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).