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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. 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 ( 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 .

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