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3.3. Finding Integer and Rational Roots 91

1. (a) Show that the rational number a/b is a zero of q(t) if and only
   if

cnan + c”-la”b + c”-2an-‘b2 + ..a + club”-l + cob” = 0.

(b) If the zero a/b of q(t) is written in lowest terms, show that c,, is

a multiple of b and that co is a multiple of a.

2. Prove that any rational zero of a manic polynomial with integer co
   efficients must in fact be an integer.


3. For each polynomial, determine all of its rational zeros. Recall that,
   having found one rational zero, you can divide the polynomial by a
   suitable linear factor and deal with a polynomial of lower degree to
   determine the other rational zeros.

(a) 5t3 - 4t2 + 3t + 2

(b) 6t3 + 13t2 - 22t - 8

(c) 3t4 + 5t3 + 2t2 - 6t - 4

(d) t4 - t3 - 32t2 - 62t - 56.

4. If a/b is a rational zero of q(t) written in lowest terms. Then, as we
   have seen, a I CO. Another way of expressing this is to say that -a is
   a divisor of q(0). There is a useful generalization to this:

if q(t) is a polynomial over Z with rational zero a/b, ihen

bt - a is a divisor of q(t) over Z, and, for each integer m,

bm - a must be a divisor of q(m).

In this exercise, this result will be first applied and then demon-

strated.

(a) Consider the polynomial 6t3 + 13t2 - 22t - 8. Show that any

positive rational zero must be one of

1, l/2, l/3, l/6, 2, 213, 4, 4/3, 8, 813.

Evaluate the polynomial at t = -2, -1, 1, 2, and apply the

result with these values of m to eliminate all but two of the

rationals in the list as a possible zero of the polynomial.

(b) Show that, if a/b is a zero of the polynomial, then

q(t) = (bt - a)(r,-It”-’ + r”-2t”-’ +... + rlt + ro)

where the coefficients rn, rn-1,... , r1, rs are rational numbers

satisfying

cn = brn-l