给定包含R根的向量，如何计算多项式的系数

Question

I want to compute the coefficients of a polynomial based on a vector containing the roots. I first defined a vector of coefficients:

pol <- c(0,1,2,3,4) 

and computed the roots

roots <- polyroot(pol)

to have a test result. Then i tried the following:

result <- 1
for (n in 1:(length(roots))){
result <- c(result, 0) + c(0,-roots[n]*result)
}

But the my result is the following:

result 
[1] 1.00+0i 0.75+0i 0.50+0i 0.25+0i 0.00+0i

What am I missing here?

Answer 1

Notice that

identical(polyroot(pol), polyroot(pol / 4))
# [1] TRUE

That is, by going from a polynomial to its roots you lose information about the coefficient of the highest degree term (in this case, 4). For instance, 2x^2-x=2x(x-1/2), but also x^2-x/2=x(x-1/2), so that the roots are the same and we only normalized the first polynomial with respect to the quadratic term. So,

Re(result) * 4
# [1] 4 3 2 1 0

gives the result but also requires the knowledge of tail(pol, 1) .