polynomial

Generally speaking, the term polynomial refers to an expression in which there are many terms (which is what ‘polynomial’ means) being added together. A polynomial equation will have the sum of the terms set equal to a constant, typically 0.

Algebra/Calculus/Precalculus

A polynomial of a single real variable is a function of the form

\[p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n\]

where the \(a_i\) (called the coefficients) are real (or usually, rational) constants, some of which may be zero, and the exponents are positive integers. The highest exponent is called the degree of the polynomial, and the coefficient \(a_n\) on the highest degree term is called the leading coefficient. Polynomials of degree two are called quadratic polynomials, of degree 3 cubic, of degree 4 quartic, and those of degree 5 are called quintic. A polynomial of degree 1 is called a monic polynomial or linear function.

Polynomials of a single real variable with rational coefficients may be factored into a product of monic polynomials over the complex numbers, or into a product of monic and quadratic polynomials over the real numbers. There are general formulas for factoring quadratic, cubic, and quartic polynomials, but there is no general formula for factoring quintic or higher-degree polynomials.

More generally, a polynomial may be in several variables \(x_1,x_2,\ldots,x_k\), and may be thought of as a sum of the form

\[p(x)=\sum_{i=0}^{\infty} a_ix_1^{e_{i,1}}x_2^{e_{i,2}}\cdots x_k^{e_{i,k}}\]

where all but finitely many of the \(a_i\) are 0. In this case the degree of the polynomial is the highest sum of exponents appearing in any term. For example, \(2x^4\) and \(3xy^2z\) are both fourth-degree terms.

Abstract Algebra

A (formal) polynomial is a sum of the form

\[p(x)=\sum_{i=0}^{\infty} a_ix^i\]

The \(a_i\) are called coefficients, and are elements of some commutative ring \(R\). We say that \(p\) is a polynomial over \(R\), or with coefficients in \(R\). The \(x\) is just a formal symbol. Only finitely many of the \(a_i\) can be non-zero, and if \(a_n\) is the last non-zero coefficient, \(n\) is called the degree of \(p\).

Polynomials can be added and multiplied in the natural way, by using the cummutative and distributive laws of addition and multiplication in \(R\). Explicitly, if \(p\) and \(q\) are polynomials over the same ring, and the \(a_i\) and \(b_i\) are the coefficients of \(p\) and \(q\), respectively, then

\[p(x)+q(x)=\sum_{i=0}^{\infty}(a_i+b_i)x_i\]

and

\[p(x)q(x)=\sum_{i=0}^{\infty}\left( \sum_{j=0}^i a_jb(i-j)x^i\right)\]

These operations allow us to define \(R[x]\), the polynomial ring over \(R\).

The substitution principle allows us evaluate a polynomial \(p\) on any element of \(R\), and we can use this to define a function corresponding to \(p\), thereby capturing the informal notion of a polynomial.