Sum of Roots of Polynomial/Proof 1

Proof
Let the roots of $P$ be $z_1, z_2, \ldots, z_n$.

Then $P$ can be written in factored form as:
 * $\ds a_n \prod_{k \mathop = 1}^n \paren {z - z_k} = a_n \paren {z - z_1} \paren {z - z_2} \cdots \paren {z - z_n}$

Multiplying this out, $P$ can be expressed as:
 * $a_n \paren {z^n - \paren {z_1 + z_2 + \cdots + z_n} z^{n - 1} + \cdots + \paren {-1}^n z_1 z_2 \cdots z_n} = 0$

where the coefficients of $z^{n - 2}, z^{n - 3}, \ldots$ are more complicated and irrelevant.

Equating powers of $z$, it follows that:
 * $-a_n \paren {z_1 + z_2 + \cdots + z_n} = a_{n - 1}$

from which:
 * $z_1 + z_2 + \cdots + z_n = - \dfrac {a_{n - 1} } {a_n}$