theta=var('theta')
R=var('R')
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R=1.75
A=parametric_plot( (R^3*cos(3*theta) + 3*R*cos(theta)+1,R^3*sin(3*theta) + 3*R*sin(theta)), (theta, 0,2*pi))
R=1.76
B=parametric_plot( (R^3*cos(3*theta) + 3*R*cos(theta)+1,R^3*sin(3*theta) + 3*R*sin(theta)), (theta, 0,2*pi))
C=parametric_plot( (0.01*cos(theta),0.01*sin(theta)), (theta, 0,2*pi))
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show(A)
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complex_roots(x^3+3*x+100)
[(-4.426307243864840?, 1), (2.213153621932420? - 4.20644111605260?*I, 1), (2.213153621932420? + 4.20644111605260?*I, 1)] [(-4.426307243864840?, 1), (2.213153621932420? - 4.20644111605260?*I, 1), (2.213153621932420? + 4.20644111605260?*I, 1)] |
x = polygen(ZZ)
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from sage.rings.polynomial.complex_roots import complex_roots
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sqrt(0.161092677313043^2+1.754380959783722^2)
1.76176144887313 1.76176144887313 |
A=parametric_plot3d( (R^3*cos(3*theta) + 3*R*cos(theta)+100,R^3*sin(3*theta) + 3*R*sin(theta),R), (theta, 0,2*pi),(R,4.4,4.5))
B=parametric_plot3d((0,0,R),(R,0,7),color='red')
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show(A+B)
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solve(x^3+3*x+1,x)
[x == -1/2*(1/2*sqrt(5) - 1/2)^(1/3)*(I*sqrt(3) + 1) + 1/2*(-I*sqrt(3) + 1)/(1/2*sqrt(5) - 1/2)^(1/3), x == -1/2*(1/2*sqrt(5) - 1/2)^(1/3)*(-I*sqrt(3) + 1) + 1/2*(I*sqrt(3) + 1)/(1/2*sqrt(5) - 1/2)^(1/3), x == (1/2*sqrt(5) - 1/2)^(1/3) - 1/(1/2*sqrt(5) - 1/2)^(1/3)] [x == -1/2*(1/2*sqrt(5) - 1/2)^(1/3)*(I*sqrt(3) + 1) + 1/2*(-I*sqrt(3) + 1)/(1/2*sqrt(5) - 1/2)^(1/3), x == -1/2*(1/2*sqrt(5) - 1/2)^(1/3)*(-I*sqrt(3) + 1) + 1/2*(I*sqrt(3) + 1)/(1/2*sqrt(5) - 1/2)^(1/3), x == (1/2*sqrt(5) - 1/2)^(1/3) - 1/(1/2*sqrt(5) - 1/2)^(1/3)] |
plot(x^3+3*x+1,x,-3,4)
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