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

and let p/q be a rational root reduced to lowest terms.

a. Substitute p/q for x in the equation and show that the equation can be written asb. 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 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.

Transcribed Image Text:

anx" + an-1x²-1 + an-2xn-2 + + ax + ao = 0,