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Line 1: − + − Examples of polynomials include 0, x and 5xy<sup>2</sup>. Every element of R is also a polynomial, a [[constant]].+ + + − Polynomials can be evaluated to return an element of the ring R.+ − + + + + + + + + + + + +
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A '''polynomial''' is an [[Algebra|algebraic]] mathematical object. It is any element of a polynomial ring, that is a [[Ring_(mathematics)|ring]] R with one or more [[variable]]s adjoined. Note that R need not be [[Commutativity|commutative]], nor have [[identity]], nor be [[Associativity|associative]].
A polynomial in one variable <math>x</math> is a function <math>f(x)</math> of the form:
<math>
f(x) = a_n x^n+a_{n-1}x^{n-1}+...+a_1 x + a_0
</math>
In elementary mathematics, the coefficients <math>a_i</math> are typically chosen to be real or complex numbers. However, it makes sense to define a polynomial with coefficients in any [[ring]].
Some [[algorithm]]s are said to perform in polynomial time. There are algorithms which can factor polynomials in polynomial time.
The largest power of <math>x</math> that appears in the polynomial is called the degree of the polynomial.
*Example: <math>f(x) = 4x^3-3x+1</math> is a degree 3 polynomial with integer coefficients.
A polynomial in two variables <math>x,y</math> is, similarly, a finite sum
<math>
\sum a_{ij}x^i y^j
</math>,
where the coefficients <math>a_{ij}</math> are elements of some [[ring]]. Polynomials in 3 or more variables are defined similarly.
Some [[algorithm]]s are said to perform in polynomial time. There are algorithms which can factor polynomials in polynomial time.{{fact}}
[[Category:Algebra]]
[[Category:Algebra]]