# Polynomials

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Polynomials are among the most common expressions in **algebra**. Each is just the sum of one or more powers of x, with each power multiplied by various numbers. In formal language, a polynomial in one **variable**, x, is the sum of terms ax^{k} where k is a non-negative integer and a is a constant. Polynomials are to algebra about what **integers** or whole numbers are to **arithmetic**. They can be added, subtracted, multiplied, and factored. **Division** of one polynomial by another may leave a remainder.

There are various words that are used in conjunction with polynomials. The **degree** of a polynomial is the **exponent** of the highest power of x. Thus the degree of

is 3. The leading **coefficient** is the coefficient of the highest power of x. Thus the leading coefficient of the above equation is 2. The constant **term** is the term that is the coefficent of x^{0} (=1). Thus the constant term of the above equation is 2 whereas the constant term of x^{3} + 5x^{2 }+ x is 0.

The most general form for a polynomial in one variable is

where a_{n}, a_{n-1},..., a_{1}, a_{0} are **real numbers**. They can be classified according to degree. Thus a first degree polynomial, a_{1}x + a_{2}, is, is linear; a second degree polynomial a x^{2} 1 + a_{2}x + a_{3} is quadratic; a third degree polynomial, a x^{3} + a 2 3 2x + a_{1}x+a_{0} is a cubic and so on. An irreducible or prime polynomial is one that has no factors of lower degree than a constant. For example, 2x^{2} + 6 is an irreducible polynomial although 2 is a factors. Also x^{2} + 1 is irreducible even though it has the factors x + i and x - i that involve **complex numbers**. Any polynomial is the product of of irreducible polynomials just as every integer is the product of **prime numbers**.

A polynomial in two variables, x and y, is the sum of terms, ax^{k}y^{m} where a is a real number and k and m are non-negative integers. For example,
is a polynomial in x and y. The degree of such a polynomial is the greatest of the degrees of its terms. Thus the degree of the above equation is 4 - both from x^{3}y (3 + 1 = 4) and from x^{2}y^{2} (2 + 2 = 4).

Similar definitions apply to polynomials in 3, 4, 5 ellipsevariables but the term "polynomial" without qualification usually refers to a polynomial in one variable.

A polynomial equation is of the form P = 0 where P is a polynomial. A polynomial **function** is one whose values are given by polynomial.

## Resources

### Books

Bittinger, Marvin L, and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications.* 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Larson, Ron. *Precalculus.* 5th ed. New York: Houghton Mifflin College, 2000.

Roy Dubisch