# Polynomial Function

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A Polynomial Function is a smooth function that restricts itself to constant nonnegative integer exponentiation.

**Context:**- It can (typically) have a Polynomial Function Degree, e.g. a first-order polynomial (a linear function), or second-order polynomial function.
- It can be a member of a Polynomial Equation.

**Example(s):**- A Squaring Function, such as [math]\displaystyle{ f(x) = x^2 }[/math].
- A Quadratic Function.
- [math]\displaystyle{ f(x) = \frac{1}{3} \times x^2 −2.3 \times x^1 + 6 \times x^0 }[/math].
- [math]\displaystyle{ f(x) = cx^2 + bx + a }[/math], where [math]\displaystyle{ a, b, c }[/math] are Mathematical Constants.
- …

**Counter-Example(s):**- a Linear Function.
- an Exponential Function, such as [math]\displaystyle{ f(x) = 3^x }[/math]
- ?[math]\displaystyle{ f(x) = cx^2 + bx + a }[/math], where [math]\displaystyle{ a, b, c }[/math] are free parameters.
- ?[math]\displaystyle{ f(x) = x^2 −2.3x + 6.1 + x^{-1} }[/math].

**See:**Polynomial Interpolation, Polynomial Complexity Function.

## 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
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**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
*a*_{0},*a*_{1},*a*_{2}, ...,*a*are constant coefficients. …_{n} Polynomial functions are a class of functions having many important properties. They are all continuous, smooth, entire, computable, etc.

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