Program of Lessons

# 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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