They are just powers of 3
Algebra
Level
3
What is the remainder when the polynomial is divided by ?
NMTC problem.
None of these choices
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1 solution
We know , y − 1 divides y n − 1 for all positive integers n . Replacing y by x 2 we see that, x 2 divides x 2 n − 1 for all positive integers n .
Therefore , x 2 4 3 + x 8 1 + x 2 7 + x 9 + x 3 + x
= x 2 4 3 − x + x 8 1 − x + x 2 7 − x + x 9 − x 3 − x + 5 x + x
= x ( x 2 4 2 − 1 ) + x ( x 8 0 − 1 ) + x ( x 2 6 − 1 ) + x ( x 8 − 1 ) + x ( x 2 − 1 ) + 6 x
Now , ( x 2 4 2 − 1 ) , ( x 8 0 − 1 ) , ( x 2 6 − 1 ) , ( x 8 − 1 ) and x ( x 2 − 1 ) are exactly divisible by ( x 2 − 1 ) .
Therefore , 6 x is the remainder.