Question

with distinct nodes, prove there is at most one polynomial of degree ≤ 2n + 1 that interpolates the data. Remember the Fundamental Theorem of Algebra says a nonzero polynomial has number of roots ≤ its degree. Also, Generalized Rolle’s Theorem says if r0 ≤ r1 ≤ . . . ≤ rm are roots of g ∈ C m[r0, rm], then there exists ξ ∈ (r0, rm) such that g (m) (ξ) = 0.

1. (25 pts) Given the table of data f(r) f(xo) f(r.) f(rn) with distinct nodes, prove there is at most one polynomial of degree interpolates the data. 2n 1 that Remember the Fundamental Theorem of Algebra says a nonzero polynomial has num ber of roots its degree. Also, Generalized Rolle's Theorem says if To S r . .. £ Tm are roots of g E C"'[ro,Tm], then there exists ξ E (ro, rm) such that g(m)(ξ)-0.

Answer 1

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