Question

Let P(x) be a polynomial with real coefficients and let b>0 . Use the Division Algorithm to write P(x)=(x-b) • Q(x)+r . Suppose that r ≥ 0 and that all the coefficients in Q(x) are nonnegative. Let z > b (b) Prove the first part of the Upper and Lower Bounds Theorem.

Answer 1

To prove the first part of the Upper and Lower Bounds Theorem, we need to show that the polynomial P(x) has a lower bound and an upper bound. We are given that P(x) can be divided by (x-b) with a remainder r, where r is greater than or equal to 0. This means that P(x) is decreasing as x approaches infinity, as the remainder r is nonnegative.

Step-by-step explanation:

To prove the first part of the Upper and Lower Bounds Theorem, we need to show that the polynomial P(x) has a lower bound and an upper bound. We are given that P(x) can be divided by (x-b) with a remainder r, where r is greater than or equal to 0. This means that P(x) is decreasing as x approaches infinity, as the remainder r is nonnegative. We also know that all the coefficients in Q(x) are nonnegative, so the polynomial Q(x) is also nonnegative. Therefore, the polynomial P(x) has a lower bound of b and no upper bound.