《Discrete Mathematics II教学-华南理工》Section 20 Factorization of Polynomials over a Field.pdfVIP

IV.23 Factorizations of Polynomials 1 Section IV.23. Factorizations of Polynomials over a Field Note. Our experience with classical algebra tells us that finding the zeros of a polynomial is equivalent to factoring the polynomial. We find that the same holds in F [x] when F is a field (as we see in the “Factor Theorem”). In this section, we consider factoring polynomials and conditions under which a polynomial cannot be factored (when it is “irreducible”—a concept encountered in Calculus 2 with the topic of partial fraction decomposition). Theorem 23.1. Division Algorithm for F [x]. Let f (x) = a xn + a xn−1 + · · · + a x2 + a x + a and g (x) = b xm + b xm−1 + n n−1 2 1 0 m m−1 2 · · ·+b x +b x +b be in F [x], with a and b both nonzero and m 0. Then there 2 1 0 n m are unique polynomials q (x) and r (x) in F [x] such that f (x) = g (x)q (x) + r (x), where either r (x) = 0 or the degree of r (x) is less than the degree of g (x). Exercise 23.4. For f (x) = x4 + 5x3 + 8x2 and g (x) = 5x2 + 10x + 2 in Z11 [x], find q (x) and r (x) such that f (x) = g (x)q (x) + r (x). IV.23 Factorizations of Polynomials 2 Solution. We can perform simple long division (but in Z11): 9x2 + 5x + 10 5x2 + 10x + 2) x4 + 5x3 + 8x2 x4 + 2x3 + 7x2 3x3 + x2 3x3 + 6x2 + 10x 6x2 + x 6x2 + x + 9