Viète's Formulas/Examples/Monic Polynomial

Example of Use of Viète's Formulas
Let:

Let $U$ be the set of $N$ roots of equation $\map P x = 0$.

Then:

Proof
Let:


 * $ U = \set {x_1,\ldots,x_N}$

Translate Viète's Formulas from notation $a_0$ to $a_N$:

Let $N-k = j$ define a change of index.

Then $k = N-j$.

Apply the change of index:

 Common Errors 

The mundane task of applying Viète's Formulas can produce fatal errors in calculations and in proofs:


 * * Wrong number $N$ of roots.


 * * An off-by-one index error, e.g, $1\le k \le N$ instead of $0 \le k \le N-1$.


 * * Swapping $k$ and $N-k$ in $b_k$ and $e_{N-k}$.

Realized in a quadratic equation example, the sum and product of the roots might be reversed in order with possibly the wrong signs.
 * Product of Roots of Quadratic Equation