What Are Polynomial Functions?

Polynomial functions are mathematical expressions involving variables raised to whole number powers, all combined using addition, subtraction, and multiplication. The general form looks like this:

\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \]

where \( a_n, a_{n-1}, \ldots, a_0 \) are constants, and \( n \) is a non-negative integer called the degree.

Key Features

• Degree: The highest exponent of the variable.
• Leading Coefficient: The coefficient of the term with the highest degree.
• End Behavior: How the graph behaves as \( x \) approaches infinity or negative infinity.

Why Are They Important?

Polynomial functions model real-world situations, like predicting profits, tracking the trajectory of a ball, or understanding population growth.

Graphing Polynomial Functions

The shape of a polynomial graph depends on its degree and leading coefficient. Higher-degree polynomials have more turning points and can cross the x-axis more times.

Working with Polynomials

You can add, subtract, multiply, and even divide polynomials. Polynomial division is especially useful for simplifying expressions.