Scaled generalized Bernstein polynomials and containment regions for the zeros of polynomials

Let P n ( x ) = ∑ k = 0 n β k b k , n ( x ; α , β ) , β n ≠ 0 , where b k , n ( x ; α , β ) ≔ ( α + x ) k ( β - x ) n - k , k = 0 , 1 , … , n , α , β ∈ C , α ≠ - β , a generalized scaled Bernstein polynomial. Extending a result of Winkler [A resultant matrix for scaled Bernstein polynomials, Linear Algebra Appl. 319 (2000) 179–191] a companion matrix for P n is given. The application of some matrix methods to this companion matrix yields regions for the zeros of P n .