Vieta's formulas

Theorem 14.1.4 (Vieta’s Formula For Higher Degree Polynomials) In a polynomial

anx n + an−1x n−1 + ... + a1x + a0 = 0

with roots

r1, r2, r3, ...rn

the following holds:

r1 + r2 + r3 + ... + rn (the sum of all terms) = − an−1 an

r1r2 + r1r3 + .. + rn−1rn (the sum of all products of 2 terms) = an−2 an

r1r2r3 + r1r2r4 + ... + rn−2rn−1rn (the sum of all products of 3 terms) = − an−3 an

. . .

r1r2r3 . . . rn (the sum of all products of n terms) = (−1)n a0 an

Note that the negative and positive signs alternate. When summing the products for odd number of terms, we will have a negative sign otherwise we will have a positive sign.