Polynomial Function

A Polynomial Function is an Algebraic Function that can be expressed as a finite sum of terms, each consisting of a constant coefficient multiplied by a variable raised to a nonnegative integer power.

• AKA: Polynomial, Polynomial Mapping, Polynomial Expression Function.
• Context:
  • It can typically be expressed in the form [math]\displaystyle{ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 }[/math].
  • It can typically have a Polynomial Degree determined by its highest power term.
  • It can typically exhibit Smooth Function properties with continuous derivatives of all orders.
  • It can typically be uniquely determined by its coefficients and degree.
  • It can typically have at most n Complex Roots for a polynomial of degree n (by the Fundamental Theorem of Algebra).
  • ...
  • It can often serve as Approximation Functions in Taylor series expansions and interpolation tasks.
  • It can often be evaluated efficiently using Horner's method or other polynomial evaluation algorithms.
  • It can often form a Vector Space under addition and scalar multiplication.
  • It can often be factored into Linear Factors over the complex numbers.
  • ...
  • It can range from being a Constant Polynomial Function to being a High-Degree Polynomial Function, depending on its polynomial degree.
  • It can range from being a Univariate Polynomial Function to being a Multivariate Polynomial Function, depending on its variable count.
  • It can range from being a Dense Polynomial Function to being a Sparse Polynomial Function, depending on its nonzero coefficient count.
  • It can range from being a Monic Polynomial Function to being a General Polynomial Function, depending on its leading coefficient.
  • It can range from being a Homogeneous Polynomial Function to being a Non-Homogeneous Polynomial Function, depending on its term degree uniformity.
  • ...
  • It can satisfy Algebraic Properties including:
    • Closure Under Addition and Closure Under Multiplication.
    • Closure Under Differentiation and Closure Under Integration.
    • Closure Under Composition with other polynomial functions.
    • Division Algorithm properties for polynomial division.
  • It can exhibit Analytical Properties such as:
    • Continuity on the entire real line.
    • Differentiability of all orders.
    • Local Behavior determined by Taylor expansions.
    • Global Behavior dominated by the leading term.
  • ...
• Example(s):
  • Constant Polynomial Functions, such as:
    • Zero Polynomial: [math]\displaystyle{ f(x) = 0 }[/math].
    • Non-Zero Constant Polynomial: [math]\displaystyle{ f(x) = c }[/math] where [math]\displaystyle{ c \neq 0 }[/math].
  • Linear Polynomial Functions (degree 1), such as:
    • Identity Function: [math]\displaystyle{ f(x) = x }[/math].
    • Affine Function: [math]\displaystyle{ f(x) = ax + b }[/math] where [math]\displaystyle{ a \neq 0 }[/math].
  • Quadratic Polynomial Functions (degree 2), such as:
    • Squaring Function: [math]\displaystyle{ f(x) = x^2 }[/math].
    • General Quadratic Function: [math]\displaystyle{ f(x) = ax^2 + bx + c }[/math] where [math]\displaystyle{ a \neq 0 }[/math].
    • Parabolic Function: [math]\displaystyle{ f(x) = x^2 - 4x + 3 }[/math].
  • Cubic Polynomial Functions (degree 3), such as:
    • Cubing Function: [math]\displaystyle{ f(x) = x^3 }[/math].
    • General Cubic Function: [math]\displaystyle{ f(x) = ax^3 + bx^2 + cx + d }[/math] where [math]\displaystyle{ a \neq 0 }[/math].
  • Higher-Degree Polynomial Functions, such as:
    • Quartic Polynomial Function: [math]\displaystyle{ f(x) = x^4 - 2x^2 + 1 }[/math].
    • Quintic Polynomial Function: [math]\displaystyle{ f(x) = x^5 - 5x^3 + 4x }[/math].
    • General n-th Degree Polynomial: [math]\displaystyle{ f(x) = \sum_{i=0}^n a_i x^i }[/math].
  • Special Polynomial Functions, such as:
    • Chebyshev Polynomials of the first kind and second kind.
    • Legendre Polynomials used in orthogonal function systems.
    • Hermite Polynomials appearing in quantum mechanics.
    • Laguerre Polynomials used in mathematical physics.
    • Bernstein Polynomials used in Bézier curves.
  • Multivariate Polynomial Functions, such as:
    • Bivariate Polynomial: [math]\displaystyle{ f(x,y) = x^2 + xy + y^2 }[/math].
    • Homogeneous Polynomial: [math]\displaystyle{ f(x,y,z) = x^3 + y^3 + z^3 - 3xyz }[/math].
    • Symmetric Polynomial: [math]\displaystyle{ f(x,y) = x^2 + y^2 + 2xy }[/math].
  • Piecewise Polynomial Functions, such as:
    • Spline Functions for interpolation tasks.
    • B-spline Functions for curve fitting.
  • ...
• Counter-Example(s):
  • Rational Functions containing variables in the denominator, such as [math]\displaystyle{ f(x) = \frac{1}{x} }[/math].
  • Exponential Functions with variable exponents, such as [math]\displaystyle{ f(x) = 2^x }[/math].
  • Logarithmic Functions, such as [math]\displaystyle{ f(x) = \log(x) }[/math].
  • Trigonometric Functions, such as [math]\displaystyle{ f(x) = \sin(x) }[/math].
  • Root Functions with non-integer exponents, such as [math]\displaystyle{ f(x) = x^{1/2} }[/math].
  • Power Functions with negative exponents, such as [math]\displaystyle{ f(x) = x^{-2} }[/math].
  • Transcendental Functions that cannot be expressed as finite algebraic expressions.
• See: Polynomial Equation, Polynomial Ring, Polynomial Interpolation, Polynomial Approximation, Polynomial Degree, Root of a Polynomial, Factorization, Polynomial Division, Polynomial Evaluation Algorithm, Taylor Polynomial, Orthogonal Polynomial, Polynomial Complexity.

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References

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Polynomial
  • In mathematics, a polynomial is a finite length expression constructed from variables (also known as indeterminates) and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x and also because its third term contains an exponent that is not a whole number.
  • Polynomials are one of the most important concepts in algebra and throughout mathematics and science. They are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics, and are used in calculus and numerical analysis to approximate other functions. Polynomials are used to construct polynomial rings, one of the most powerful concepts in algebra and algebraic geometry.

• http://en.wikipedia.org/wiki/Polynomial#Polynomial_functions
  • A polynomial function is a function that can be defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies
    • : [math]\displaystyle{ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 \, }[/math]
  • for all arguments [math]\displaystyle{ x }[/math], where [math]\displaystyle{ n }[/math] is a non-negative integer and a₀, a₁,a₂, ..., a_n are constant coefficients. …

     Polynomial functions are a class of functions having many important properties. They are all continuous, smooth, entire, computable, etc.

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