Third Millennium International Mathematical Olympiad 2009
Grade 11-12
Problem 6

The derivatives of two polynomials P(x) and Q(x) are divisible by x2009. Prove that the same is true of the derivative of their product.

Solution

|Up| |Contact| |Front page| |Contents|

Copyright © 1996-2018 Alexander Bogomolny

Let P'(x) = p(x) x²⁰⁰⁹ and Q'(x) = q(x) x²⁰⁰⁹. Then

	(PQ)'	= P'Q + PQ'
		= px2009 Q + Pq x2009
		= x2009(pQ + Pq).

Which proves the assertion.

|Up| |Contact| |Front page| |Contents|

Copyright © 1996-2018 Alexander Bogomolny