What is a Polynomial?

A polynomial is a mathematical expression of finite length. It is composed of both variables and constants. Those variables and constants can be added, subtracted, multiplied, and divided. They can also be raised to exponents, as long as those exponents are whole numbers.

In mathematics and in science, polynomials are extremely important. They are used to create sales models in business and to model physical phenomena in physics and in chemistry. Polynomial functions also form the basis for much of calculus; derivatives and integrals of polynomial functions provide information to scientists, economists, doctors, and others about rates of change.

Polynomials take the form of a_nxⁿ+...+a₂x²+a₁x+a₀, and are arranged in terms, which are sometimes called monomials. A term is one section of a polynomial that is being multiplied together, and is typically composed of a constant multiplied by an exponent that is being raised to a power. For example, 3x² is a term, and 3x²+2x+5 is a polynomial composed of three terms. Terms are ordered from highest to lowest according to degree, the number of the exponent on a variable.

As many high-schoolers learn, polynomials are often used in equations, in which two polynomials are set equal to each other. Generally, the goal of a polynomial equation is finding the value or values of the variable or variables. Solving these equations can give such information as time or distance in practical, physics-related scenarios.

Graphs are often used in the study of polynomial functions, which take the form of f(x)= a_nxⁿ+...+a₂x²+a₁x+a₀. The value of the variable, x, determines the value of the function as a whole, f(x). Graphs of polynomial functions can range in shape from parabolas to intricate series of curves depending on the degree and complexity of the function. Such visual representations make understanding the meaning of the function far easier, as they plot all of the values of f(x) based on the values of x in a given range.

Multivariate polynomials involve more than one variable. They can involve any number of variables, and generally become more complex as the number increases. Generally, little attention is given to multivariate polynomials in high school. They are usually presented in upper-level college calculus classes that deal with three dimensional shapes or analyses of many different forms of combined data.

Polynomials have been used for a very long time, and are integral to modern mathematics. Their many forms set the foundation for the representation of countless models in business, science, economics, and other fields.