Prove: Upper and Lower Bounds Theorem Let P(x) be a polynomial with real coefficients, and let b > 0. Use the Division Algorithm to write.
Suppose that and that all the coefficients in Q(x) are nonnegative.
Let z > b.
(a) Show that .
(b) Prove the first part of the Upper and Lower Bounds Theorem.
(c) Use the first part of the Upper and Lower Bounds Theorem to prove the second part. (Hint: Show that if P(x) satisfies the second part of the theorem, the P(-x) satisfies the first part].
