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 (c) Use the first part of the Upper and Lower Bounds Theorem to prove the second part. IHint: Show that if P(x) satisfies the second part of the theorem, then P(-x) satisfies the first part.

Answer 1

To prove the second part of the Upper and Lower Bounds Theorem, we can use the first part. By substituting -x into the given equation for P(x), we can show that P(-x) satisfies the first part of the theorem.

Step-by-step explanation:

To prove the second part of the Upper and Lower Bounds Theorem, we can use the first part. Let's assume that P(x) satisfies the second part of the theorem (r ≥ 0 and all coefficients in Q(x) are nonnegative). We need to show that P(-x) satisfies the first part of the theorem (r ≥ 0 and all coefficients in Q(-x) are nonnegative).

To show this, we can substitute -x into the given equation P(x) = (x-b) · Q(x) + r. This gives us P(-x) = (-x-b) · Q(-x) + r. Since b > 0, we have -x-b < 0, which means that (-x-b) · Q(-x) will have nonnegative coefficients as long as Q(-x) has nonnegative coefficients.

Therefore, if P(x) satisfies the second part of the theorem, then P(-x) satisfies the first part, as required.