Notes on Zeros, Factors, and Graphs of Polynomial Functions

Introduction

• Topic: Zeros, factors, and graphs of polynomial functions.
• Aim: Understand the connection among real rational zeros, factors, and graphs of polynomial functions.

Definitions

• Polynomial Function of Degree N: Form: P(x) where degree is N.
• Zero/Root: A real number C such that P(C) = 0.
  • Also known as a solution to the equation P(x) = 0.
• Factor: The quantity (x - C) is a factor of P(x).
  • Means it divides evenly into the polynomial function.
• X-Intercept: The point (C, 0) is the x-intercept of the graph of P(x).

Example 1: Degree Four Polynomial Function

• Zeros/Roots:
  • x = 0 (multiplicity 1)
  • x = 3 (multiplicity 1)
  • x = -1 (multiplicity 2)
• Leading Coefficient: -2

Finding X-Intercepts

• X-intercepts correspond to the real zeros:
  • (0, 0)
  • (3, 0)
  • (-1, 0)

Writing in Factored Form

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

Graphing the Polynomial Function

• Multiplicity Effects:
  • Odd multiplicity (1): Graph crosses x-axis.
  • Even multiplicity (2): Graph touches x-axis and turns.
• At x = -1: Touches (even).
• At x = 0 and x = 3: Crosses (odd).

Example 2: Degree Three Polynomial Function

• Finding X-Intercepts:
  • X-intercepts:
    • (-2, 0) (crosses)
    • (1, 0) (touches)

Finding Real Zeros with Multiplicities

• Zeros:
  • x = -2 (multiplicity 1)
  • x = 1 (multiplicity 2, even)

Finding the Polynomial Function

• Form: P(x) = A(x + 2)(x - 1)^2
• Use point (2, -2) to find A:
  • -2 = A(4)(1)
  • A = -1/2
• Therefore, P(x) = -1/2(x + 2)(x - 1)^2

Example 3: Given Polynomial Equation

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

Finding X-Intercepts

• X-intercepts occur at:
  • x = 1, 3, -4, -2
• Coordinates:
  • (1, 0), (3, 0), (-4, 0), (-2, 0)

Zeros with Multiplicities

• Zeros:
  • x = 1, x = 3 (multiplicity 1)
  • x = -4 (multiplicity 1)
  • x = -2 (multiplicity 2)

Conclusion

• The relationships among x-intercepts, zeros, and solutions are closely related in polynomial functions.