Unknown

90 3. Factors and Zeros

(15) Show that

UT + up + *.. + 26; _ (?/*+I+ 1+ y-(*+‘))(y” + 1+ y-“) - 3(t + 1)

u1+ ‘u2 + *. * + U” - (t2 - 4)@ + 1)

Deduce from this that UT + uz +... + uz is divisible by u1 + u2 +.. .u,

(1 5 n).

Define polynomials v, and w, as follows:

VI-J = 0 Vl = 1

V2m = VZm-1 + Vzm-2 (m^2 1)

V2mtI = (t - 2)vzm + V2m-1 (m 1 1)

wo = 2 Wl = 1

w2, = (t - 2)2~2~-1+ wzm-2 (m 1 1)

W2m-t1 = w2m + warn-1 (m 2 1)

(16) v,+z = tv, - vn-2 wn+z = ha - w-2 (n L 2)

(17) v2m = urn (m 2 0)

(18) vzm+l(t) = (-l)*wmtl(-~) (m 2 0)

(19) U” = vnwn = ~(v”+lw”-1+ ~“-1wntl)

(20) When n = 2’, u, = plp2. “p, where pi(t) = t and pk = pzBl -^2

(k 2 1).

3.3 Finding Integer and Rational Roots: Newton’s

Method of Divisors

A first step in factoring polynomials over Q is to use the Factor Theorem
to locate linear factors by finding rational zeros. It is straightforward to
see that all but finitely many rationals can be rejected as possible zeros.
Special techniques will narrow down the possibilities even further.

Exercises

In these exercises, q(t) will represent the polynomial

cntn + C,-ltn-l + C,-p--2 $ *.. + qt + co

with integer coefficients ci.