Vieta’s formulas consist of a set of equations that connect the coefficients of a polynomial to its roots. Named after the French mathematician François Viète, these formulas provide a powerful tool for understanding the relationship between the roots of a polynomial and its coefficients.

For a polynomial of degree $n$ with roots $r_1, r_2, \ldots, r_n$, Vieta’s formulas can be expressed as follows:

1. The sum of the roots is equal to the negative of the coefficient of the $(n-1)$th power term divided by the coefficient of the $n$th power term: $r_1 + r_2 + \ldots + r_n = -\frac{a_{n-1}}{a_n}$

2. The product of the roots is equal to the constant term divided by the coefficient of the $n$th power term, multiplied by $(-1)^n$: $r_1 r_2 \cdots r_n = (-1)^n \frac{a_0}{a_n}$

3. The sum of the products of the roots taken two at a time corresponds to the coefficient of the $(n-2)$th power term divided by the coefficient of the $n$th power term: $r_1 r_2 + r_1 r_3 + \ldots + r_{n-1} r_n = \frac{a_{n-2}}{a_n}$

4. The sum of the products of the roots taken three at a time is equal to the negative of the coefficient of the $(n-3)$th power term divided by the coefficient of the $n$th power term: $r_1 r_2 r_3 + r_1 r_2 r_4 + \ldots + r_{n-2} r_{n-1} r_n = -\frac{a_{n-3}}{a_n}$

5. This pattern continues, culminating in the sum of the products of the roots taken $n$ at a time, which is given by: $r_1 r_2 \cdots r_n = (-1)^{n-1} \frac{a_0}{a_n}$

Vieta’s formulas are particularly useful for determining the roots of a polynomial when its coefficients are known, as well as for finding the coefficients when the roots are provided. Additionally, these formulas have applications in various areas of mathematics, including complex analysis and number theory.