Overview

This topic focuses on the structure, behavior, and analysis of polynomial and rational functions. You'll learn how to find zeros, analyze end behavior, factor polynomials, and graph rational functions with asymptotes and discontinuities. These skills are essential for understanding advanced algebraic models.

Key Concepts and Structures

• Polynomial Functions: Functions written in the form f(x) = a_nx^n + ... + a_1x + a_0. Degree determines end behavior.
• Degree and Leading Coefficient: These determine the graph’s long-term direction (positive/negative, even/odd).
• Zeros of a Polynomial: Solutions to f(x) = 0. Also called roots or x-intercepts.
• Multiplicity: Describes repeated roots:
  • Odd multiplicity: graph crosses the x-axis
  • Even multiplicity: graph touches and turns at the x-axis
• Factoring: Use methods like grouping, synthetic division, or special patterns (difference of squares).
• End Behavior: Analyze leading term to determine how the function behaves as x → ±∞.
• Rational Functions: Quotients of polynomials. Example: f(x) = (x + 2)/(x^2 - 9).
• Asymptotes:
  • Vertical: Occur at values that make the denominator zero (unless canceled).
  • Horizontal: Compare degrees of numerator and denominator.
  • Oblique (Slant): When numerator’s degree is one more than denominator’s.
• Discontinuities: Points where the function is undefined (holes or vertical asymptotes).

Step-by-Step Example

Problem: Find the zeros and end behavior of f(x) = x(x - 1)(x + 2)

Step 1: Identify zeros from factors:
x = 0, x = 1, x = -2

Step 2: Degree is 3 (odd), leading coefficient is positive → graph falls left, rises right

Final Answer: Zeros at x = -2, 0, 1. End behavior: f(x) → -∞ as x → -∞; f(x) → ∞ as x → ∞.

Quick Tip