88 3. Factors and Zeros
- Show that each of the following polynomials is irreducible over Z,
for some m. Can you deduce from this that it is irreducible over Z?
(a) t2 + t + 1
(b) 49t2 + 35t + 11
(c) 124t3 - 119t2 + 35t + 64.
- (a) Factor 63t4 - 2t3 - 79t2 + 52t - 10 over Z4, Z5, Z7, Zg.
(b) Use the factorizations of the polynomial in (a) over the finite
domains to determine a factorization over Z. - To demonstrate how cumbersome a “sure-fire” method can be, con-
sider the problem of factoring the quintic polynomial
f(t) = lot5 + 3t4 - 38t3 - 5t2 - 6t + 3.
One strategy is to note that any factorization over Z leads to a numer-
ical factorization of the possible values that j(t) can assume. Thus,
knowing, say, f(1) d e t ermines a finite set of values which any factors
might assume at 1.
(a) Verify that f(-1) = 35, f(0) = 3, f(1) = -33.
(b) Supposing that f(t) is reducible over Z, we can assume that
there is a factor g(t) of degree at most 2. Verify that, if g(-1) =
u, g(0) = v, g(1) = w, then
2g(t) = (w + u - 2v)P + (w - U)t + 2v.
(c) Show that u I 35, v I 3, w I 33, so that there are 8 x 4 x 8 = 256
possible choices of (u, v, w) to examine in determining g(t).
(d) Show that w + u - 2v is an even divisor of 20.
(e) With no loss of generality, we can assume v = 1 or v = 3 (why?).
Show that, if v = 1, the possible values of w + u are -18, -8,
-2, 0,4, 6, 12,22, and that if v = 3, the possible values of w+u
are -14, -4, 2, 4, 8, 10, 16, 26.
(f) Use (c), (4, (e) t o fi n d candidates for a factor g(t). Does one of
them work?
Exploration
E.29. Let a, b, n be positive integers. Investigate under what conditions
the polynomial
t2--t+a
is a factor of t” + t + b over Z.