Newton's Identities give a recursive relation between the power sums of the roots of a polynomial, meaning the sums of each root raised to successive whole-number powers, and the elementary symmetric polynomials in those same roots, which up to sign are exactly the polynomial's own coefficients. Named for Isaac Newton, they let either family of quantities be computed from the other without ever finding the roots themselves, and are used throughout algebra and combinatorics.
Facts
StatementNewton's identities relate power sums of the roots of a monic polynomial to its elementary symmetric polynomials, expressing the sums of k-th powers of the roots in terms of the coefficients without finding the roots. 1 Classification
Statement Form Connections
Named After
Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)
Proved By
Sources
1. Newton's identities, Wikipedia
Lead, second sentence
Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots.
History section, discovery sentence
These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard.
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