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86 3. Factors and Zeros

Because of the symmetry of the coefficients, one might try to find two

symmetrically related factors, each of degree 4.

(a) Show that, if there is a factorization of the form

t” + 98t4 + 1 = (t4 + ai3 + bt2 + ct + d)(t4 + kt3 + mt2 + nt + r)

thenwemusthaver=d=lorr=d=-1

O=a+k

O=m+ak+b

O=n+ma+bk+c

98=2d+an+bm+ck

O=ad+bn+cm+kd

O=bd+cn+dm

0 = d(c + n).

(b) Show that k = - a, n = -c, a(b - m) = 0, c2 = (b + m)d = a’d.

(c) Show that a # 0, so that d = 1, c2 = a2, b = m.

(d) Determine a factorization of ts + 98t4 + 1 as a product of two

polynomials of degree 4.

7. Test the following polynomials for irreducibility over Q. Factor all
   the reducible polynomials as far as you can.

(a) 7t - 8

(b) 2t2 + 2t - 1

(c) 4t2 + 4t - 1

(d) 28t2 + llt - 24

(e) 28t2 - llt + 24

(f) 2t3 + 3t2 - 21t - 6

(g) (t2 + 2t)’ - 5(t2 + 2t) + 6

(h) t4 + 2t3 + t2 + t + 1

(i) 3t4 - 2t3 - t2 - 3t - 1

(j) 4t5 - 15t3 + 5t2 + 15t - 9

(k) t6 - t4 - t2 + 1

(1) t3 - t2 - 24t - 36

(m) t3 - 7t” + 13t - 15

(n) t5 - t3 - 3t2 - 2t - 1