Vieta’s formula relates the coefficients of polynomial to the sum and product of their roots, as well as the products of the roots taken in groups. Vieta’s formula describes the relationship of the roots of a polynomial with its coefficients. Consider the following example to find a polynomial with given roots. (Only discuss real-valued polynomials, i.e. the coefficients of polynomials are real numbers). Let’s take a quadratic polynomial. Given two real roots and , then find a polynomial. Consider the polynomial . Given the roots, we can also write it as . Since both equation represents the same polynomial, so equate both polynomial
Simplifying the above equation, we get
Comparing the coefficients of both sides, we get For , , For , , For constant term, , Which gives, ,
Equation (1) and (2) are known as Vieta’s Formulas for a second degree polynomial. In general, for an degree polynomial, there are n different Vieta’s Formulas. They can be written in a condensed form as For
The following examples illustrate the use of Vieta’s formula to solve a problem. Examples:
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