Polynomials

views updated

Polynomials

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 nonnegative 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

2x³ + 5x²–x + 2

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 x⁰ (= 1). Thus the constant term of the above equation is 2, whereas the constant term of x³ + 5x² + x is 0.

The most general form for a polynomial in one variable is

a_nxⁿ + a_n – 1x^{n – 1} + ... + a₁x + a₀

where a_n, a_n–1, ..., a₁, a₀ are real numbers. They can be classified according to degree. Thus a first-degree polynomial, a₁x + a₂, is linear; a second-degree polynomial a₁x² + a₂x + a₃ is quadratic; a third-degree polynomial, a₃x³ + a₂x² + a₁x + a₀ 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, 2x² + 6 is an irreducible polynomial although 2 is a factor. Also x² + 1 is irreducible even though it has the factors x + iandx–i that involve complex numbers. All polynomials are the product 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^ky^m where a is a real number and k and m are non-negative integers. For example,

x³y + 3x²y² + 3xy–4x + 5y–12

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 x³y(3 + 1←= 4) and from x²y² (2 + 2 = 4).

Similar definitions apply to polynomials in 3, 4, 5 . . . variables, 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.