# Viète's Formulas/Examples

## Examples of Use of Viète's Formulas

### Coefficients of Cubic

Consider the cubic equation:

$x^3 + a_1 x^2 + a_2 x^1 + a_3 = 0$

Let its roots be denoted $x_1$, $x_2$ and $x_3$.

Then:

 $\displaystyle x_1 + x_2 + x_3$ $=$ $\displaystyle -a_1$ $\displaystyle x_1 x_2 + x_2 x_3 + x_3 x_1$ $=$ $\displaystyle a_2$ $\displaystyle x_1 x_2 x_3$ $=$ $\displaystyle -a_3$

### Coefficients of Quartic

Consider the quartic equation:

$x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4 = 0$

Let its roots be denoted $x_1$, $x_2$, $x_3$ and $x_4$.

Then:

 $\displaystyle x_1 + x_2 + x_3 + x_4$ $=$ $\displaystyle -a_1$ $\displaystyle x_1 x_2 + x_2 x_3 + x_3 x_4 + x_4 x_1 + x_1 x_3 + x_2 x_4$ $=$ $\displaystyle a_2$ $\displaystyle x_1 x_2 x_3 + x_2 x_3 x_4 + x_1 x_2 x_4 + x_1 x_3 x_4$ $=$ $\displaystyle -a_3$ $\displaystyle x_1 x_2 x_3 x_4$ $=$ $\displaystyle a_4$

### Two Numbers whose Sum is $4$ and whose Product is $8$

Let $z_1$ and $z_2$ be two numbers whose sum is $4$ and whose product is $8$.

Then:

 $\displaystyle z_1$ $=$ $\displaystyle 2 + 2 i$ $\displaystyle z_2$ $=$ $\displaystyle 2 - 2 i$

### Cubic with all Equal Roots

The coefficient of $x$ in the expansion of cubic $\paren {x - 1}^3$ is $3$.

### Monic Polynomial Formulas

Let:

 $\displaystyle \map P x$ $=$ $\displaystyle x^N + \displaystyle \sum_{k \mathop = 0}^{N - 1} b_k x^k$ Monic polynomial of degree $N$.

Let $U$ be the set of $N$ roots of equation $\map P x = 0$.

Then:

 $\displaystyle b_k$ $=$ $\displaystyle \paren {-1}^{N - k} \, \map {e_{N - k} } U, \quad 0 \le k \le N - 1$ Definition of Elementary Symmetric Function $\map {e_m} U$