# Root locus problem

The problem consists of two questions: a) Find k so that one of the roots is at s=-2 $$(s+1)^5+k=0$$ b) for the value of k you just found find the rest roots using root locus.

Regarding the first question s=-2 can be a solution if k=1. Whenever I use root locus there is a parameter in the equation. Since we consider k=1 in the b question how am I supposed to draw a root locus?

Update: This is a problem from an exam sheet and computers can't be used. What are the ways of finding the 4 other roots by hand? I can draw the root locus easily and I can see the s=-2 pole's movement and position when k becomes 1. What happens in the other 4 branches(poles) of the root locus though? Has the pole travelled the same distance in each of them?

I could also see that we have 2 conjugate pairs of poles but the system is too complicated to solve.

You need to plot the root locus assuming k is a parameter, and then from that root locus find the roots when $k=1$ (or one of the roots is at -2).
To find out the values of the roots when $k=1$, solve $(s+1)^5+1=0$. This will give the values $\{-2.,-1.30902\pm 0.951057 i,-0.190983\pm 0.587785 i\}$.
In general, for any specific value $kval$ of the parameter solve $(s+1)^5+kval=0$ to get the roots for that value.