Polynomial Functions

Multiplicity of Zeros

Relating a zero's multiplicity to the graph's behavior at that x-intercept.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

End Behavior & Zeros

Not Every Zero Crosses the Axis

f(x) = (x − 2)(x − 3) has a zero at x = 2 that the graph crosses cleanly. Before reading on, predict what happens instead at the zero of g(x) = (x − 2)²(x − 3) — does the graph still cross at x = 2, or could a repeated factor make it behave differently?

Definition — Multiplicity

The number of times a factor (x − a) appears in a polynomial's factored form is the multiplicity of the zero at x = a. A zero with odd multiplicity crosses the x-axis there; a zero with even multiplicity touches the x-axis and turns back without crossing.

Near x = 2, the factor (x − 2) is small, and it's this small factor's power that controls the graph's local behavior — the other factors stay roughly constant nearby and don't affect the sign change. A small number raised to an odd power keeps the same sign as the number itself (negative stays negative), so the function's sign genuinely flips as x crosses 2, meaning the graph crosses. A small number raised to an even power is always positive, so the function's sign doesn't flip — the graph touches x = 2 and turns back, like a parabola's own vertex behavior locally.

Worked Example — Identifying Multiplicity and Predicting Graph Behavior

For f(x) = (x + 1)³(x − 4)², describe the behavior at each zero. At x = −1, the factor (x + 1) has multiplicity 3 (odd), so the graph crosses the x-axis there. At x = 4, the factor (x − 4) has multiplicity 2 (even), so the graph touches the x-axis and turns back without crossing.

Worked Example — Writing a Polynomial from Given Zero Behavior

Write a polynomial with a zero at x = 0 where the graph crosses, and a zero at x = 3 where the graph only touches. A crossing zero needs odd multiplicity (use multiplicity 1); a touching zero needs even multiplicity (use multiplicity 2): f(x) = x(x − 3)².

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

A multiplicity of exactly 1 crosses like an ordinary line through that point; a multiplicity of 2 flattens and touches like a parabola's vertex; a multiplicity of 3 crosses but flattens noticeably as it passes through, like a cubic's inflection point — higher multiplicities flatten the crossing or touch even further.

Common Mistakes

Assuming every zero crosses the x-axis, regardless of the factor's power.
Check the multiplicity of each zero individually — an even multiplicity means the graph touches and turns back rather than crossing through.
Miscounting a factor's multiplicity, such as reading (x − 2)² as multiplicity 1 instead of 2.
Multiplicity is the exponent on the factor itself — (x − 2)² means the factor (x − 2) appears twice, giving multiplicity 2, not 1.

Key Takeaways

A zero's multiplicity is how many times its factor appears in the polynomial's factored form.
Odd multiplicity crosses the x-axis; even multiplicity touches and turns back.
Near a zero, the corresponding small factor controls the local graph behavior, since the other factors stay roughly constant nearby.

Summary

Multiplicity refines what a zero actually looks like on the graph. The next lesson turns from graphing polynomials to dividing them, extending long division from numbers to polynomial expressions.

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