Polynomials

Definition and Structure of Polynomials

A polynomial is an algebraic expression consisting of one or more terms, where each term is a product of a constant (called a coefficient) and a non-negative integer power of a variable. Polynomials are fundamental objects in algebra and are used to model a wide variety of real-world and mathematical situations.

• General Form: , where are constants and is a non-negative integer.

• Examples:

  • $7$ (a constant polynomial)

Degree and Leading Coefficient of a Polynomial

Key Definitions

• Degree of a Polynomial: The highest power of the variable in the polynomial. For example, the degree of is 5.

• Leading Coefficient: The coefficient of the term with the highest degree. In , the leading coefficient is 4.

Identifying the degree and leading coefficient is essential for understanding the behavior of polynomial functions, especially for graphing and analyzing their end behavior.

Graphing Polynomial Functions

Basic Principles

Graphing a polynomial function involves plotting its curve on the coordinate plane. The shape of the graph depends on the degree and leading coefficient:

• Intercepts: Points where the graph crosses the axes (x-intercepts and y-intercept).

• Turning Points: Points where the graph changes direction. A polynomial of degree can have up to turning points.

• Symmetry: Even-degree polynomials may be symmetric about the y-axis; odd-degree polynomials may be symmetric about the origin.

Example: The graph of is a parabola opening upwards, while is a cubic curve opening downwards on the right.

End Behavior of Polynomial Functions

Determining End Behavior

The end behavior of a polynomial function describes how the values of the function behave as approaches positive or negative infinity. The degree and leading coefficient determine this behavior:

• Even Degree: If the leading coefficient is positive, both ends of the graph rise to infinity. If negative, both ends fall to negative infinity.

• Odd Degree: If the leading coefficient is positive, the left end falls and the right end rises. If negative, the left end rises and the right end falls.

Examples:

• : As , (both ends up).

• : As , ; as , (left up, right down).