4.2 Factorization of Polynomials over a Field

1. a. f = a(a−1f).
2. a. Yes, since a ≠ 0, f(b) = 0if and only if af(b) = 0.
3. a. f(1) = 0; indeed f = (x − 1)(x2x + 2) over any field.
4.
a. Irreducible because it has no roots in img
c. img in img. Not irreducible.
e. Irreducible, because it has no root in img
5.
img
Every polynomial of odd degree in img has a root in img–see Exercise 9.
7. f = [x − (1 − i)][x − (1 + i)][xi][x + i] = (x2 − 2x + 2)(x2 + 1) = x4 − 2x3 + 3x2 − 2x + 2.
The polynomial f2 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 img Now equate coefficients.
9. Assume f = anxn + an−1xn−1 + ...

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