2023 AMC 10B Problem 12

Below is the professionally curated solution for Problem 12 of the 2023 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 10B solutions, or check the answer key.

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Concepts:polynomial parity inequality

Difficulty rating: 1560

12.

When the roots of the polynomial $P(x) = \prod_{i=1}^{10} (x - i)^i$ are removed from the real number line, what remains is the union of $11$ disjoint open intervals. On how many of those intervals is $P(x)$ positive?

$3$

$7$

$6$

$4$

$5$

Solution:

For $x \gt 10,$ every factor $(x - i)^i$ is positive, so $P(x) \gt 0.$ Now move left. Crossing $x = i$ flips the sign only when $i$ is odd, that is at $i = 9, 7, 5, 3, 1.$ So the eleven intervals, right to left, carry signs $+, +, -, -, +, +, -, -, +, +, -.$ Six are positive. Therefore, the answer is C.

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