# 4.2 Factorization of Polynomials over a Field

1. a. f = a(a

^{−1}f).2. a. Yes, since a ≠ 0, f(b) = 0if and only if af(b) = 0.

3. a. f(1) = 0; indeed f = (x − 1)(x

^{2}− x + 2) over any field.4.

a. Irreducible because it has no roots in

c. in . Not irreducible.

e. Irreducible, because it has no root in

5.

^{∗}Every polynomial of odd degree in has a root in –see Exercise 9.

7. f = [x − (1 − i)][x − (1 + i)][x − i][x + i] = (x

^{2}− 2x + 2)(x^{2}+ 1) = x^{4}− 2x^{3}+ 3x^{2}− 2x + 2.The polynomial f

^{2}has the same roots, albeit of different multiplicities.8. a. As f is monic, we may assume that both factors are monic (Exercise 6). Hence Now equate coefficients.

9. Assume f = a

_{n}x^{n}+ a_{n−1}x^{n−1}+ ...Get *Introduction to Abstract Algebra, Solutions Manual, 4th Edition* now with O’Reilly online learning.

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