A quick and clever method for solving nonlinear systems of equations

[royzizzle]

if we are given a polynomial s^6+as^5+bs^4+cs^3+ds^2+es+k

(if a,b,c,d are known) what is a clever method to solving for the value k if we are given the following:

the above polynomial is equal to the following(zeta is given as some constant, say 1 for simplicity):
(s+alpha)(s+beta)(s+gamma)(s+lamda)(s^2+2w*zeta*s+w^2)

we would have to multiply out this expression that separate out coefficients for s and equation to a,b,c, and d

we then have a system of 6 nonlinear equations to solve for 6 unknowns

what is the fastest way to do this by hand within a 20 minute timeframe?

 

[Stephen Tashi]

we would have to multiply out this expression that separate out coefficients for s and equation to a,b,c, and d

If you only want to know the value of the constant term "k", you'd only have to think about all the terms in the product that can be formed without have any variable "s" in them. You wouldn't have to multiply out the whole product.

If you only want to know the value of the constant term "k", you could substitute s = 0 in the second function.