Polynomial Functions and Theorems

📋 Course Outline

1. Polynomial Function Structure
2. Recognizing Polynomial Functions
3. Rational Root Theorem
4. Remainder Theorem
5. Factor Theorem and Zeros

📖 1. Polynomial Function Structure

🔑 Key Concepts & Definitions

• Polynomial expression: A polynomial expression in x of degree n has the form $a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_1x+a_0$, where $a_n\ne0$, n is a nonnegative integer, and all coefficients are real numbers.

📝 Essential Points

📌 For a polynomial expression, the degree n is a nonnegative integer and the leading coefficient a_n is not zero.

📖 2. Recognizing Polynomial Functions

★ Must-know

• A polynomial function cannot contain:
  • a negative exponent
  • a fractional exponent
  • a variable in the denominator
  • a variable inside a radical
  • a decimal exponent

Further detail

• The expression $2x^{-2}+3x+5$ is not a polynomial because it contains a negative exponent.

• The expression $2x^{1/2}+3x+5$ is not a polynomial because it contains a fractional exponent.

• An expression with a variable in the denominator or inside a radical is not a polynomial function.

💡 Memory Hook

Polynomial terms use whole-number exponents; non-polynomial terms use forbidden exponents or variable denominators.

📖 3. Rational Root Theorem