3.3. Finding Integer and Rational Roots 93
Divide cl + bso by a. If the result is not an integer, a/b is not a zero.
Otherwise, let the quotient be sr, and write sr under cz. Beneath the
line, put cg + bsl. Continue on to get a table:co Cl c2 c3 *a.
so Sl s2co cl + bso c2 + bsl c3 + bs2 ..a
Each si is equal to (ci+bsi-1)/a. Stop if some si fails to be an integer,
for then a/b is not a zero. If all the si turn out to be integers, then
for a/b to be a zero, c, must equal -bs,-1, and the last term below
the line must be 0.(a) Justify this algorithm.
(b) The method is mainly used to check for integer zeros (b = l),
which explains why the coefficients are listed in ascending rather
than descending order in the top row of the table. Devise an
alternative algorithm which takes the coefficients in the opposite
order.
(c) Two candidates for rational zeros of 2t3 - lit’ + 2t + 15 are 3
and 3/2. Verify that the respective tables for these are15 2 -11 2 15 2 -11^2
5 5 4 -115 7 15 12 -3 0Construct the tables for the candidates 5 and 512. Identify zeros
of 2t3 - lit’ + 2t + 15 and factor this polynomial over Z.- Find all rational zeros of the following polynomials:
(a) 4t3 - 22t2 + 7t + 15
(b) 40t3 + 25t2 + 9t - 9
(c) 5tz - 12t + 4
(d) t3 - 9t2 + llt + 21
(e) 8t3 + 20t2 - 18t - 45
(f) 6t4 + t3 - 66t2 + 30t + 56.- Write as a product of irreducible factors over Z:
18t5 - 48t4 + 23t3 + 174t2 - 171t - 60.- Find all the zeros of the polynomial
24t5 + 143t4 - 136t3 + 281t2 + 36t - 140.