Overview
This lecture explains polynomial functions, focusing on how to identify the degree, leading coefficient, and constant term, as well as how to rewrite polynomials in standard form.
Definition and Structure of Polynomial Functions
- A polynomial function has the form: ( p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 ) where ( a_n \neq 0 ).
- The exponents ( n ) are non-negative integers; the coefficients ( a_0, a_1, ..., a_n ) are real numbers.
- The term ( a_n x^n ) is the leading term; ( a_n ) is the leading coefficient.
- ( a_0 ) is the constant term.
Notation and Standard Form
- Polynomials may be denoted as ( p(x) ), ( f(x) ), ( y ), etc.; the variable and letter do not change the function’s properties.
- The standard form arranges terms by decreasing exponents (highest to lowest).
- Example: ( f(x) = 2x^3 + 5x^2 + 7x - 5 ) is in standard form.
Identifying Polynomials and Non-Polynomials
- Valid polynomials have non-negative integer exponents, no variables in denominators, and no variable radicals.
- Expressions with negative/fractional exponents, radicals, or variables in the denominator are not polynomials.
Rewriting and Combining Like Terms
- Arrange terms in decreasing exponent order for standard form.
- Combine like terms (same exponent) when simplifying.
- Use distributive property (FOIL) to expand factored polynomials before arranging.
Determining Key Features
- Leading term: term with the highest exponent.
- Leading coefficient: number in the leading term.
- Degree: highest exponent in the polynomial.
- Constant term: term without a variable.
Examples
- ( f(x) = 4x^3 - 16x - 4 + x^4 - x^2 ) → Standard form: ( x^4 + 4x^3 - x^2 - 16x - 4 ); degree: 4, leading coefficient: 1, constant: -4.
- ( f(x) = 3x^2 + 8x + 5 ): degree 2 (quadratic), leading coefficient: 3, constant: 5.
Key Terms & Definitions
- Polynomial function — An expression with non-negative integer exponents and real-number coefficients.
- Standard form — Arrangement with terms in order of decreasing exponent.
- Leading term — The term with the highest exponent.
- Leading coefficient — The coefficient of the leading term.
- Degree — The value of the highest exponent.
- Constant term — Term not multiplied by a variable.
Action Items / Next Steps
- Practice rewriting polynomials in standard form.
- Identify leading term, leading coefficient, degree, and constant from examples.
- Review restrictions to distinguish polynomials from non-polynomials.