User:JayBeeEll/sandbox

• NB I have not dealt with the problem of sourcing the things I've changed, most of which are focused on composition and division. Moderately good sourcing on division.
• Should we say things about degrees of sums, products, and compositions?
• Nothing about subtraction yet
• Should we add anything about composition where the outer function has multiple variables? Like, f(g(x, y), h(x, z), t) or whatever. Maybe no.

Factorization

All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible constant. In the case of the field of complex numbers, the irreducible factors are linear. Over the real numbers, they have the degree either one or two. Over the integers and the rational numbers the irreducible factors may have any degree. For example, the factored form of

$5x^{3}-5$

is

$5(x-1)\left(x^{2}+x+1\right)$

over the integers and the reals and

$5(x-1)\left(x+{\frac {1+i{\sqrt {3}}}{2}}\right)\left(x+{\frac {1-i{\sqrt {3}}}{2}}\right)$

over the complex numbers.

The computation of the factored form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms are available in most computer algebra systems.

Calculus

Main article: Calculus with polynomials

Calculating derivatives and integrals of polynomial functions is particularly simple. The derivative of the polynomial $P=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{2}x^{2}+a_{1}x+a_{0}=\sum _{i=0}^{n}a_{i}x^{i}$ with respect to x is the polynomial $$na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+\ldots +2a_{2}x+a_{1}=\sum _{i=1}^{n}a_{i}\cdot i\cdot x^{i-1}.$$ Similarly, the general antiderivative (or indefinite integral) of $P$ is $${\frac {a_{n}x^{n+1}}{n+1}}+{\frac {a_{n-1}x^{n}}{n}}+\ldots +{\frac {a_{2}x^{3}}{3}}+{\frac {a_{1}x^{2}}{2}}+a_{0}x+c=c+\sum _{i=0}^{n}{\frac {a_{i}x^{i+1}}{i+1}}$$ where c is an arbitrary constant. For example, antiderivatives of x² + 1 have the form ⁠1/3⁠x³ + x + c.

For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number p, or elements of an arbitrary ring) then the formula for the derivative can still be interpreted formally, with the coefficient ka_k understood to mean the sum of a_k with itself, k times. For example, over the integers modulo p, the derivative of the polynomial x^p + x is the polynomial 1.