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88 3. Factors and Zeros

14. 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.

15. (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.


16. 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.