Zeros, Factors, and Graphs of Polynomial Functions

Introduction

• Focus on the connections among real rational zeros, factors, and graphs of polynomial functions.

Definition of Key Terms

• Polynomial Function of Degree N: Given in a certain form where P(C) = 0.
  • C is a Zero/Root: If substituting C into the function results in zero, C is a zero or root of P(x).
  • Solution to P(x) = 0: X = C is a solution to the polynomial equation.
  • Factor: The quantity X - C divides evenly into P(x).
  • X-Intercept: Point (C, 0) on the graph of P(x).

Example 1: Degree Four Polynomial Function

• Given: Zeros/roots at:
  • X = 0 (Multiplicity 1)
  • X = 3 (Multiplicity 1)
  • X = -1 (Multiplicity 2)
• Leading Coefficient: -2

Finding X-Intercepts

• X-Intercepts occur at real zeros:
  • At X = 0: (0, 0)
  • At X = 3: (3, 0)
  • At X = -1: (-1, 0)

Writing in Factored Form

• Polynomial Function:
  • P(x) = -2(x - 0)(x - 3)(x + 1)^2
  • Simplified: P(x) = -2x(x - 3)(x + 1)^2

Graphing the Polynomial Function

• Three X-Intercepts present.
• Multiplicity Effects:
  • Odd multiplicity: Graph crosses X-axis (X = 0, X = 3).
  • Even multiplicity: Graph touches X-axis and turns (X = -1).

Example 2: Degree Three Polynomial Function

• Given: Graph of a degree three polynomial.

Finding X-Intercepts

• X-Intercepts are at:
  • X = -2 (crosses)
  • X = 1 (touches)

Finding Real Zeros and Their Multiplicities

• X = -2: Multiplicity 1 (crosses the axis).
• X = 1: Multiplicity 2 (touches the axis).

Equation of the Polynomial Function

• P(x) = A(x + 2)(x - 1)^2
  • Using point (2, -2) to find A:
    1. Substitute: -2 = A(2 + 2)(2 - 1)^2
    2. Solve for A: A = -1/2
  • Final Form: P(x) = -1/2(x + 2)(x - 1)^2

Example 3: Given Polynomial Equation

• Polynomial Function for verification:
  • P(x) = 0.25(x - 1)(x - 3)(x + 4)(x + 2)^2

Verifying Solutions

• X-Intercepts: Locations where P(x) = 0:
  • X = 1, 3, -4, -2
  • Corresponding Points: (1, 0), (3, 0), (-4, 0), (-2, 0)

Zeros of the Polynomial Function

• Zeros (including multiplicity):
  • X = 1, X = 3 (Multiplicity 1)
  • X = -4 (Multiplicity 1)
  • X = -2 (Multiplicity 2)

Conclusion

• X-Intercepts, Zeros, and Solutions are closely related.
• Visualizing the graph helps in understanding the relationships and properties of polynomial functions.