Difference between revisions of "Polynomial"
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| − | 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> | ||
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</math> | </math> | ||
| − | In elementary mathematics, the | + | 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. | ||
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</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. | ||
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| + | 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]] | ||
Revision as of 01:06, July 17, 2008
A polynomial in one variable 
is a function 
of the form:
In elementary mathematics, the coefficients 
are typically chosen to be real or complex numbers. However, it makes sense to define a polynomial with coefficients in any ring.
The largest power of that appears in the polynomial is called the degree of the polynomial.
- Example:
is a degree 3 polynomial with integer coefficients.
A polynomial in two variables 
is, similarly, a finite sum
,
where the coefficients 
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 
. This process is called "evaluating the polynomial at b".
Some algorithms are said to perform in polynomial time. There are algorithms which can factor polynomials in an amount of time that is a polynomial nk.
