Math Gold Medalist

Lor

2023 AMC 10B 

Problem 12


When the roots of the polynomial

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

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? (2 different versions of this, some people had answer choices C and D flipped on the test)

$\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$

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   Solution