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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 the Francois Viete. The most 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 $\left(n-k\right)^{th}$  coefficient $a_{n-k}$  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)

Solved Examples

Example: This is a polynomial: $P(x)=5x^{3}+4x^{2}-2x+1$ 

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

$a_{3}=5$
$a_{2}=4$
$a_{1}=-2$
$a_{0}=1$

Since x is a variable, It can be evaluate the polynomial for some values of x. All this means is that I can plug in different values for x so that my polynomial simplifies to a single number:

$P\left(1\right)=5\cdot\left(1\right)^{3}+4\cdot\left(1\right)^{2}-2\cdot\left(1\right)+1=8$

$P\left(0\right)=5\cdot\left(0\right)^{3}+4\cdot\left(0\right)^{2}-2\cdot\left(0\right)+1=1$

$P\left(-1\right)=5\cdot\left(-1\right)^{3}+4\cdot\left(-1\right)^{2}-2\cdot\left(-1\right)+1=2$