# 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. x m – a m < 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 x m – a m is divided both ( 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 :  ( x 2 – a 2 ) : ( x – a ) = x + a ;

( x 3 – a 3 ) : ( x – a ) = x 2 + a x+ a 2 ;

( x 5 – a 5 ) : ( x – a ) = x 4 + a x 3 + a 2 x 2 + a 3 x + a 4 .