public-notes/Vieta’s Formulas.md

#Math #Algebra

Let polynomial a be:

a = c_n \prod _{i = 0}^n (x - r_i)

where r_i is a root of a, and c_n is the leading coefficient of a.

We can also represent a as:

a = \sum _{i = 0}^n c_i x^i

By expanding the first definition of a, we can define c_i by:

c_{n-i} = (-1)^i c_n\sum _{sym}^i r

This is through the nature of multiplying binomials, with the coefficient resulting in the sum of all possible combinations of i roots multiplied together, or the $i$th elementary symmetric sum of set r. We also have to multiply by the negative sign, resulting in (-1)^i

We can refactor to state:

\sum _{sym}^i r = (-1)^i \frac {c_{n-i}} {c_n}