#Polynomial and Nonlinear Functions: Your Ultimate Guide 🚀

Hey there! Let's get you prepped for the AP SAT (Digital) Math section with a deep dive into polynomial and nonlinear functions. Think of this as your cheat sheet for tonight – quick, clear, and designed to make everything click. We'll cover the key concepts, throw in some memory aids, and make sure you're feeling confident for tomorrow. Let's do this!

#Polynomial Functions and Their Graphs

# Defining Polynomial Functions

• Polynomial functions are in the form: $$f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ$$ where:
  • a₀, a₁, ..., aₙ are real numbers.
  • n is a non-negative integer.
• Degree: The highest power of x (e.g., in x³ + 2x² - 5x + 1, the degree is 3).
• Leading Coefficient: The coefficient of the term with the highest degree (e.g., in 3x³ + 2x² - 5x + 1, it's 3).
• Classification:
  • Linear: Degree 1
  • Quadratic: Degree 2
  • Cubic: Degree 3
  • And so on...
• End Behavior: Determined by the degree and the sign of the leading coefficient.
  • Even degree, positive leading coefficient: Both ends go up. ⬆️⬆️
  • Odd degree, positive leading coefficient: Left end goes down, right end goes up. ⬇️⬆️
  • Negative leading coefficient: Flips the end behavior.

# Characteristics of Polynomial Graphs

• Turning Points: Maxima and minima. The number of turning points is at most one less than the degree.
• Multiplicity of Zeros (Roots):
  • Odd Multiplicity: Graph crosses the x-axis.
  • Even Multiplicity: Graph touches the x-axis but doesn't cross.
• Fundamental Theorem of Algebra: A polynomial of degree n has exactly n complex roots (counting multiplicity).

# Analyzing Polynomial Functions

• Factoring: Find the ...