Unknown

92 3. Factors and Zeros

~“-1 = br”-2 - urn-l

c,,-2 = br,-3 - ar,,-2

...

c2 = bq - ar2

cl = bra - arl

co = -are.

(c) In (b), suppose that a/b is written in lowest terms. It is known

that a, b and the ci are integers, and also that a I CO and b I cn.

By looking at the equations relating the ri and the ci in turn

from the top, prove that

k-1 is an integer

br,-2 is an integer

b2rn-3 is an integer

...

b”-‘ro is an integer.

and conclude that each ri is a rational whose denominator di-

vides b”-l. Similarly, by looking at the equations in turn from

the bottom, show that each ri is a rational whose denominator

divides an-r. Deduce from this each ri is an integer.

(d) Complete the proof of the result.

5. Newton’s Method of Divisors. The set of equations involving ci, ri, a
   and b in Exercise 4 is the basis of an algorithm known as Newton’s
   Method of Divisors. We wish to check whether a/b is a root of q(t).
   Write in a row the coefficients of the polynomial, beginning with the
   constant coefficient:

co Cl c2 c3 ... C”.

Divide CO by a. If the result is not an integer, then we do not have a

zero. If the result is an integer, let co/a = so (this is -ro in Exercise

4); write this integer under cr. Draw a line across. Beneath the line,

put the sum of cl1 and bso:

co Cl c2 ...

so

co cl + bso