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. |
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Copyright © 1996-2015 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.
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Copyright © 1996-2015 Alexander Bogomolny
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