Describe the Vieta's formulas for polynomials.

Vieta's formulas are a set of equations that relate the coefficients of a polynomial to its roots.

Vieta's formulas are named after the French mathematician François Viète. They are a set of equations that relate the coefficients of a polynomial to its roots. For a polynomial of degree n with roots r1, r2, ..., rn, Vieta's formulas state that:

- The sum of the roots is equal to the opposite of the coefficient of the (n-1)th power term divided by the coefficient of the nth power term: r1 + r2 + ... + rn = -a(n-1)/a(n)
- The product of the roots is equal to the constant term divided by the coefficient of the nth power term: r1r2...rn = (-1)^n * an-1 / an
- The sum of the products of the roots taken two at a time is equal to the coefficient of the (n-2)th power term divided by the coefficient of the nth power term: r1r2 + r1r3 + ... + rn-1rn = a(n-2)/a(n)
- The sum of the products of the roots taken three at a time is equal to the opposite of the coefficient of the (n-3)th power term divided by the coefficient of the nth power term: r1r2r3 + r1r2r4 + ... + rn-2rn-1rn = -a(n-3)/a(n)
- And so on, up to the sum of the products of the roots taken n at a time, which is equal to (-1)^(n-1) * an-1 / an.

Vieta's formulas are useful for finding the roots of a polynomial when its coefficients are known, or for finding the coefficients of a polynomial when its roots are known. They are also used in many other areas of mathematics, such as complex analysis and number theory.