Chapter 3.4, Problem 92E

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

$a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+\cdots +a_{1}x+a_0=0,$

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

$a_n{p}^n+a_{n-1}{p}^{n-1}q+a_{n-2}{p}^{n-2}{q}^2+\cdots +a_{1}p{q}^{n-1}=-a_0{q}^n.$

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 a_n.