Polynomial

In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. Here, simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i.e., they have derivatives of all finite orders.

Definition

For given constants (i.e., numbers) a0, …, an in some field (possibly but not limited to R or C) with an non-zero, for n > 0, then a polynomial (function) of degree n is a function of the form

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:f(x) = a_0 + a_1 x + cdots + a_{n - 1} x^{n - 1} + a_n x^n.

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More concisely, the polynomial can be written in sigma notation as

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: f(x) = sum_{i = 0}^{n} a_{i} x^{i}.

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The constants a0, …, an are called the coefficients of the polynomial. a0 is called the constant coefficient and an is called the leading coefficient. When the leading coefficient is 1, the polynomial is called monic or normed.

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Each summand ai xi of the polynomial is called a term. A polynomial with one, two or three terms is called monomial, binomial or trinomial respectively.

Related Topics:
Monomial - Binomial

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Polynomial functions of

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• degree 0 are called constant functions (excluding the zero polynomial, which has indeterminate degree),
• degree 1 are called linear functions,
• degree 2 are called quadratic functions,
• degree 3 are called cubic functions,
• degree 4 are called quartic functions and
• degree 5 are called quintic functions.