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reinstating formatting and concepts lost by previous editor.
A polynomial in one variable <math>x</math> is a function <math>f(x)</math> of the form:
A '''polynomial''' in one [[variable]] <math>x</math> is a [[function]] <math>f(x)</math> of the form:
<math>
<math>
</math>
</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]].
In elementary mathematics, the [[coefficient]]s <math>a_i</math> are typically chosen to be [[real]] or [[complex number]]s. However, it makes sense to define a polynomial with coefficients in any [[ring]].
The largest power of <math>x</math> that appears in the polynomial is called the degree of the polynomial.
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.
*Example: <math>f(x) = 4x^3-3x+1</math> is a degree 3 polynomial with integer coefficients.
</math>,
</math>,
where the coefficients <math>a_{ij}</math> are elements of some [[ring]]. Polynomials in 3 or more variables are defined similarly.
where the coefficients <math>a_{ij}</math> are elements of some [[ring]]. Polynomials in 3 or more variables are defined similarly.
When we substitute a value ''b'' into the polynomial ''f(x)'' then we get an actual number as output. This number is the value <math>a_nb^n +a_{n-1}b^{n-1}+...+ a_2 b^2+a_1b +a_0</math>. This process is called "evaluating the polynomial at ''b''".
Some [[algorithm]]s are said to perform in polynomial time. There are algorithms which can factor polynomials in an amount of time that is a polynomial n<sup>k</sup>.
[[Category:Algebra]]
[[Category:Algebra]]