Sum of Roots of Polynomial/Proof 2

Theorem

Let $P$ be the polynomial equation:

$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$

such that $a_n \ne 0$.

The sum of the roots of $P$ is $-\dfrac {a_{n - 1} } {a_n}$.

Proof

From Viète's Formulas:

$\ds a_{n - k} = \paren {-1}^k a_n \sum_{1 \mathop \le i_1 \mathop < \dotsb \mathop < i_k \mathop \le n} z_{i_1} \dotsm z_{i_k}$

for $k = 1, 2, \ldots, n$.

The result follows for $k = 1$.

$\blacksquare$