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Vieta’s classical formulae explicitly determine the coefficients of a polynomial  p  ∈ [ x ] in terms of the roots of  p , where  is any commutative ring. In this paper, Vieta’s formulae are obtained for slice-regular polynomials over the noncommutative algebra of quaternions, by an argument which essentially relies on induction, without invoking quasideterminants or noncommutative symmetric functions.

Vieta’s formulae for regular polynomials of a quaternionic variable