AMC 10A 2025 (Problem 9)
Let . For how many real numbers does the graph of pass through the point ?
Let .
We don't need the exact roots, only how many real roots there are.
We don't need the exact roots, only how many real roots there are.
By Toolkit 91 — Intermediate Value Theorem (IVT):
If is continuous on and and have opposite signs (that is, ), then there exists some such that .
If is continuous on and and have opposite signs (that is, ), then there exists some such that .
Use the Fundamental Theorem of Algebra (real consequence):
A cubic polynomial with real coefficients has at most three real roots.
A cubic polynomial with real coefficients has at most three real roots.
Therefore, can have at most three real roots.
From Hints 6 and 7, there is one real root in each interval:
From Hints 9 and 10, has exactly three real roots.
(C)