I will post a challenging math problem each day. The level of difficulty will vary, but most problems should not require any specialized knowledge beyond calculus. Problems are also posted on Twitter using the hashtag #mathpotd.
Tuesday, September 29, 2009
Polynomial remainder
Find the remainder when x^1000 is divided by x^4 - x^2 + 1.
The answer is x^2 - 1.
ReplyDeletex^1000 = (x^4)(x^996 - 1) + x^4
= (x^4)(x^996 - 1) + (x^4 - x^2 + 1) + (x^2 - 1).
But x^996 - 1 is divisible by x^6 + 1, and
x^6 + 1 = (x^4 - x^2 + 1)(x^2 + 1).
Therefore the remainder is x^2 - 1.