Vieta’s Formula – Vieta Problem solution with Solved Example

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What is Vieta?

In mathematics, Vieta’s formula is related to the coefficients of polynomial of sum and product of its roots. It was named after the scientist François Vièteand it is frequently used in algebra as well. The other name for Vieta’s Formula is the Viete’s Law where a set of equations together are related to the root or coefficient of polynomial.

Vieta Formula

Here, the Vieta’s formula is related to the coefficient or roots of the polynomials. It could also be defined as the coefficients of polynomials or sum or product of their roots or the product of roots taken in the group. Here is the basic Vieta’s formula in general form with polynomial for degree n.

\[\large P\left(x\right)=a_{n}x^{n}+a_{n-1}x^{n-1}+….+a_{1}x+a_{0}\]

Equivalently stated, the (n−k)th coefficient an-k is related to a signed sum of all possible subproducts of roots, taken k at-a-time:

\[\large \sum_{1\,\leq\,i_{1}\,<\,i_{2}\,….i_{k}\,\leq\,n}r_{i1}\,r_{i2}….\,r_{ik}=\left(-1\right)^{k}\frac{a_{n-k}}{a_{n}}\]

for k = 1, 2, …, n (where we wrote the indices $i_{k}$ in increasing order to ensure each sub product of roots is used exactly once)

This is necessary to learn by students where you have to calculate the multiple roots for a polynomial equation. As discussed earlier, the Vieta’s formula was discovered by the French mathematician François Viète that is relating the sum or product of roots of a polynomial to its coefficients. The simplest use of Vieta’s formula is done in Quadratics. They are helpful in solving complicated algebraic polynomials having multiple roots where roots are not easy to derive. For example, Vieta’s formula could also serve as the shortcut to find solutions of sum or product of their roots quickly.