This exercise outlines a proof of the rational roots theorem. At one point in the proof

Chapter 12, Problem 41

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This exercise outlines a proof of the rational roots theorem. At one point in the proof, we will need to rely on the following fact, which is proved in courses on number theory. FACT FROM NUMBER THEORY Suppose that A, B, and C are integers and that A is a factor of the number BC. If A has no factor in common with C (other than 1), then A must be a factor of B. (a) Let A 2, B 8, and C 5. Verify that the fact from number theory is correct here. (b) Let A 20, B 8, and C 5. Note that A is a factor of BC, but A is not a factor of B. Why doesnt this contradict the fact from number theory? (c) Now were ready to prove the rational roots theorem. We begin with a polynomial equation with integer coefficients: anxn an1xn1 p a1x a0 0 1n 1, an 02 x5We assume that the rational number p/q is a root ofthe equation and that p and q have no common factorsother than 1. Why is the following equation now true?an apq bn an1 apq bn1 p a1 apq b a0 0an(d) Show that the last equation in part (c) can be writtenSince p is a factor of the left-hand side of this lastequation, p must also be a factor of the right-handside. That is, p must be a factor of a0qn. But since pand q have no common factors, neither do p and qn.Our fact from number theory now tells us that p mustbe a factor of a0, as we wished to show. (The proof thatq is a factor of an is carried out in a similar manner.)