Viète's Formulas/Examples

Examples of Use of Viète's Formulas

Sum of Roots of Quadratic Equation

Let $P$ be the quadratic equation $a x^2 + b x + c = 0$.

Let $\alpha$ and $\beta$ be the roots of $P$.

Then:

$\alpha + \beta = -\dfrac b a$

Product of Roots of Quadratic Equation

Let $P$ be the quadratic equation $a x^2 + b x + c = 0$.

Let $\alpha$ and $\beta$ be the roots of $P$.

Then:

$\alpha \beta = \dfrac c a$

Coefficients of Cubic

Consider the cubic equation:

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

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

Then:

\(\ds x_1 + x_2 + x_3\)

\(=\)

\(\ds -a_1\)

\(\ds x_1 x_2 + x_2 x_3 + x_3 x_1\)

\(=\)

\(\ds a_2\)

\(\ds x_1 x_2 x_3\)

\(=\)

\(\ds -a_3\)

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:

\(\ds x_1 + x_2 + x_3 + x_4\)

\(=\)

\(\ds -a_1\)

\(\ds x_1 x_2 + x_2 x_3 + x_3 x_4 + x_4 x_1 + x_1 x_3 + x_2 x_4\)

\(=\)

\(\ds a_2\)

\(\ds x_1 x_2 x_3 + x_2 x_3 x_4 + x_1 x_2 x_4 + x_1 x_3 x_4\)

\(=\)

\(\ds -a_3\)

\(\ds x_1 x_2 x_3 x_4\)

\(=\)

\(\ds a_4\)

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:

\(\ds z_1\)

\(=\)

\(\ds 2 + 2 i\)

\(\ds z_2\)

\(=\)

\(\ds 2 - 2 i\)

Cubic with all Equal Roots

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

Monic Polynomial Formulas

Let:

$\ds \map P x = x^N + \sum_{k \mathop = 0}^{N - 1} b_k x^k$

be a monic polynomial of degree $N$.

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

Then:

$b_k = \paren {-1}^{N - k} \map {e_{N - k} } U, \quad 0 \le k \le N - 1$

where $\map {e_m} U$ denotes an elementary symmetric function.