Newton's Relations

Let $s_i$ be the sum of the products of distinct Roots $r_j$ of the Polynomial equation of degree $n$

$$a_n x^n + a_{n-1} x^{n-1} + \ldots+a_1x+a_0 = 0,$$ (1)

where the roots are taken $i$ at a time (i.e., $s_i$ is defined as the Elementary Symmetric Function $\Pi_i(r_1, \ldots, r_n)$) $s_i$ is defined for $i=1$, ..., $n$. For example, the first few values of $s_i$ are

$$s_1 = r_1+r_2+r_3+r_4+\ldots$$ (2)

$$s_2 = r_1r_2+r_1r_3+r_1r_4+r_2r_3+\ldots$$ (3)

$$s_3 = r_1r_2r_3+r_1r_2r_4+r_2r_3r_4+\ldots,$$ (4)

and so on. Then

$$s_i=(-1)^i {a_{n-i}\over a_n}.$$ (5)

This can be seen for a second Degree Polynomial by multiplying out,

$$a_2x^2+a_1x+a_0=a_2(x-r_1)(x-r_2)=a_2[x^2-(r_1+r_2)x+r_1r_2],$$ (6)

so

$$s_1 = \sum_{i=1}^2 r_i=r_1+r_2=-{a_1\over a_2}$$ (7)

$$s_2 = \sum_{\scriptstyle i,j=1\atop\scriptstyle i\not=j}^2 r_ir_j=r_1r_2={a_0\over a_2},$$ (8)

and for a third Degree Polynomial,

$a_3x^3+a_2x^2+a_1x+a_0=a_3(x-r_1)(x-r_2)(x-r_3)$
$= a_3[x^3-(r_1+r_2+r_3)x^2+(r_1r_2+r_1r_3+r_2r_3)x-r_1r_2r_3],\quad$	(9)

so

$$s_1 = \sum_{i=1}^3 r_i = -{a_2\over a_3}$$ (10)

$$s_2 = \sum_{\scriptstyle i,j\atop\scriptstyle i\not=j}^3 r_ir_j = r_1r_2+r_1r_3+r_2r_3={a_1\over a_3}$$ (11)

$$s_3 = \sum_{\scriptstyle i,j,k\atop\scriptstyle i\not=j\not=k}^3 r_ir_jr_k = r_1r_2r_3 =-{a_0\over a_3}.$$ (12)

See also Elementary Symmetric Function

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 1-2, 1959.