# 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

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## Also see

- Newton-Girard Identities, also known as Newton's Identities

## Source of Name

This entry was named for Isaac Newton.

## Historical Note

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

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Newton's identities**(I. Newton, 1707)