CoCalc -- 1543.sagews

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x=var('x')
g(x)=x^3+(3*x)+1

solve(g(x)==0, 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)]

plot(g(x),x,(-5,5))

solve(g(g(x))==0, x)

[x == -1/2*(I*sqrt(3) + 1)*(-1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) + 1)*2^(1/3) + 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) + 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) + 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) - 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3) + 1/2*(-I*sqrt(3) + 1)/(-1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) + 1)*2^(1/3) + 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) + 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) + 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) - 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3), x == -1/2*(-I*sqrt(3) + 1)*(-1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) + 1)*2^(1/3) + 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) + 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) + 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) - 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3) + 1/2*(I*sqrt(3) + 1)/(-1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) + 1)*2^(1/3) + 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) + 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) + 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) - 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3), x == (-1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) + 1)*2^(1/3) + 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) + 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) + 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) - 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3) - 1/(-1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) + 1)*2^(1/3) + 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) + 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) + 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) - 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3), x == -1/2*(I*sqrt(3) + 1)*(1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) - 1)*2^(1/3) - 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) + 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(-2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) - 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) - 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) + 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3) + 1/2*(-I*sqrt(3) + 1)/(1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) - 1)*2^(1/3) - 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) + 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(-2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) - 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) - 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) + 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3), x == -1/2*(-I*sqrt(3) + 1)*(1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) - 1)*2^(1/3) - 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) + 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(-2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) - 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) - 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) + 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3) + 1/2*(I*sqrt(3) + 1)/(1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) - 1)*2^(1/3) - 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) + 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(-2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) - 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) - 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) + 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3), x == (1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) - 1)*2^(1/3) - 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) + 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(-2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) - 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) - 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) + 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3) - 1/(1/8*((sqrt(5) - 1)^(2/3)*(I*sqrt(3) - 1)*2^(1/3) - 2*(sqrt(5) - 1)^(1/3)*2^(2/3) + 2*I*sqrt(3) + 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/8*sqrt(-2*I*(sqrt(5) - 1)^(4/3)*sqrt(3) - 4*I*(sqrt(5) - 1)*2^(1/3)*sqrt(3) - 2*(sqrt(5) - 1)^(4/3) + 4*(sqrt(5) - 1)*2^(1/3) + 12*(sqrt(5) - 1)^(2/3)*2^(2/3) - 8*I*(sqrt(5) - 1)^(1/3)*sqrt(3) + 4*I*2^(1/3)*sqrt(3) - 8*(sqrt(5) - 1)^(1/3) - 4*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3), x == -1/2*(I*sqrt(3) + 1)*(1/4*((sqrt(5) - 1)^(2/3)*2^(1/3) - (sqrt(5) - 1)^(1/3)*2^(2/3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/4*sqrt((sqrt(5) - 1)^(4/3) - 2*(sqrt(5) - 1)*2^(1/3) + 3*(sqrt(5) - 1)^(2/3)*2^(2/3) + 4*(sqrt(5) - 1)^(1/3) + 2*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3) + 1/2*(-I*sqrt(3) + 1)/(1/4*((sqrt(5) - 1)^(2/3)*2^(1/3) - (sqrt(5) - 1)^(1/3)*2^(2/3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/4*sqrt((sqrt(5) - 1)^(4/3) - 2*(sqrt(5) - 1)*2^(1/3) + 3*(sqrt(5) - 1)^(2/3)*2^(2/3) + 4*(sqrt(5) - 1)^(1/3) + 2*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3), x == -1/2*(-I*sqrt(3) + 1)*(1/4*((sqrt(5) - 1)^(2/3)*2^(1/3) - (sqrt(5) - 1)^(1/3)*2^(2/3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/4*sqrt((sqrt(5) - 1)^(4/3) - 2*(sqrt(5) - 1)*2^(1/3) + 3*(sqrt(5) - 1)^(2/3)*2^(2/3) + 4*(sqrt(5) - 1)^(1/3) + 2*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3) + 1/2*(I*sqrt(3) + 1)/(1/4*((sqrt(5) - 1)^(2/3)*2^(1/3) - (sqrt(5) - 1)^(1/3)*2^(2/3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/4*sqrt((sqrt(5) - 1)^(4/3) - 2*(sqrt(5) - 1)*2^(1/3) + 3*(sqrt(5) - 1)^(2/3)*2^(2/3) + 4*(sqrt(5) - 1)^(1/3) + 2*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3), x == (1/4*((sqrt(5) - 1)^(2/3)*2^(1/3) - (sqrt(5) - 1)^(1/3)*2^(2/3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/4*sqrt((sqrt(5) - 1)^(4/3) - 2*(sqrt(5) - 1)*2^(1/3) + 3*(sqrt(5) - 1)^(2/3)*2^(2/3) + 4*(sqrt(5) - 1)^(1/3) + 2*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3) - 1/(1/4*((sqrt(5) - 1)^(2/3)*2^(1/3) - (sqrt(5) - 1)^(1/3)*2^(2/3) - 2)*2^(1/3)/(sqrt(5) - 1)^(1/3) + 1/4*sqrt((sqrt(5) - 1)^(4/3) - 2*(sqrt(5) - 1)*2^(1/3) + 3*(sqrt(5) - 1)^(2/3)*2^(2/3) + 4*(sqrt(5) - 1)^(1/3) + 2*2^(1/3))*2^(2/3)/(sqrt(5) - 1)^(1/3))^(1/3)]

plot(g(g(g(x))),x,(-2,2))

x^27