Polynomial

In mathematics, a polynomial is a special kind of mathematical expression that uses indeterminates, also called variables, and coefficients. These expressions only use addition, subtraction, multiplication, and exponentiation with whole number exponents, and they have a limited number of terms. For example, something like ( x^2 - 4x + 7 ) is a polynomial with one variable, while ( x^3 + 2xyz^2 - yz + 1 ) shows a polynomial with three variables.

Polynomials are very important in many areas of math and science. They help us create polynomial equations to solve all sorts of problems, from simple word problems to complex scientific questions. Polynomials also define polynomial functions, which are used in fields like chemistry, physics, economics, and social science. In addition, they play a key role in calculus and numerical analysis by helping to approximate more complicated functions.

Beyond basic uses, polynomials are essential in advanced math. They are used to build polynomial rings and algebraic varieties, which are important ideas in algebra and algebraic geometry. These concepts help mathematicians understand the deeper structures and patterns in many mathematical problems.

Etymology

The word polynomial joins two different parts: the Greek poly, meaning "many", and the Latin nomen, meaning "name". It came from the term binomial by changing the Latin root bi- to the Greek poly-. This shows that a polynomial is a sum of many terms, called monomials. People first used the word polynomial in the 17th century.

Notation and terminology

In polynomials, the letter x is called a variable or an indeterminate. When we talk about x by itself, it doesn’t have a fixed number value—it’s just a symbol. But when we use a polynomial to create a function, x stands for the input value, and we call it a variable.

We can write a polynomial in two ways: either as P or as P(x). For example, we might say “let P(x) be a polynomial.” This shorter way of writing helps us talk about the polynomial and its variable together in one phrase. Both ways mean the same thing, just written differently to make reading formulas easier.

Definition

A polynomial is a type of math expression made from numbers and variables (also called indeterminates). You create it by adding, multiplying, and raising variables to powers that are whole numbers (like 2 or 3, but not fractions or negatives). For example, x<sup>2</sup> − 4x + 7 is a polynomial with one variable, x.

Polynomials can have many variables. Another example with three variables is x<sup>3</sup> + 2xyz<sup>2</sup> − yz + 1. Each part of the polynomial is called a term, and each term has a number (its coefficient) multiplied by some variables raised to powers.

Classification

Further information: Degree of a polynomial

A polynomial is made up of terms. Each term has a number (called a coefficient) and one or more letters (called variables or indeterminates). The degree of a term is how many letters are raised to powers in that term, added together. For example, in the term (-5x^{2}y), the degree is (2 + 1 = 3).

Polynomials can have one or more terms. A polynomial with just one term is called a monomial, with two terms a binomial, and with three terms a trinomial. For example, (3x^{2} - 5x + 4) is a trinomial. Polynomials with many terms are called multinomials.

Small-degree polynomials have special names: a degree-zero polynomial is a constant, degree-one is linear, degree-two is quadratic, and degree-three is cubic. Higher degrees don’t have commonly used names, but you might hear quartic (degree four) or quintic (degree five).

Operations

Polynomials can be added and subtracted by grouping their terms together and combining similar terms. For example, adding two polynomials means adding each matching term and simplifying the result. The same process works for subtraction.

Polynomials can also be multiplied. This is done by multiplying each term of one polynomial by every term of the other polynomial, then combining and simplifying the results. The product of two polynomials will always be another polynomial.

Polynomial functions

See also: Ring of polynomial functions

A polynomial function is a special kind of function made by using a polynomial. This means you take a polynomial — an expression with numbers and letters combined using addition, subtraction, multiplication, and raising to whole number powers — and use it to create a function. For example, the function ( f(x) = x^3 - x ) is a polynomial function because it follows these rules.

Polynomial functions can have one or more variables. They are smooth and continuous, meaning their graphs have no breaks or sharp corners. When you graph a polynomial function, the shape depends on its degree — the highest power of the variable. A degree 0 polynomial is just a flat line, degree 1 is a straight line, degree 2 makes a U-shaped curve called a parabola, and higher degrees create more complex curves.

Equations

Main article: Algebraic equation

A polynomial equation is an equation where a polynomial equals zero. For example, 3x² + 4x − 5 = 0 is a polynomial equation. When we solve these equations, we are looking for values of the variables that make the equation true. These values are called solutions.

In algebra, we learn ways to solve equations with one variable and degrees up to two, like using the quadratic formula. For higher degrees, there are no simple formulas, but we can still find answers using special methods or computers. The number of solutions a polynomial equation can have is related to its degree, which tells us the highest power of the variable in the polynomial.

Polynomial expressions

Polynomials can also be used in special ways by replacing the usual numbers with other mathematical ideas. For example, trigonometric polynomials use functions like sine and cosine in their calculations. These are helpful for studying repeating patterns and are used in many areas of science and engineering.

Another type is matrix polynomials, where the variables in a polynomial are replaced with matrices. This helps solve complex equations that involve matrices instead of simple numbers. There are also exponential polynomials, which involve exponential functions combined with polynomial terms.

Related concepts

Main article: Rational function

A rational function is made by dividing two polynomials. While polynomials work for any value, a rational function only works when the bottom part (denominator) is not zero.

Main article: Laurent polynomial

Laurent polynomials are similar to regular polynomials, but they can include negative exponents.

Main article: Formal power series

Formal power series are like polynomials but can have an unlimited number of terms. They follow the same rules as polynomials, even though they cannot always be fully written down.

Polynomial ring

A polynomial is a special kind of math expression made up of numbers (called coefficients) and letters (called variables). These expressions use only addition, subtraction, multiplication, and raising to whole number powers. For example, x² − 4x + 7 is a polynomial.

Polynomial rings are collections of these polynomials. When we take a set of numbers and add a new variable, we can build a polynomial ring. This helps us create new number systems from existing ones. For example, we can use polynomial rings to build complex numbers from real numbers.

Applications

Main article: Positional notation

Polynomials are very useful in mathematics. One example is in how we write numbers. In our usual decimal system, the number 45 really means 4 × 10¹ + 5 × 10⁰. This is a polynomial way of showing the number!

Polynomials also help us understand and approximate other kinds of functions. For example, a big idea in calculus is that complicated functions can often look like polynomials up close. This helps mathematicians study them more easily.

Polynomials are also used in many other areas. They can tell us about important features of graphs and matrices, and even how long some computer algorithms take to run.

History

Main articles: Cubic function § History, Quartic function § History, and Abel–Ruffini theorem § History

Finding the solutions to equations with polynomials is one of the oldest topics in math. People used to write these problems out in words instead of using the symbols we use today. For example, an old problem from China around 200 BCE talked about selling different types of crops to make a certain amount of money.

The symbols we use now, like the plus sign and the equal sign, started appearing in the 1500s. A math book from 1557 by Robert Recorde is the first known use of the equal sign. René Descartes, in 1637, helped us understand how to graph polynomial equations and introduced ways to use letters for numbers and variables.