2014/043) For any polynomial P(x) show that P(a) - P(b) is divisible by a-b

 Proof:
Let p(x) =  t(n)x^n + t(n-1)x^{n-1} + ...  + t(0)
Then
 p(a) =  t(n)a^n + t(n-1)a^{n-1} + ...  + t(0)
 p(b) =  t(n)b^n + t(n-1)b^{n-1} + ...  + t(0)

So  p (a) – p(b) =  t(n)(a^n- b^n)  +t(n-1)(a^(n-1)-b^(n-1)  + ....  + t(1)(a-b)

As each of the a^k-b^k  is divisible by a- b so  p(a) – p(b) is divisible by a-b.

As a corollary


If p(x) has integer coefficients and P(0) and P(1) are odd it does not have any integer root.
This is so because P(even) – p(0) is even and P(odd) – p(1) is even so neither can be zero