Zeros, Factors, and Multiplicity

A zero is where f(x) = 0, and it comes from a factor (x - c) in the polynomial. Multiplicity is how many times that factor repeats. If multiplicity is odd, like 1 or 3, the graph crosses straight through the axis, because the sign of f(x) actually switches from one side of c to the other. If multiplicity is even, like 2 or 4, the graph touches the axis and turns back around, because the factor squared (or to the fourth power) stays non-negative on both sides, so f(x) doesn't change sign.

Near a zero of multiplicity 1, the crossing looks roughly like a straight line. At multiplicity 3 or higher odd numbers, the graph still crosses, but it flattens out and almost pauses at the axis before continuing through, giving it an S-shaped wiggle. At even multiplicities, raising the power from 2 to 4 doesn't change the bounce behavior itself, but it flattens the curve more tightly against the axis before it turns, because higher powers shrink faster near zero.

This connects directly to building and reading polynomials in factored form: once you know every zero and its multiplicity, you can sketch the whole shape of the graph without plotting points, and you can go the other way too, writing a factored polynomial from a description of its zeros. It also sets up end behavior, since after accounting for all the local bounces and crossings, the leading term still decides where the graph goes far from the origin.