Multiplicity
SAT glossary · Math
Definition
The number of times a factor repeats at a zero. Odd multiplicity means the graph crosses the x-axis there; even multiplicity means it touches and turns back. Multiplicities sum to the degree, so a degree-n polynomial may have fewer than n distinct zeros.
Multiplicity in plain English
Multiplicity tells you how many times the SAME factor repeats when you fully factor a polynomial and set it to find its zeros (x-intercepts). If a zero has odd multiplicity (it repeats 1, 3, 5... times), the graph actually CROSSES through the x-axis right there. If it has even multiplicity (2, 4, 6... times), the graph just TOUCHES the x-axis and bounces back up (or down) without ever crossing it. Since all the multiplicities of a polynomial's zeros have to add up to its total degree, a polynomial can easily have FEWER distinct zeros than its degree suggests — for example, a degree-4 polynomial might only touch the x-axis at 2 different spots if one of them repeats twice.
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