explicit : truefalse

domain : absolute, rational, integer, real, or parametric.

dropmultiplicity : truefalse

Solve for a polynomial in x.  Polynomial uses factor, compoly, and explicit root formulae to write the roots explicitly where possible.

If not possible, a list of indexed RootOf will be returned.

The behavior of Polynomial is controlled by the option explicit, or by the environment variable _EnvExplicit.  In all cases the option, if specified, overrides the environment variable. has three possible behaviors depending on the option

By default (if the option explicit not specified and _EnvExplicit is not set) explicit roots are calculated for polynomials of degree 2 and 3 but not for polynomials higher degree (unless they factor or decompose). Implicit roots that do not involve non-numeric symbols are given as indexed RootOfs.

If explicit is specified as an option (or _EnvExplicit=true) then explicit roots are computed when possible.

If explicit=false is specified as an option (or _EnvExplicit=false) then no attempt is made to compute explicit roots, and unspecialized RootOf expressions are returned.

The domain option can be used to restrict the roots returned. Using domain=real or domain=integer will return only real or integer roots respectively.  domain=absolute will return all the roots and domain=rational will return the roots which lie in the same field as the coefficients of f in the same way as roots; in particular if f is a polynomial with integer coefficients, domain=rational will return only the roots which are rational numbers.  domain=parametric will return a piecewise expression giving a discussion of different cases.

If the option dropmultiplicity is specified, only one copy of each root is returned.

$\mathrm{with}\left(\mathrm{SolveTools}\right)\:$

$\mathrm{Polynomial}\left(0\,x\right)$

$\left[x\right]$

$\mathrm{Polynomial}\left(1\,x\right)$

$\mathrm{Polynomial}\left(x^2\,x\right)$

$\left[0\,0\right]$

$\mathrm{Polynomial}\left(x^2-1\,x\right)$

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

$\mathrm{Polynomial}\left(x^2+1\,x\,\mathrm{explicit}=\mathrm{false}\right)$

$\left[\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\,\mathrm{label}=\mathrm{\_L1}\right)\right]$

$\mathrm{Polynomial}\left(x^5+2x+1\,x\right)$

$\left[\mathrm{RootOf}\left({\mathrm{\_Z}}^5+2\mathrm{\_Z}+1\,\mathrm{index}=1\right)\,\mathrm{RootOf}\left({\mathrm{\_Z}}^5+2\mathrm{\_Z}+1\,\mathrm{index}=2\right)\,\mathrm{RootOf}\left({\mathrm{\_Z}}^5+2\mathrm{\_Z}+1\,\mathrm{index}=3\right)\,\mathrm{RootOf}\left({\mathrm{\_Z}}^5+2\mathrm{\_Z}+1\,\mathrm{index}=4\right)\,\mathrm{RootOf}\left({\mathrm{\_Z}}^5+2\mathrm{\_Z}+1\,\mathrm{index}=5\right)\right]$

$\mathrm{Polynomial}\left(x^5+2x+1\,x\,\mathrm{explicit}=\mathrm{false}\right)$

$\left[\mathrm{RootOf}\left({\mathrm{\_Z}}^5+2\mathrm{\_Z}+1\,\mathrm{label}=\mathrm{\_L2}\right)\right]$

$\mathrm{f1}≔\mathrm{expand}\left({\left(x-1\right)}^4\left(\mathrm{eval}\left(z^4-z-1\,z=x^3+x\right)\right)\right)\:$

$\mathrm{Polynomial}\left(\mathrm{f1}\,x\,\mathrm{domain}=\mathrm{integer}\right)$

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

$\mathrm{Polynomial}\left(\mathrm{f1}\,x\,\mathrm{domain}=\mathrm{integer}\,\mathrm{dropmultiplicity}\right)$

$\left[1\right]$

$\mathrm{Polynomial}\left(\mathrm{f1}\,x\,\mathrm{domain}=\mathrm{rational}\right)$

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

$\mathrm{Polynomial}\left(\mathrm{f1}\,x\,\mathrm{domain}=\mathrm{real}\right)$

$\left[\mathrm{RootOf}\left({\mathrm{\_Z}}^{12}+4{\mathrm{\_Z}}^{10}+6{\mathrm{\_Z}}^8+4{\mathrm{\_Z}}^6+{\mathrm{\_Z}}^4-{\mathrm{\_Z}}^3-\mathrm{\_Z}-1\,\mathrm{-0.5542396980}\right)\,\mathrm{RootOf}\left({\mathrm{\_Z}}^{12}+4{\mathrm{\_Z}}^{10}+6{\mathrm{\_Z}}^8+4{\mathrm{\_Z}}^6+{\mathrm{\_Z}}^4-{\mathrm{\_Z}}^3-\mathrm{\_Z}-1\,0.7679130647\right)\,1\,1\,1\,1\right]$

$\mathrm{SolveTools}:-\mathrm{Polynomial}\left(a{x}^2-\left(b+a\right)x+b\,x\right)$

$\left[\frac{b}{a}\,1\right]$

$\mathrm{Polynomial}\left(a{x}^2-\left(b+a\right)x+b\,x\,\mathrm{domain}=\mathrm{parametric}\right)$

$\left\{\begin{array}{cc}\left\{\begin{array}{cc}\left[x\right]& b=0\\ \left[1\right]& \mathrm{otherwise}\end{array}\right.& a=0\\ \left[1\,\frac{b}{a}\right]& \mathrm{otherwise}\end{array}\right.$