A cubic has simple real root \(4\) and nonreal roots \(-1~+~2i\) and \(-1~-~2i\). Describe its \(x\)-intercepts, behavior at the real root, and how all roots account for the degree.

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Show a crossing at x=4 but claim the one real root makes the polynomial degree 1.

Show the graph touching and turning at x=4, treating the simple root as even multiplicity.

The only x-intercept is (4,0), and the graph crosses there because the root is simple. Roots -1±2i are nonreal; all three roots fill degree 3.

Show x-intercepts at 4, -1+2i, and -1-2i, treating the nonreal roots as real-axis points.

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Curriculum context

Course: Math III
Standard: N-CN.9
Category: Number and Quantity
Domain: The Complex Number System
Objective: Know the Fundamental Theorem of Algebra and connect it to polynomial roots.
Problem type: Connect root records to visible graph behavior