Editing Polynomial (section)

=== Calculus ===
{{Main|Calculus with polynomials}}
Calculating [[derivative]]s and integrals of polynomials is particularly simple, compared to other kinds of functions.
The [[derivative]] of the polynomial <math display="block">P = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_2 x^2 + a_1 x + a_0 = \sum_{i=0}^n a_i x^i</math> with respect to {{mvar|x}} is the polynomial 
<math display="block"> n a_n x^{n - 1} + (n - 1)a_{n - 1} x^{n - 2} + \dots + 2 a_2 x + a_1 = \sum_{i=1}^n i a_i x^{i-1}.</math>
Similarly, the general [[antiderivative]] (or indefinite integral) of <math>P</math> is
<math display="block"> \frac{a_n x^{n + 1}}{n + 1} + \frac{a_{n - 1} x^{n}}{n} + \dots + \frac{a_2 x^3}{3} + \frac{a_1 x^2}{2} + a_0 x + c = c + \sum_{i = 0}^n \frac{a_i x^{i + 1}}{i + 1}</math>
where {{mvar|c}} is an arbitrary constant.  For example, antiderivatives of {{math|''x''<sup>2</sup> + 1}} have the form {{math|{{sfrac|3}}''x''<sup>3</sup> + ''x'' + ''c''}}.

For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers [[modular arithmetic|modulo]] some [[prime number]] {{math|''p''}}, or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient {{math|''ka''<sub>''k''</sub>}} understood to mean the sum of {{mvar|k}} copies of {{math|''a''<sub>''k''</sub>}}. For example, over the integers modulo {{math|''p''}}, the derivative of the polynomial {{math|''x''<sup>''p''</sup> + ''x''}} is the polynomial {{math|1}}.<ref name=Barbeau-2003-pp64-65>{{harvnb|Barbeau|2003|pp=[https://books.google.com/books?id=CynRMm5qTmQC&pg=PA64 64]–5}}</ref>