?In this exercise, we lead you through the steps involved in the proof of the Rational

Chapter 3, Problem 92

(choose chapter or problem)

In this exercise, we lead you through the steps involved in the proof of the Rational Zero Theorem. Consider the  polynomial equation

\(a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\ldots+a_{1} x+a_{0}=0\),

and let \(\frac{p}{q}\) be a rational root reduced to lowest terms.

a. Substitute \(\frac{p}{q}\) for x in the equation and show that the equation can be written as

\(a_{n} p^{n}+a_{n-1} p^{n-1} q+a_{n-2} p^{n-2} q^{2}+\ldots+a_{1} p q^{n-1}=-a_{0} q^{n}\).

b. Why is p a factor of the left side of the equation?

c. Because p divides the left side, it must also divide the right side. However, because \(\frac{p}{q}\) is reduced to lowest terms, p and q have no common factors other than -1 and 1. Because p does divide the right side and has no factors in common with qn, what can you conclude?

d.  Rewrite the equation from part (a) with all terms containing q on the left and the term that does not have a factor of q on the right. Use an argument that parallels parts (b) and (c) to conclude that q is a factor of an.

Equation Transcription:

Text Transcription:

a_nx^n+a_n-1x&n-1+a_n-2x^n-2+...+a_1x+a_0=0

p/q

a_np&n+a_n-1p^n-1q+a_n-2p^n-2q^2+...+a_1pq^n-1=-a_0q^n