The roots function computes the exact roots of a polynomial over the rationals or an algebraic number field. The roots are returned as a list of pairs of the form $\[\[\mathrm{r1},\mathrm{m1}\],...,\[\mathrm{rn},\mathrm{mn}\]\]$ where $\mathrm{ri}$ is a root of the polynomial a with multiplicity $\mathrm{mi}$, that is, ${\left(x-\mathrm{ri}\right)}^{\mathrm{mi}}$ divides a.

The call roots(a) returns roots over the field implied by the coefficients present.  For example, if all the coefficients are rational, then the rational roots are computed.  If a has no roots in the implied coefficient field, then an empty list is returned.  This assumes that a is a univariate polynomials.

The call roots(a, K) computes the roots of a over the algebraic number field defined by K. Here K must be a single RootOf, or a list or set of RootOfs, or a single radical, or a list or set of radicals.  For example, if I is given as the second argument, then roots looks for the roots of a over the complex rationals.

The calls roots(a, x) and roots(a, x, K) are equivalent to the above if a is univariate in x. Otherwise, it treats the other indeterminates in a as parameters, and finds all roots as above and ignoring symbolic roots.

$\mathrm{roots}\left(2x^3+11x^2+12x-9\right)$

$\left[\left[\mathrm{-3}\,2\right]\,\left[\frac{1}{2}\,1\right]\right]$

$\mathrm{roots}\left(x^4-4\right)$

$\mathrm{roots}\left(x^4-4\,x\right)$

$\mathrm{roots}\left(x^3+\left(-6-b-a\right)x^2+\left(6a+5+5b+ab\right)x-5a-5ab\,x\right)$

$\left[\left[5\,1\right]\right]$

$\mathrm{roots}\left(x^4-4\,\mathrm{sqrt}\left(2\right)\right)$

$\left[\left[\sqrt{2}\,1\right]\,\left[-\sqrt{2}\,1\right]\right]$

$\mathrm{roots}\left(x^4-4\,\left\{\mathrm{I}\,\mathrm{sqrt}\left(2\right)\right\}\right)$

$\left[\left[\mathrm{-I}\sqrt{2}\,1\right]\,\left[\sqrt{2}\,1\right]\,\left[\mathrm{I}\sqrt{2}\,1\right]\,\left[-\sqrt{2}\,1\right]\right]$

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

$\mathrm{alias}\left(\mathrm{\beta }=\mathrm{RootOf}\left(x^2+2\right)\right)\:$

$\mathrm{roots}\left(\left(x^4-4\right)\left(x-a\right)\,x\,\mathrm{\alpha }\right)$

$\left[\left[-\mathrm{\alpha }\,1\right]\,\left[\mathrm{\alpha }\,1\right]\right]$

$\mathrm{roots}\left(x^4-4\,\left\{\mathrm{\alpha }\,\mathrm{\beta }\right\}\right)$

$\left[\left[-\mathrm{\alpha }\,1\right]\,\left[-\mathrm{\beta }\,1\right]\,\left[\mathrm{\alpha }\,1\right]\,\left[\mathrm{\beta }\,1\right]\right]$