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 Viete’s formula is quadratics and are used specifically in algebra.

Basic formula of Vieta’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)th coefficient

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

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

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 + 1 

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’s 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

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