Polynomial

In mathematics, a polynomial is a special type of mathematical expression. It uses indeterminates, also called variables, and coefficients. Polynomials only use addition, subtraction, multiplication, and exponentiation with whole number exponents. They also have a limited number of terms. For example, ( x^2 - 4x + 7 ) is a polynomial with one variable. Another example, ( 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 many problems. These range from simple word problems to complex scientific questions. Polynomials also define polynomial functions. These are used in fields like chemistry, physics, economics, and social science. They also play a key role in calculus and numerical analysis. This is because they help to approximate more complicated functions.

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

Etymology

The word polynomial combines two parts: the Greek poly, meaning "many", and the Latin nomen, meaning "name". It started from the word binomial by changing the Latin root bi- to the Greek poly-. This tells us that a polynomial is a sum of many parts, called monomials. People began using the word polynomial in the 17th century.

Notation and terminology

In polynomials, the letter x is called a variable or an indeterminate. When we use x by itself, it doesn’t have a fixed number—it’s just a symbol. But when we use a polynomial to make 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 math expression made from numbers and variables. You build it by adding, multiplying, and raising variables to whole number powers, like 2 or 3. For example, x<sup>2</sup> − 4x + 7 is a polynomial with one variable, x.

Polynomials can have many variables. Here’s an example with three variables: x<sup>3</sup> + 2xyz<sup>2</sup> − yz + 1. Each part of the polynomial is called a term. Each term has a number, called its coefficient, multiplied by variables raised to powers.

Classification

Further information: Degree of a polynomial

A polynomial is made of terms. Each term has a number (called a coefficient) and one or more letters (called variables). 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. One with two terms is a binomial. One with three terms is 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 way 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 from a polynomial. A polynomial is an expression with numbers and letters. You can add, subtract, multiply, and raise them to whole number powers. For example, the function ( f(x) = x^3 - x ) is a polynomial function.

Polynomial functions can have one or more variables. Their graphs are smooth and have no breaks or sharp corners. The shape of the graph depends on its degree — the highest power of the variable. A degree 0 polynomial is a flat line. Degree 1 is a straight line. Degree 2 makes a U-shaped curve called a parabola. Higher degrees create more complex curves.

Equations

Main article: Algebraic equation

A polynomial equation is when a polynomial equals zero. For example, 3x² + 4x − 5 = 0 is a polynomial equation. When we solve these, we look for numbers that make the equation true. These numbers are called solutions.

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

Polynomial expressions

Polynomials can be used in special ways by using different mathematical ideas instead of numbers.

For example, trigonometric polynomials use functions like sine and cosine. They help us study repeating patterns and are used in science and engineering.

Another type is matrix polynomials, where the variables are replaced with matrices. This helps solve hard equations that use matrices instead of simple numbers. There are also exponential polynomials, which mix exponential functions with polynomial terms.

Related concepts

Main article: Rational function

A rational function is made by dividing two polynomials. Polynomials work for any value, but 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.

Polynomial ring

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

Polynomial rings are groups 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 how we write numbers. In our usual decimal system, the number 45 means 4 × 10¹ + 5 × 10⁰. This shows the number using a polynomial!

Polynomials also help us understand other kinds of functions. For example, a big idea in calculus is that complicated functions can often look like polynomials up close. This makes them easier for mathematicians to study.

Polynomials are used in many other areas too. They can tell us about 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 answers to equations with polynomials is a very old math topic. People used to write problems in words, not with symbols like we do today. For example, an old problem from China around 200 BCE talked about selling 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 draw graphs of polynomial equations and showed ways to use letters for numbers and variables.