Overview
This lesson explains how to find and interpret the multiplicity of roots (zeros) in a polynomial using synthetic division, factoring, and graph analysis.
Finding Rational Roots
- For f(x) = x³ - 3x - 2, possible rational roots are ±1 and ±2.
- Synthetic division shows x = 2 is a root.
- Dividing by (x - 2) gives x² + 2x + 1, which factors to (x + 1)(x + 1).
Understanding Multiplicity
- The zeros are x = 2 and x = -1 (repeated).
- A repeated root, like x = -1, has multiplicity 2.
- Multiplicity is shown by the exponent: (x + 1)².
Polynomial as Linear Factors
- f(x) = (x - 2)(x + 1)².
- x = 2 is a single root; x = -1 is a double root.
- Do not confuse the value of a root with its multiplicity.
Graph Behavior and Multiplicity
- At x = -1, the graph touches but does not cross the x-axis.
- Even multiplicity means the graph touches the axis at that root.
Key Terms & Definitions
- Root (Zero): Value where the polynomial equals zero.
- Multiplicity: Number of times a root occurs.
- Synthetic Division: Method for dividing by a linear factor.
- Linear Factor: Expression like (x - r), where r is a root.
Action Items / Next Steps
- Practice finding roots and their multiplicities.
- Observe how repeated roots affect polynomial graphs.