is a zero of Q ( x ) then x = t will also be a zero of P ( x ) .

This fact is simple enough to check directly.  First, if x = t is a zero of Q ( x ) then we know that,

                                                                      Q (t) = 0

As that is what it means to be a zero.  Thus, if x = t is to be a zero of P (x) then all we have to do is illustrates that P (t) =0 and that's in fact quite simple. Following it is,

P (t) = (t - r) Q (t) = (t - r) (0) = 0 and hence x = t is a zero of P (x).