Newton's Identities

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Theorem

Let $\PP$ be the polynomial equation:

$\PP: \quad a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x^1 + a_0 = 0$

Let $s_k$ be the sum of the $k$th powers of the roots of $\PP$.

Then Newton's Identities are:

\(\ds a_n s_1 + a_{n - 1}\) \(=\) \(\ds 0\)
\(\ds a_n s_2 + a_{n - 1} s_1 + 2 a_{n - 2}\) \(=\) \(\ds 0\)
\(\ds \) \(\cdots\) \(\ds \)
\(\ds a_n s_k + a_{n - 1} s_{k - 1} + \cdots + a_{n - k + 1} s_1 + k a_{n - k}\) \(=\) \(\ds 0\)

where $a_i$ is taken to be $0$ if $i < 0$.


Proof



Also see


Source of Name

This entry was named for Isaac Newton.


Historical Note

Newton's Identities were published by Isaac Newton in $1707$.


Sources