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Algebraic

 MakeMonic
 rewrite a RootOf in terms of a monic RootOf

 Calling Sequence MakeMonic(r)

Parameters

 r - algebraic expression

Description

 • MakeMonic expresses a RootOf in terms of a monic RootOf. It pulls the leading coefficient of the defining polynomial out of the RootOf and into the denominator.
 • MakeMonic only works on indexed, labeled, or one-argument RootOfs. For RootOfs with a numerical approximation or a range as selector, MakeMonic returns the input unchanged.
 • Nested RootOfs that are indexed, labeled, or one-argument, are handled recursively.
 • If r is not a RootOf or not of type algext, it is returned unchanged.
 • For indexed RootOfs, the leading coefficient is pulled out only if its signum is $1$ or $-1$.

Examples

 > Algebraic:-MakeMonic(RootOf(y*x^3-1,x));
 $\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{-}{{y}}^{{2}}\right)}{{y}}$ (1)
 > Algebraic:-MakeMonic(1+RootOf(y*x^3-1,x));
 ${1}{+}{\mathrm{RootOf}}{}\left({y}{}{{\mathrm{_Z}}}^{{3}}{-}{1}\right)$ (2)
 > Algebraic:-MakeMonic(RootOf(2*sin(x)-1));
 ${\mathrm{RootOf}}{}\left({2}{}{\mathrm{sin}}{}\left({\mathrm{_Z}}\right){-}{1}\right)$ (3)
 > Algebraic:-MakeMonic(RootOf(y*x^3-1,x,index=1));
 ${\mathrm{RootOf}}{}\left({y}{}{{\mathrm{_Z}}}^{{3}}{-}{1}{,}{\mathrm{index}}{=}{1}\right)$ (4)
 > Algebraic:-MakeMonic(RootOf(y*x^3-1,x,index=1)) assuming y>0;
 $\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{-}{{y}}^{{2}}{,}{\mathrm{index}}{=}{1}\right)}{{y}}$ (5)
 > Algebraic:-MakeMonic(RootOf(y*x^3-1,x,index=1)) assuming y<0;
 ${-}\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{+}{{y}}^{{2}}{,}{\mathrm{index}}{=}{1}\right)}{{y}}$ (6)
 > Algebraic:-MakeMonic(RootOf(5*y^2-RootOf(3*x^2+1,index=1),index=1));
 $\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{15}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{3}{,}{\mathrm{index}}{=}{1}\right){,}{\mathrm{index}}{=}{1}\right)}{{15}}$ (7)

Note that RootOf itself tries to get rid of non-integral leading coefficients by inverting them:

 > RootOf(RootOf(_Z^2-2)*_Z^2-2);
 ${\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}\right)\right)$ (8)

In the next example, the leading coefficient is not invertible, and after pulling it out there is a zero divisor in the denominator:

 > f := RootOf(RootOf(x^2-x)*y^2-2);
 ${f}{≔}{\mathrm{RootOf}}{}\left({\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{\mathrm{_Z}}\right){}{{\mathrm{_Z}}}^{{2}}{-}{2}\right)$ (9)
 > Algebraic:-MakeMonic(f);
 $\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{\mathrm{_Z}}\right)\right)}{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{\mathrm{_Z}}\right)}$ (10)