When the roots of the polynomial <img alt="\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot \cdot (x-10)^{10}\]" height="20" src="https://latex.artofproblemsolving.com/c/0/c/c0c2500605267cf44fde5a74bcb0986b11626bc5.png" width="365" /> are removed from the number line, what remains is the union of <img alt="$11$" height="12" src="https://latex.artofproblemsolving.com/c/6/8/c6878713578626763c38433b3f4c8c2205ad0c15.png" width="17" /> disjoint open intervals. On how many of these intervals is <img alt="$P(x)$" height="18" src="https://latex.artofproblemsolving.com/6/e/f/6efaaf74e576202fad965134bc49c6a34aac4c30.png" width="37" /> positive?

Question No: 416

When the roots of the polynomial

$P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot \cdot (x-10)^{10}$

are removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive?

Editorial

Tags: Algebra

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