Problem

A quartic has simple real roots \(-1\) and \(2\) and nonreal roots \(3~+~i\) and \(3~-~i\). Describe its \(x\)-intercepts, behavior at each real root, and how all roots account for the degree.

Big Picture

What this problem is really about

Separate the root record into real and nonreal entries before interpreting the graph. Only real roots can become x-intercepts; odd multiplicity produces a crossing, while even multiplicity produces a touch, with higher multiplicity often flattening the graph nearby. Nonreal roots remain absent from the real-axis picture but still count, with multiplicity, toward the polynomial's degree.

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Preview only

Intercepts (-1,0) and (2,0); the graph crosses at both simple roots. Roots 3±i are nonreal and not x-intercepts; all four roots fill degree 4.

Show x-intercepts at -1, 2, 3+i, and 3-i, treating all four roots as real-axis points.

Show crossings at -1 and 2 but claim those two real roots make the polynomial degree 2.

Show the graph touching and turning at x=-1 and x=2, treating both simple roots as even multiplicity.

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A quartic has simple real roots \(-1\) and \(2\) and nonreal roots \(3~+~i\) and \(3~-~i\). Describe its \(x\)-intercepts, behavior at each real root, and how all roots account for the degree.

What this problem is really about

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