Noncommutative Vieta Theorem and Symmetric Functions

There are two ways to generalize basic constructions of commutative algebra for a noncommutative case. More traditional way is to define commutative functions like trace or determinant over noncommuting variables. Beginning with [6] this approach was widely used by different authors, see for example [5], [15], [14], [12], [11], [7]. However, there is another possibility to work with purely noncommutative objects without using trace or determinant or passing to a quotient space or quotient algebra started in [9] and [10]. Let us compare these two approaches on a simplest example a classical Vieta theorem which, of course, is a starting point in a theory of symmetric functions. Consider an algebraic equation