In Exercise 35.12, we considered polynomial division. In this problem, you are asked to

Chapter 36, Problem 36.22

(choose chapter or problem)

In Exercise 35.12, we considered polynomial division. In this problem, you are asked to develop the concept of polynomial gcd. Polynomials in this problem may be assumed to have rational coefficients. a. Let p and q be nonzero polynomials. Write a careful definition for common divisor and greatest common divisor of p and q. In this context, greatest refers to the degree of the polynomial. b. Show, by giving an example, that there need not be a unique gcd of two nonzero polynomials. c. Let d be a greatest common divisor of nonzero polynomials p and q. Prove that there exist polynomials a and b such that ap C bq D d. d. Give a careful definition of relatively prime for nonzero polynomials. e. Prove that two nonzero polynomials p and q are relatively prime if and only if there exist polynomials a and b such that ap C bq D 1. f. Let p D x 4 3x2 1 and q D x 2 C 1. Show that p and q are relatively prime by finding polynomials a and b such that ap C bq D 1.