A polynomial has real root \(3\) with multiplicity \(3\). Describe the \(x\)-intercept, local graph behavior, and number of degree slots contributed by this root.

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Place an additional unspecified nonreal root on the real x-axis as a second intercept.

At x=3 the graph crosses without flattening and the root occupies only one degree slot.

The x-intercept is (3,0). Multiplicity 3 is odd, so the graph crosses with local flattening and the root occupies three degree slots.

At x=3 the graph touches and turns, the behavior for an even-multiplicity root.

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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