Unknown

172 5. Approximation and Location of Zeros

(d) Deduce that, if r is any positive zero of f(t), then

r<l+ I-

M

c

-.

Qn

13. Suppose f(t) = Cakt k is a polynomial of degree n over R which
    satisfies the hypotheses of Exercise 12. Let

Si = C(C4j : j > i,aj > 0)~

the sum of all positive coefficients of powers oft greater than the ith;

R = max{]ai]/si : ai < 0).

(a) Verify that tm = (t - l)(tm-’ + tmw2 + 3 .. + t + 1) + 1.

(b) Consider the polynomial

g(t) = 3t4 + 2t3 - 8t2 + t - 7.

By (a) applied to powers of t with positive coefficients, verify

that

g(t) = 3(t-l)t3+(-8+5(t-l))t2+5(t-l)t+(-7+6(t-1))+6.

Deduce that g(r) > 0 if r - 1 > max{8/5,7/6} = 8/5.

(c) Show that, in general, if r > 1, then

f(r) > E{[Odi + (r - l)si]ri : Ui < 0)

and deduce that, if f(r) = 0, then r < 1 + R.

14. Apply the results of Exercises 12 and 13 to determine upper bounds
    for the real zeros of

(a) t’l + t” - 3t5 + t4 + t3 - 2t2 + t - 2

(b) t7 - t6 + t5 + 2t4 - 3t3 + 4t2 + t - 2.

15. Rolle’s Theorem. At the end of the seventeenth century, Michel Rolle
    gave a method of locating intervals which contain zeros of a real poly-
    nomial p(t). If p(t) is hard to deal with, determine a real polynomial
    q(t) whose derivative is equal to p(t), i.e. q’(t) = p(t). It may happen
    that it is easier to determine where the zeros of q(t) might be, perhaps
    because q(t) is factorable.
    Rolle’s result is that, if a and b are distinct real zeros of q(t), then
    the open interval (a, b) will contain at least one real zero of p(t). The
    proof of this result is sketched in Exploration E.28 at the end of
    Section 2.4.