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"
Do not worry about your difficulties in mathematics, I assure you that mine are greater
". Einstein, Albert (1879-1955)
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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 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.
Example:
( 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
.
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