A polynomial is an expression built from variables and numbers using only addition, subtraction, and multiplication, with whole-number exponents. Learn its parts (terms, coefficients, degree), the names for one, two, and three-term polynomials, and what makes an expression NOT a polynomial.
What a polynomial is
A polynomial is an expression built from variables and numbers, combined using only addition, subtraction, and multiplication, with whole-number exponents (0, 1, 2, 3, ...). An expression like 3x² + 2x − 5 is a polynomial; an expression with a variable in a denominator, under a square root, or with a negative exponent is not.
A polynomial is built from terms, each with a coefficient and an exponent; its degree is the highest exponent.
The parts of a polynomial
A polynomial is made of terms separated by + or − signs. Each term has a coefficient (the number multiplying the variable) and, unless it's the constant term, an exponent. In 3x² + 2x − 5: the terms are 3x², 2x, and −5; the coefficients are 3 and 2; and −5 is the constant term (no variable attached). Understanding these parts is the foundation for later work on polynomial components.
The degree of a polynomial
The degree of a polynomial is its highest exponent. 3x² + 2x − 5 has degree 2, since the highest power of x is 2. The degree tells you the polynomial's general shape and behavior, and it's the first thing to check before evaluating an polynomial expression.
Types of polynomials by number of terms
Polynomials are named by how many terms they have: a monomial has one term (5x), a binomial has two (x + 3), and a trinomial has three (x² + 2x + 1). A polynomial with four or more terms is just called a polynomial. Recognizing these characteristics helps you spot equivalent expressions of polynomials more quickly.
What is NOT a polynomial
An expression is not a polynomial if it has a variable in a denominator (like 1/x), a variable under a radical (like √x), or a negative or fractional exponent (like x⁻¹). These operations break the "whole-number exponent" rule that defines a polynomial.