Polynomial Functions Overview

Turning Points and Polynomial Degrees

• A polynomial function of degree ( n ) has at most ( n-1 ) turning points.
• Odd degree functions can have an even number of turning points.
• Even degree functions can have an odd number of turning points.

Polynomial Function Types

• Degree 0: Constant
• Degree 1: Linear (Name: Linear)
• Degree 2: Quadratic (Name: Quadratic)
• Degree 3: Cubic (Name: Cubic)
• Degree 4: Quartic (Name: Quartic)
• Degree 5: Quintic (Name: Quintic)

Quadratic Functions

• Standard form: ( ax^2 + bx + c )
• Degree of a quadratic: 2
• Leading coefficient: ( a )

Classifying Polynomials by Terms

• Monomial: 1 term
• Binomial: 2 terms
• Trinomial: 3 terms
• Polynomial: More than 3 terms

End Behavior of Polynomials

• Focus on the leading term's degree and coefficient to determine ( y ) values as ( x \to \ ) or ( x \to - ).
  • Even degree + positive leading coefficient: Both ends go to ( + )
  • Even degree + negative leading coefficient: Both ends go to ( - )
  • Odd degree + positive leading coefficient: Left to right goes from ( - ) to ( + )
  • Odd degree + negative leading coefficient: Left to right goes from ( + ) to ( - )

Polynomial Properties

• A polynomial in one variable can be expressed as ( an x^n + a{n-1} x^{n-1} + … + a2 x^2 + a1 x + a_0 ).
• The degree of a polynomial is the highest exponent, and the leading coefficient is the coefficient of that term.
• Zeros of a function are values ( x ) for which ( f(x) = 0 ).
• According to the Fundamental Theorem of Algebra, every polynomial of degree ( n > 0 ) has at least one root in the complex numbers.