Viète's Formulas/Examples

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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 $\map {e_m} U$