If x1, x2 are the roots of a quadratic polynomial x2 + ax + b, then we have
(x - x1)(x - x2) = x2 + ax + b.
By expanding the left side and compare with the right side, we get
a = - (x1 + x2), b = x1x2.
Since the polynomial is specified by the two coefficients a and b, we see that the transformation
roots (x1, x2) → polynomial x2 + ax + b
is essentially the same as
R2 → R2, (x1, x2) → (x1 + x2, x1x2)
As an exercise for yourself, how would you define the transformation Root-Polynomial: R3 → R3 for three roots of polynomials of degree 3?
Finally, you may try the general case Root-Polynomial: Rn → Rn.