What Is a Polynomial?

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 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 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.

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.

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.

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• Algebra