Fundamental Theorem of Algebra

2244 days ago by calkin

theta=var('theta') R=var('R') 
       
 
       
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)) 
       
show(A) 
       
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) 
       
 
       
from sage.rings.polynomial.complex_roots import complex_roots 
       
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') 
       
show(A+B) 
       
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)