Vieta's Formulas

Introduction

Vieta's Formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after the French mathematician François Viète and are very useful in algebra, especially for solving polynomial equations.

General Polynomial

Consider a general polynomial of degree $ n $:

$$ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0 $$

Where $ a_n, a_{n-1}, \dots, a_1, a_0 $ are real coefficients and $ a_n \neq 0 $.

For the polynomial $ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0 $ with roots $ x_1, x_2, \dots, x_n $, Vieta's Formulas are:

The sum of the roots (taken one at a time) is given by:

$$ x_1 + x_2 + \cdots + x_n = -\frac{a_{n-1}}{a_n} $$

The sum of the products of the roots taken two at a time is given by:

$$ x_1 x_2 + x_1 x_3 + \cdots + x_{n-1} x_n = \frac{a_{n-2}}{a_n} $$

In general, the sum of the products of the roots taken $ k $ at a time is:

$$ \sum_{1 \leq i_1 < i_2 < \cdots < i_k \leq n} x_{i_1} x_{i_2} \cdots x_{i_k} = (-1)^k \frac{a_{n-k}}{a_n} $$

The product of all the roots is:

$$ x_1 x_2 \cdots x_n = (-1)^n \frac{a_0}{a_n} $$

Examples

Example 1

Consider the polynomial $ 2x^3 - 6x^2 + 2x - 1 = 0 $. Apply Vieta's Formulas:

The sum of the roots is:

$$ x_1 + x_2 + x_3 = -\frac{-6}{2} = 3 $$

The sum of the products of the roots taken two at a time is:

$$ x_1 x_2 + x_1 x_3 + x_2 x_3 = \frac{2}{2} = 1 $$

The product of all the roots is:

$$ x_1 x_2 x_3 = (-1)^3 \frac{-1}{2} = \frac{1}{2} $$

Example 2

For the polynomial $ x^2 - 5x + 6 = 0 $:

The sum of the roots:

$$ x_1 + x_2 = -\frac{-5}{1} = 5 $$

The product of the roots:

$$ x_1 x_2 = \frac{6}{1} = 6 $$

Exercises

Exercise 1

Solve the polynomial equation $ x^3 + 2x^2 - 5x - 6 = 0 $ using Vieta's Formulas.

Exercise 2

Find the roots of the polynomial $ 3x^2 - 11x + 6 = 0 $ and verify using Vieta's Formulas.