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multiplicity

How many times a zero repeats — controls whether a graph crosses or bounces there.

A zero c of a polynomial f has multiplicity k if (x-c)^k is the highest power of (x-c) dividing f. Odd multiplicity crosses the x-axis; even multiplicity bounces off it without crossing.

Examples

  • f(x) = (x+1)^3(x-2) has a zero at x=-1 with multiplicity 3 (crosses, flattening) and at x=2 with multiplicity 1 (crosses cleanly).

See also

x-intercept Factor Theorem
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