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
.