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fricas
digits 20

\label{eq1}20(1)
fricas
-- n:=x^3+a1*x^2+a2*x+a3 ::Polynomial Fraction Integer
Q:=(3*a2-a1^2)/9

\label{eq2}{{1 \over 3}\  a 2}-{{1 \over 9}\ {{a 1}^{2}}}(2)
Type: Polynomial(Fraction(Integer))
fricas
R:=(9*a1*a2-27*a3-2*a1^3)/54

\label{eq3}-{{1 \over 2}\  a 3}+{{1 \over 6}\  a 1 \  a 2}-{{1 \over{27}}\ {{a 1}^{3}}}(3)
Type: Polynomial(Fraction(Integer))
fricas
S:=(R+(Q^3+R^2)^(1/2))^(1/3)

\label{eq4}\root{3}\of{{{9 \ {\sqrt{{{27}\ {{a 3}^{2}}}+{{\left(-{{18}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{54}\ {\sqrt{3}}}}(4)
Type: Expression(Integer)
fricas
T:=(R-(Q^3+R^2)^(1/2))^(1/3)

\label{eq5}\root{3}\of{{-{9 \ {\sqrt{{{27}\ {{a 3}^{2}}}+{{\left(-{{18}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{54}\ {\sqrt{3}}}}(5)
Type: Expression(Integer)
fricas
x1:=S+T-a1/3

\label{eq6}{\left(
\begin{array}{@{}l}
\displaystyle
{3 \ {\root{3}\of{{{9 \ {\sqrt{{{27}\ {{a 3}^{2}}}+{{\left(-{{1
8}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{5
4}\ {\sqrt{3}}}}}}+ 
\
\
\displaystyle
{3 \ {\root{3}\of{{-{9 \ {\sqrt{{{27}\ {{a 3}^{2}}}+{{\left(-{{18}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{5
4}\ {\sqrt{3}}}}}}- 
\
\
\displaystyle
a 1 
(6)
Type: Expression(Integer)
fricas
x2:=-(S+T)/2-a1/3 + %i*sqrt(3)*(S-T)/2

\label{eq7}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({3 \  i \ {\sqrt{3}}}- 3 \right)}\ {\root{3}\of{{{9 \ {\sqrt{{{2
7}\ {{a 3}^{2}}}+{{\left(-{{18}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{5
4}\ {\sqrt{3}}}}}}+ 
\
\
\displaystyle
{{\left(-{3 \  i \ {\sqrt{3}}}- 3 \right)}\ {\root{3}\of{{-{9 \ {\sqrt{{{27}\ {{a 3}^{2}}}+{{\left(-{{18}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{54}\ {\sqrt{3}}}}}}- 
\
\
\displaystyle
{2 \  a 1}
(7)
Type: Expression(Complex(Integer))
fricas
x3:=-(S+T)/2-a1/3 - %i*sqrt(3)*(S-T)/2

\label{eq8}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{3 \  i \ {\sqrt{3}}}- 3 \right)}\ {\root{3}\of{{{9 \ {\sqrt{{{27}\ {{a 3}^{2}}}+{{\left(-{{18}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{54}\ {\sqrt{3}}}}}}+ 
\
\
\displaystyle
{{\left({3 \  i \ {\sqrt{3}}}- 3 \right)}\ {\root{3}\of{{-{9 \ {\sqrt{{{27}\ {{a 3}^{2}}}+{{\left(-{{18}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{54}\ {\sqrt{3}}}}}}- 
\
\
\displaystyle
{2 \  a 1}
(8)
Type: Expression(Complex(Integer))
fricas
a5:=x^3+a1*x^2+a2*x+a3 ::Polynomial AlgebraicNumber

\label{eq9}{{x}^{3}}+{a 1 \ {{x}^{2}}}+{a 2 \  x}+ a 3(9)
Type: Polynomial(AlgebraicNumber?)
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a6:=(x-x11)  ::Polynomial AlgebraicNumber;
Type: Polynomial(AlgebraicNumber?)
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a7:=monicDivide(a5,a6,x) ;
Type: Record(quotient: Polynomial(AlgebraicNumber?),remainder: Polynomial(AlgebraicNumber?))
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a77:=a7.quotient;
Type: Polynomial(AlgebraicNumber?)
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a78:=a7.remainder;
Type: Polynomial(AlgebraicNumber?)
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qu1 :=eval(a77,x11,x1)

\label{eq10}{\left(
\begin{array}{@{}l}
\displaystyle
{9 \ {{\root{3}\of{{{9 \ {\sqrt{{{27}\ {{a 3}^{2}}}+{{\left(-{{18}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{5
4}\ {\sqrt{3}}}}}^{2}}}+ 
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{18}\ {\root{3}\of{{-{9 \ {\sqrt{{{27}\ {{a 3}^{2}}}+{{\left(-{{18}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{5
4}\ {\sqrt{3}}}}}}+ 
\
\
\displaystyle
{9 \  x}+{3 \  a 1}
(10)
Type: Expression(Integer)
fricas
rem1:=eval(a78,x11,x1)

\label{eq11}{\left(
\begin{array}{@{}l}
\displaystyle
{
\begin{array}{@{}l}
\displaystyle
9 \  \cdot 
\
\
\displaystyle
{\root{3}\of{{-{9 \ {\sqrt{{{27}\ {{a 3}^{2}}}+{{\left(-{{18}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{54}\ {\sqrt{3}}}}}\  \cdot 
\
\
\displaystyle
{{\root{3}\of{{{9 \ {\sqrt{{{27}\ {{a 3}^{2}}}+{{\left(-{{18}\  a 1 \  a 2}+{4 \ {{a 1}^{3}}}\right)}\  a 3}+{4 \ {{a 2}^{3}}}-{{{a 1}^{2}}\ {{a 2}^{2}}}}}}+{{\left(-{{27}\  a 3}+{9 \  a 1 \  a 2}-{2 \ {{a 1}^{3}}}\right)}\ {\sqrt{3}}}}\over{{54}\ {\sqrt{3}}}}}^{2}}
(11)
Type: Expression(Integer)
fricas
eval(rem1,[a3=1.0, a2=1.0, a1=1.0])

\label{eq12}-{0.2 E - 19}(12)
Type: Expression(Float)

How about this:

fricas
pkg:= SOLVEFOR(UP('x,Complex Float), Complex Float)

\label{eq13}\hbox{\axiomType{PolynomialSolveByFormulas}\ } (\hbox{\axiomType{UnivariatePolynomial}\ } (x , \hbox{\axiomType{Complex}\ } (\hbox{\axiomType{Float}\ })) , \hbox{\axiomType{Complex}\ } (\hbox{\axiomType{Float}\ }))(13)
Type: Type
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root := aCubic(1,1,1,1)$pkg

\label{eq14}-{0.9999999999_9999999997}(14)
Type: Complex(Float)
fricas
qfactor := monicDivide(x^3 + x^2 + x + 1,x - root)

\label{eq15}\begin{array}{@{}l}
\displaystyle
\left[{quotient ={{{x}^{2}}+{{0.3 E - 19}\  x}+{0.9999999999_9999999997}}}, \: \right.
\
\
\displaystyle
\left.{remainder ={0.6 E - 19}}\right] 
(15)
Type: Record(quotient: UnivariatePolynomial(x,Float),remainder: UnivariatePolynomial(x,Float))
fricas
qfactor.quotient

\label{eq16}{{x}^{2}}+{{0.3 E - 19}\  x}+{0.9999999999_9999999997}(16)
Type: UnivariatePolynomial(x,Float)
fricas
qfactor.remainder

\label{eq17}0.6 E - 19(17)
Type: UnivariatePolynomial(x,Float)


myeq := (x-9)^3-(x/2)^2+13<em>x 
 




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