Given the following cubic equation
$\lambda^3+( \sigma+b+1)\lambda^2+(r+\sigma)b\lambda+2\sigma b(r-1)=0$
And given the fact that $\sigma=10$ and $b=\frac{8}{3}$
Suppose that that there are $3$ roots to the following equation $\lambda_{1}$,$\lambda_{2}$ and $\lambda_{3}$, and suppose that for some unknown values of $r$ ($r$ is to be taken as always greater than $1$) we have all $3$ roots that are real and negative and as we increase the value of $r$ to above the certain critical value, call this value $r*$, the three root changes to become one real and two complex conjguate roots with the real part of the complex conjugate being negative. Find this critical value of $r*$ in which the complex conjugate roots start to appear.
For a quadratic equation, it is simple I just need the quadratic formula and study how the roots behave, but im unsure how to do it if the equation is in cubic form. What I think of doing is to write the cubic equation in this form $(\lambda-\lambda_{1}$)($\lambda-\lambda_{2}$)($\lambda-\lambda_{3}$) and then expanding out the terms and then comparing it with the original expression. Could anyone explain how to do this. Thanks
You need the discriminant of the polynomial. For a cubic polynomial with real coefficients, it is positive if the roots are three distinct real numbers, and negative if there is one real root and two complex conjugate roots.