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Root Multiplicity in Polynomials

Sep 14, 2025

Overview

This lesson covers the concept of root multiplicity in polynomial functions and how it affects the graph, especially at the x-axis.

Multiplicity of Roots/Zeros

  • The multiplicity of a root describes how many times a zero occurs in a polynomial function.
  • If the multiplicity is odd, the graph crosses the x-axis at that root.
  • If the multiplicity is even, the graph touches (is tangent to) the x-axis but does not cross at that root.
  • Multiplicity is determined by the exponent of the factor in the polynomial.

Example 1: y = (x + 2)²(x + 1)³(x - 1)⁴(x - 2)

  • Roots: x = -2 (multiplicity 2), x = -1 (multiplicity 3), x = 1 (multiplicity 4), x = 2 (multiplicity 1).
  • At x = -2: Multiplicity 2 (even), graph is tangent to x-axis.
  • At x = -1: Multiplicity 3 (odd), graph crosses x-axis.
  • At x = 1: Multiplicity 4 (even), graph is tangent to x-axis.
  • At x = 2: Multiplicity 1 (odd), graph crosses x-axis.

Example 2: y = (x + 3)(x - 2)²(x + 1)³

  • Roots: x = -3 (multiplicity 1), x = -1 (multiplicity 3), x = 2 (multiplicity 2).
  • At x = -3: Multiplicity 1 (odd), graph crosses x-axis.
  • At x = -1: Multiplicity 3 (odd), graph crosses x-axis.
  • At x = 2: Multiplicity 2 (even), graph is tangent to x-axis.

Key Terms & Definitions

  • Multiplicity — The number of times a root is repeated in a polynomial's factorization.
  • Root/Zero — A value of x where the polynomial is equal to zero.
  • Tangent to x-axis — The graph only touches, but does not cross, the x-axis at a root.
  • Crosses x-axis — The graph moves from positive to negative or vice versa through the x-axis at a root.

Action Items / Next Steps

  • Practice identifying roots and their multiplicities from polynomial equations.
  • For homework, graph polynomial functions and label root behaviors based on multiplicity.