Vieta's Formulas

Vieta's formulas provide a relationship between the roots and coefficients of a polynomial, which are particularly useful in in computing roots when some are known, or in calculating the norm and trace.

Consider the cubic polynomial \(aX^3 + bX^2 + cX + d\) with roots \(\alpha_1, \alpha_2, \alpha_3\). Consider then that this polynomial can be written in the form

\[ a(X - \alpha_1)(X - \alpha_2)(X - \alpha_3).\]

Expanding this we have

\[\begin{align*} & a(X - \alpha_1)(X - \alpha_2)(X - \alpha_3) \\ =& a(X^3 - \alpha_1 X^2 - \alpha_2 X^2 - \alpha_3 X^2 + \alpha_1\alpha_2 X + \alpha_2\alpha_3 X + \alpha_1\alpha_3 X - \alpha_1\alpha_2\alpha_3) \\ =& aX^3 - a(\alpha_1 + \alpha_2 + \alpha_3)X^2 + a(\alpha_1\alpha_2 + \alpha_2\alpha_3 + \alpha_1\alpha_3)X - a(\alpha_1\alpha_2\alpha_3). \\ \end{align*}\]

We can then equate coefficients as follows

Example

For a cubic polynomial \(aX^3 + bX^2 + cX + d\) with roots \(\alpha_1, \alpha_2, \alpha_3\) we have that

\[ \begin{align*} \alpha_1 + \alpha_2 + \alpha_3 &= -\frac{b}{a} \\ \alpha_1\alpha_2 + \alpha_2\alpha_3 + \alpha_1\alpha_3 &= \frac{c}{a} \\ \alpha_1\alpha_2\alpha_3 &= -\frac{d}{a}. \\ \end{align*}\]

This then becomes a special case of the formula below. Since this expression is much uglier when written in full generality, notice the pattern above which motivates the general formula.

In particular, consider that in the expansion, for each term, we choose either the indeterminate \(X\) or the root \(\alpha_i\) from each set of parenthesis. If on the right we have the coefficient of the \(k^{\text{th}}\) power of \(X\), that means in \(k\) terms the \(X\) was chosen, while in \(n - k\) terms the root was chosen. This means we have the negative sign on the right precisely when \(n - k\) is odd. Similarly, on the left hand side we have the sum of all possible products of \(n - k\) roots, since we chose \(n - k\) roots.

Vieta's Formulas

For the polynomial

\[ a_nX^n + a_{n - 1}X^{n - 1} + \dots + a_1X + a_0\]

with roots \(\alpha_1, \dots, \alpha_n\) we have that

\[ \frac{a_k}{a_n} = (-1)^{n - k}\sum_{S \in S_{n - k}} \prod_{\alpha \in S} \alpha\]

for \(k \in \{0, \dots, n - 1\}\) where

\[ S_m = \{x \in \mathcal{P}(\{\alpha_1, \dots, \alpha_n\}) : |x| = m\}.\]