Evaluation via recursive remaindering. Let R be a ring. (a) Show that for all & R...

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Evaluation via recursive remaindering. Let R be a ring. (a) Show that for all & R and f = R[x] we have f(g) = frem (x - §). (b) Let a, b, c E R[x], with b and c monic, and suppose that c divides b. Show that a rem c = (a rem b) rem c. Hints: Recall that the quotient and remainder are unique for a, b e R[a] with b monic. Use the defining equality a = qb+r with degr < deg b for both parts. For part (a), investigate what happens when you evaluate the defining equality at §. Evaluation via recursive remaindering. Let R be a ring. (a) Show that for all & R and f = R[x] we have f(g) = frem (x - §). (b) Let a, b, c E R[x], with b and c monic, and suppose that c divides b. Show that a rem c = (a rem b) rem c. Hints: Recall that the quotient and remainder are unique for a, b e R[a] with b monic. Use the defining equality a = qb+r with degr < deg b for both parts. For part (a), investigate what happens when you evaluate the defining equality at §.

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a Recall that given any polynomial x in Rx we can write it as x qxxa r for some qx r in Rx and r of ... View the full answer