Exercise 12.10. prove Theorem 12.3. ΤΟ do so, use induction on the number m of factors in one of the irreducible factorizations of a(x) 1. Deal with the case m 1. Observe that this means that a(x) is irreducible. 2. Perform the inductive step. For each integer k 21, show that if the result we want to prove is true for polynomials that can be factored as a product of k irreducible polynomials, then it is true for polynomials that can be factored as a product of k + 1 irreducible polynomials. Theorem 12.3. Let K be a field and let a(x) be a polynomial in Kr] of pos- itive degree. Suppose pi(z)Pm(r) and qi(x)---qn(x) are two factorizations of a(z) as a product of irreducible polynomials in K[r]. Then m n, and the order of the factors in the second factorization can be changed so that for each inder i there is a nonzero constant G such that qī(r) = cpi(r)