This function expands product and powers of rational functions with algebraic coefficients. Powers of algebraic numbers and functions are reduced and denominators are rationalized.

Algebraic functions and algebraic numbers may be represented by radicals or with the RootOf notation (see type,algnum, type,algfun, type,radnum, type,radfun).

More precisely, the output satisfies the following properties:

Positive powers, products of sums and products of positive powers are expanded. The first operand of negative powers is expanded recursively, but negative powers are not expanded. Products of negative powers are not expanded either.

Algebraic numbers and functions have been reduced modulo the minimal polynomials. See Normal for a more precise definition.

Denominators have been rationalized. In other words, RootOfs and radicals defining algebraic numbers and functions have been removed from the denominator of rational functions.

Arguments of functions have been expanded recursively. Note that, unlike expand, Expand has no effect on mathematical functions such as sin or exp.

Other objects are frozen and considered as variables, except in the cases below.

If a is a set, a list, a range, a relation, or a series, then Expand is mapped over the object.

This function can be used to normalize polynomials over algebraic number fields. If the coefficients are algebraic functions or if a is not a polynomial, Expand cannot be used to decide whether a is mathematically equal to zero. See Normal in this case.

This function does not check that the algebraic quantities are independent.

$\mathrm{s1}≔\frac{x+\mathrm{sqrt}\left(2\right)}{\left(x-\mathrm{sqrt}\left(2\right)\right)x}$

$\mathrm{s1}≔\frac{x+\sqrt{2}}{\left(x-\sqrt{2}\right)x}$

$\mathrm{evala}\left(\mathrm{Expand}\left(\mathrm{s1}\right)\right)$

$\frac{2\sqrt{2}}{x^2-2}+\frac{x}{x^2-2}+\frac{2}{x\left(x^2-2\right)}$

$\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left(y^2-y+x\,y\right)\right)\:$

$\mathrm{s2}≔{\left(\mathrm{\alpha }-y\right)}^2$

$\mathrm{s2}≔{\left(\mathrm{\alpha }-y\right)}^2$

$\mathrm{evala}\left(\mathrm{Expand}\left(\mathrm{s2}\right)\right)$

$-2\mathrm{\alpha }y+y^2+\mathrm{\alpha }-x$

$\mathrm{s3}≔{\left(x-\mathrm{RootOf}\left(x^2-4\right)\right)}^2$

$\mathrm{s3}≔{\left(x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-4\right)\right)}^2$

$\mathrm{evala}\left(\mathrm{Expand}\left(\mathrm{s3}\right)\right)$

$-2x\mathrm{RootOf}\left({\mathrm{\_Z}}^2-4\right)+x^2+4$

$\mathrm{s4}≔-\frac{1}{x-1}+\frac{x}{x-1}-1$

$\mathrm{s4}≔-\frac{1}{x-1}+\frac{x}{x-1}-1$

$\mathrm{evala}\left(\mathrm{Expand}\left(\mathrm{s4}\right)\right)$

$-\frac{1}{x-1}+\frac{x}{x-1}-1$

$\mathrm{normal}\left(\mathrm{s4}\right)$

$0$

$\mathrm{s5}≔\sin \left({\mathrm{\alpha }}^2-\mathrm{\alpha }+2x+y\right)$

$\mathrm{s5}≔\sin \left({\mathrm{\alpha }}^2-\mathrm{\alpha }+2x+y\right)$

$\mathrm{evala}\left(\mathrm{Expand}\left(\mathrm{s5}\right)\right)$

$\sin \left(x+y\right)$