Top 10 Things to Know About Polynomial Functions

1. What is a Polynomial Function?

• Definition: Sum of terms where each term is a power of x multiplied by a coefficient.
• Structure: General equation involves terms like a_n*x^n + a_(n-1)*x^(n-1) + ... + a_0.
• Components:
  • x: Variable
  • Coefficients (a): Can be real or complex numbers.
  • Exponents: Non-negative integers, decreasing to 0.
• Degree: Determined by the highest exponent (e.g., degree 4 if highest exponent is 4).
• Leading Coefficient: Coefficient of the highest power of x.
• Continuity: Polynomial functions are continuous with no domain restrictions.
• Graphical Traits: Y-intercept is the constant term, and graph is continuous.
• Finite Differences: nth differences are constant for a degree n polynomial._

2. End Behavior

• Quadrant System:
  • Quadrant 1: Top right
  • Quadrant 2: Top left
  • Quadrant 3: Bottom left
  • Quadrant 4: Bottom right
• Types of End Behaviors:
  • Even degree with positive leading coefficient: Quadrant 2 to 1
  • Even degree with negative leading coefficient: Quadrant 3 to 4
  • Odd degree with positive leading coefficient: Quadrant 3 to 1
  • Odd degree with negative leading coefficient: Quadrant 2 to 4
• Patterns: Odd degree ends in opposite directions; even degree ends in the same direction.

3. X Intercepts and Turning Points

• X Intercepts: At most n x-intercepts for a degree n polynomial.
• Turning Points: At most n-1 turning points.
• Odd Degree: At least 1 x-intercept.
• Even Degree: Can have zero x-intercepts.
• Examples: Degree 5 polynomial has at most 4 turning points.

4. Even and Odd Symmetry

• Even Functions:
  • Symmetry over the y-axis.
  • Algebraic property: f(x) = f(-x).
• Odd Functions:
  • Rotational symmetry about the origin.
  • Algebraic property: f(-x) = -f(x).
• Examples: Identifying even or odd functions based on symmetry and degree.

5. Factor Form Equation

• Structure: Product of factors.
• Roots: Parameters r are the roots.
  • Real roots correspond to x-intercepts.
• Degree: Sum of the orders of roots.
• Shape at X Intercepts:
  • Order 1: Linear
  • Order 2: Parabolic (bounces)
  • Order 3: S-shaped

6. Finding the Equation from a Graph

• Steps:
  • Identify x-intercepts and their orders.
  • Construct factors from x-intercepts.
  • Find constant factor using a known point (e.g., y-intercept).

7. Synthetic and Long Division

• Long Division: Traditional method to divide polynomials.
  • Set up is similar to numerical division.
• Synthetic Division: Shortcut method.
  • Only works for linear divisors of the form x + b.

8. Remainder, Zero, and Factor Theorems

• Remainder Theorem: Remainder is f(b) when P(x) is divided by x-b.
• Factor Theorem: x-b is a factor if f(b) = 0.
• Integral Zero Theorem: Integer roots are factors of the constant term.
• Rational Zero Theorem: Possible rational roots are factors of the constant term divided by factors of the leading coefficient.

9. Solving Polynomial Equations

• Finding Zeros:
  • Use Rational Zero Theorem for possible roots.
  • Test possible roots to find actual zeros.
• Factoring: Break down polynomials to solve equations.

10. Solving Polynomial Inequalities

• Steps:
  • Set equation to zero.
  • Factor polynomial.
  • Sketch graph to determine intervals where polynomial is below/above the x-axis.
• Solutions: State intervals of x where conditions are met (e.g., f(x) ≤ 0).