In algebra, we work with all sorts of expressions. A polynomial is a special type. Think of it as an expression built from variables and coefficients, using only addition, subtraction, and multiplication, where the variable is raised to non-negative integer powers.

Let's break down an example: $4x^3 - 2x + 7$.

This polynomial is made up of individual parts called terms, which are separated by plus or minus signs. The terms here are $4x^3$, $-2x$, and $7$.

Within each term, the number part is called the coefficient. In $4x^3$, the coefficient is 4. In $-2x$, it's -2. The term without a variable, 7, is called the constant term.

Most polynomials have variables, but not all. A number by itself, like 5, is also a polynomial. It's called a constant polynomial. You can think of it as $5x^0$, and since $x^0 = 1$, it simplifies to just 5.

The number 0 is also a polynomial, but it's a very special case called the zero polynomial because it has some unique properties that set it apart from all other constants.

We can categorize polynomials in two main ways: by how many terms they have, and by their degree.

The degree tells us the general shape and complexity of the polynomial's graph. Based on the degree, we have special names for the most common types.

We also classify polynomials by their number of terms.

An expression can be classified both ways. For example, $x^2 - 4$ is a quadratic binomial. It's quadratic because its degree is 2, and it's a binomial because it has two terms. What about $8x^3$? That's a cubic monomial.