# Vieta’s Formula

In mathematics, Vieta’s formulas are formulas that relate the coefficients of a polynomial to sums and products of its roots. It was discovered by Francois Viete. The simplest application of Vieteâ€™s formula is quadratics and are used specifically in algebra.

Basic formula of Vietaâ€™s in any general polynomial of 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)thÂ coefficient

$$\begin{array}{l}a_{n-k}\end{array}$$
Â Â is related to a signed sum of all possible subproducts of roots, taken k at-a-time:

Â $\large \sum_{1\,\leq\,i_{1}\,<\,i_{2}\,….i_{k}\,\leq\,n}r_{i_{1}}\,r_{i_{2}}….\,r_{i_{k}}=\left(-1\right)^{k}\frac{a_{n-k}}{a_{n}}$

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

### Solved Example

Example: This is a polynomial: P(x) = 5x3 + 4x2 – 2x + 1Â

The highest exponent of x is 3, so the degree is 3. P(x) has coefficients.

a3 = 5, a2 = 4, a1 = -2, a0 = 1

Since x is a variable, It can be evaluate the polynomial for some values of x. So let’sÂ can plug in different values for x so that my polynomial simplifies to a single number:

P(1) = 5(1)3 + 4(1)2 – 2(1) + 1 = 5 + 4 – 2 + 1 = 8

P(0) = 5(0)3 + 4(0)2 – 2(0) + 1 = 1

P(-1) = 5(-1)3 + 4(-1)2 – 2(-1) + 1 = -5 + 4 + 2 + 1 = 2