Theory:

Factor Theorem: If p(x) is a polynomial of degree \(n > 1\) and \(a\) is any real number, then:
 
(i)x – a is a factor of p(x), if p(a) = 0, and
 
(ii)p(a) = 0, if  is a factor of p(x).
 
This is an extension to the remainder theorem where the remainder is 0, which is p(a) = 0.
Example:
Examine whether x + 2  is factor of p(x)= x3 + 3x2 + 5x + 6.
  
To find x + 2 is a factor of p(x)= x3 + 3x2 + 5x + 6 or not.
 
As, result of factor theorem x + 2 is factor of \(p(x)\) if p(−2) = 0.
 
The zero of x + 2 is \(x+2 = 0\). That is, \(x = -2\).
 
p(–2) = (–2)3+3(–2)2+5(–2) +6p(2)=8+1210+6p(2)=18+18p(2)=0
 
Therefore, x + 2 is factor of p(x)= x3 + 3x2 + 5x + 6.
Important!
When x – a may be a factor of p(x) then p(a) = 0.