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
}
.