Polynomials and rational functions

Abstract

An expression of the form

$${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + \cdots + {a_1}x + {a_0}$$

((2.1))

where a _n , a _n−1, …, a ₁ , a ₀ are constants and n is a positive integer, is called a polynomial. The highest power of x occurring in the expression defines the degree or order of the polynomial and the a _i s are called the coefficients. If a _n ≠ 0 in Equation (2.1), the polynomial is of degree n and a _i (i = 1, 2, …, n) is the coefficient of x ⁱ . The term not involving x, namely a ₀ , is called the constant term. It is usual to write the polynomial in a systematic way either in descending powers of x as in Equation (2.1) or in ascending powers of x, when Equation (2.1) becomes

$${a_0} + {a_1}x + \cdots + {a_{n - 1}}{x^{n - 1}} + {a_n}{x^n}.$$

((2.2))

Polynomials of degree 2, 3 and 4 are called quadratics, cubics and quartics, respectively.