Find an approximation of the unique real root of $6x^3+2x^2-x-1$

Question

I'm searching a way to find an approximation of the (unique) real root of $6x^3+2x^2-x-1$ . I can use the Newton Method, but I don't know how to find a number that can allow me to start the recursive process. Is there another simpler solution?

Basically, I want to prove that the equation has only one real root, so I can solve the inequality.

Thanks a lot in advance!

Answer 1

Noticing that $f(0)=-1, f(1)=6$ allows us to conclude that there is a root in [0,1] by the IVT. So we can simply do Newton's with starting value $1/2$ to find the root, and thus solve the inequality