Question: How do you write the coefficients of a polynomial in terms of its roots?

Answer: The coefficients are plus or minus the elementary symmetric polynomials in the roots.

Here's what this means. If a₁, . . . , a_n are variables, and i = 1, . . . n, the ith elementary symmetric polynomial s_i(a₁, . . . , a_n) is the sum of all the monomials in a₁, . . . , a_n such that (1) every monomial has degree i, (2) no variable a_j occurs to a power greater than 1.

For example, here are the elementary symmetric polynomials in the variables a₁, a₂, a₃, a₄:

s₁(a₁, a₂, a₃, a₄) = a₁ + a₂ + a₃ + a₄

s₂(a₁, a₂, a₃, a₄) = a₁a₂ + a₁a₃ + a₁a₄ + a₂a₃ + a₂a₄ + a₃a₄

s₃(a₁, a, a₃, a₄) = a₂a₃a₄ + a₁a₃a₄ + a₁a₂a₄ + a₁a₂a₃

s₄(a₁, a, a₃, a₄) = a₁a₂a₃a₄

Now if f(x) has roots r₁, . . . , r_n, we have

f(x) = c₀ + c₁x¹ + . . . + c_n-1x^n-1 + xⁿ = (x - r₁) . . . (x - r_n),

and for i = 0, . . . n-1, the roots determine the coefficient c_i by

c_i = (-1)^n-i s_n-i (r₁, . . . , r_n) .