Crazy polynomial remainder!
Let a polynomial P( ) = + + + + + 1. If P( ) is divided by P( ), find its remainder.
The answer is 6.
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1 solution
P ( x ) = x 5 + x 4 + x 3 + x 2 + x + 1
P ( x 6 ) = x 3 0 + x 2 4 + x 1 8 + x 1 2 + x 6 + 1
P ( x 6 ) = ( x 3 0 − 1 ) + ( x 2 4 − 1 ) + ( x 1 8 − 1 ) + ( x 1 2 − 1 ) + ( x 6 − 1 ) + 6
Notice that ( x 3 0 − 1 ) is divisible by ( x 6 − 1 ) .
This is also true for ( x 2 4 − 1 ) , ( x 1 8 − 1 ) , ( x 1 2 − 1 ) and ( x 6 − 1 ) .
Also notice that ( x 6 − 1 ) is divisible by P ( x ) .
Hence, the remainder when P ( x 6 ) is divided by P ( x ) is 6 .