Divisibility of binomials

As consequences from Bezout’s theorem  the next criteria of divisibility  of binomials are valid:

1)	A difference of  identical powers of two numbers is divided without a remainder by a difference of these two numbers, i.e.   xm –  am    is divided by   x – a.
2)	A difference of  identical even powers of two numbers is divided without a remainder both by a difference and by a sum of these two numbers, i.e. if m – an even number, then the binomial xm –  am   is divided both by   x – a  and by   x + a. A difference of  identical odd powers of two numbers isn’t divided by a sum of these two numbers.
3)	A sum of  identical powers of two numbers is never divided by a difference of these two numbers.
4)	A sum of  identical odd powers of two numbers is divided without a remainder by a sum of these two numbers.
5)	A sum of  identical even powers of two numbers is never divided both by difference and by a sum of these two numbers.

E x a m p l e s :  ( x² – a² ) : ( x – a ) = x + a ;

                         ( x³ – a³ ) : ( x – a ) = x² + a x+ a² ;

                         ( x⁵ – a⁵ ) : ( x – a ) = x⁴ + a x³ + a² x² + a³ x + a⁴ .