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.