Answers to Exercises; Chapter 3 275Since 1
( )is divisible by p for 2 5 k 5 p - 1, the irreducibility follows
from the Eisenstein Criterion with prime p.1.20.
t3 + t2 + t + 1 = (t + l)(P + 1)
t5 + t4 + t3 + t2 + t + 1 = (t + 1)(t2 + t + l)(P -t + 1)
t7 + t6 + * *. + t + 1 = (t + l)(P + l)(P + 1).
1.21. Let u(t) be a polynomial of largest degree over D which divides both
f(t) and g(t). If g(t) d oes not divide f(t), then u(t) is a nonzero constant.
By repeated application of the division algorithm (cf. Exercise 1.6.2), we
can express u(t) in the form f(t)p(t) + g(t)q(t) for some polynomials p(t)
and q(t) over D. But then u(w) = 0, a contradiction.
1.22. Suppose if possible that g(t) h as a nonsimple zero w. Then g(w) =
g’(w) = 0. By Exercise 21, g(t) divides g’(t), which is a contradiction, since
0 5 deg g’(t) < deg g(t).
2.1. (a) (u,v) = (12,-10). 6t2 + 2t - 20 = 6t2 + 12t - lot - 20 =
(6t - lO)(t + 2) = 6t2 - lot + 12t - 20.
(b) at2 + bt + c = as1 [a^2 t^2 + u(u + v)t + Uv] = a-‘(at + u)(at + v). Let
w = gcd(a, u), a = wz, u = wy. Since a ] vwy, z ] vy. Since gcd(z, y) = 1,
z Iv. Thus
at2 + bt + c = [zt + (u/w)][wt + (v/z)].
(c) 28t2+57t+14 = 28t2+49t+8t+14 = (7t+2)(4t+7) is negative when
-7/4 < t < -2/7. 20t2 + 39t - 44 = 20t2 + 55t - 16t - 44 = (4t + 11)(5t - 4)
is negative when -1114 < t < 415.
2.2. (a) uk - bk = (a - b)(ak-’ + ake2b +... + abks2 + bk-l).
(b) a6 + bk = (a + b)(akml - akm2b +... - abkm2 + bkW1).
2.3. (a) 4t2 - 20t - 11 = (2t - 5)2 - 62 = (2t - 11)(2t + 1).
(b) 5t2 - 6t + 1 = (3t - 1)2 - 4t2 = (t - 1)(5t - 1).
(c) t4 - 47t2 + 1 = (t2 + 1>z - 49t2 = (P - 7t + 1)(t2 + 7t + 1).2.4. (a) Any reducible cubic can be factored as the product of a linear and
a quadratic. Since the leading coefficients of the factors divide that of the
polynomial, the linear factor must have the form k(t - k), where k is an
integer. Such k is a zero.
(b) Any integer zero must divide 42. A little trial and error yields the
zero 2 and the factorization (t - 2)(t” - 6t + 21).
2.6. (d) (t” + 4t3 + 8t2 - 4t + l)(t4 - 4t3 + 8t2 + 4t + 1).
2.7. (a), (b), (c), (e), (f) Irreducible.
(d) (7t + 8)(4t - 3).
(g) (t2 + 2t - 2& + 3)(t - 1).