Vieta’s Formulas

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:

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