axiom digits 20
Type: PositiveInteger axiom -- n:=x^3+a1*x^2+a2*x+a3 ::Polynomial Fraction Integer
Type: Polynomial(Fraction(Integer)) axiom R:=(9*a1*a2-27*a3-2*a1^3)/54
Type: Polynomial(Fraction(Integer)) axiom S:=(R+(Q^3+R^2)^(1/2))^(1/3)
Type: Expression(Integer) axiom T:=(R-(Q^3+R^2)^(1/2))^(1/3)
Type: Expression(Integer) axiom x1:=S+T-a1/3
Type: Expression(Integer) axiom x2:=-(S+T)/2-a1/3 + %i*sqrt(3)*(S-T)/2
Type: Expression(Complex(Integer)) axiom x3:=-(S+T)/2-a1/3 - %i*sqrt(3)*(S-T)/2
Type: Expression(Complex(Integer)) axiom a5:=x^3+a1*x^2+a2*x+a3 ::Polynomial AlgebraicNumber
Type: Polynomial(AlgebraicNumber) axiom a6:=(x-x11) ::Polynomial AlgebraicNumber; Type: Polynomial(AlgebraicNumber) axiom a7:=monicDivide(a5,a6,x) ; Type: Record(quotient: Polynomial(AlgebraicNumber),remainder: Polynomial(AlgebraicNumber)) axiom a77:=a7.quotient; Type: Polynomial(AlgebraicNumber) axiom a78:=a7.remainder; Type: Polynomial(AlgebraicNumber) axiom qu1 :=eval(a77,x11,x1)
Type: Expression(Integer) axiom rem1:=eval(a78,x11,x1)
Type: Expression(Integer) axiom eval(rem1,[a3=1.0, a2=1.0, a1=1.0])
Type: Expression(Float) ... --wyscc, Mon, 14 Nov 2005 02:53:03 -0600 reply How about this:
axiom pkg:= SOLVEFOR(UP('x,Complex Float), Complex Float)
Type: Domain axiom root := aCubic(1,1,1,1)$pkg
Type: Complex(Float) axiom qfactor := monicDivide(x^3 + x^2 + x + 1,x - root)
Type: Record(quotient: UnivariatePolynomial(x,Float),remainder: UnivariatePolynomial(x,Float)) axiom qfactor.quotient
Type: UnivariatePolynomial(x,Float) axiom qfactor.remainder
Type: UnivariatePolynomial(x,Float)
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last edited 2 years ago |
![\label{eq6}{\left(
\begin{array}{@{}l}
\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{{54}\ {\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{{54}\ {\sqrt{3}}}}}}-
\
\
\displaystyle
a 1](images/6359844075857875164-16.0px.png "\label{eq6}{\left(
\begin{array}{@{}l}
\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{{54}\ {\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{{54}\ {\sqrt{3}}}}}}-
\
\
\displaystyle
a 1")
![\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{{5
4}\ {\sqrt{3}}}}}}-
\
\
\displaystyle
{2 \ a 1}](images/3083892654891197839-16.0px.png "\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{{5
4}\ {\sqrt{3}}}}}}-
\
\
\displaystyle
{2 \ a 1}")
![\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{{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{{5
4}\ {\sqrt{3}}}}}}-
\
\
\displaystyle
{2 \ a 1}](images/9145670520910480061-16.0px.png "\label{eq8}{\left(
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\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{{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{{5
4}\ {\sqrt{3}}}}}}-
\
\
\displaystyle
{2 \ a 1}")
![\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{{54}\ {\sqrt{3}}}}}^2}}+
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{18}\ {\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{{54}\ {\sqrt{3}}}}}}+
\
\
\displaystyle
{9 \ x}+{3 \ a 1}](images/746122890872659713-16.0px.png "\label{eq10}{\left(
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\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{{54}\ {\sqrt{3}}}}}^2}}+
\
\
\displaystyle
{{\left({
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\displaystyle
{{18}\ {\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{{54}\ {\sqrt{3}}}}}}+
\
\
\displaystyle
{9 \ x}+{3 \ a 1}")
![\label{eq11}{\left(
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\displaystyle
{
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\displaystyle
9 \ \cdot
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\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{{5
4}\ {\sqrt{3}}}}}\ \cdot
\
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\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{{5
4}\ {\sqrt{3}}}}}^2}](images/2339978341145759753-16.0px.png "\label{eq11}{\left(
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9 \ \cdot
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4}\ {\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{{5
4}\ {\sqrt{3}}}}}^2}")
![\label{eq15}\begin{array}{@{}l}
\displaystyle
\left[{quotient ={{x^2}+{{0.3 E - 19}\ x}+{0.999999999999999
99997}}}, \: \right.
\
\
\displaystyle
\left.{remainder ={0.6 E - 19}}\right]](images/310944308584274319-16.0px.png "\label{eq15}\begin{array}{@{}l}
\displaystyle
\left[{quotient ={{x^2}+{{0.3 E - 19}\ x}+{0.999999999999999
99997}}}, \: \right.
\
\
\displaystyle
\left.{remainder ={0.6 E - 19}}\right]")
