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174 5. Approximation and Location of Zeros

(a) t2-t u=-1 v=2

(b) t2-t+l u=-1 v=2.

19. Let p(t) = 2t6 - 7t5 + t4 + t3 - 12t2 - 5t + 1.

(a) Verify that

p(t) = (2t - 7)t5 + (t2 - 12)t2 + (t2 - 5)t +^1

= 2t6 + t(t + l)(-7t3 + 8t2 - 7t - 5) + 1,

and deduce that all the real zeros of p(t) lie between -1 and

712.

(b) Use the Fourier-Budan Theorem to deduce that the polynomial

has

(i) exactly one zero in the interval (3, 7/2)

(ii) one or three zeros in the interval (l/8, l/4)

(iii) nil or two zeros in the interval (-1, -l/2).

(c) Calculate p(-1), p(-l/2) and p(-3/4) and refine the conclusion

in (b)(iii).

20. Prove the theorem of Fourier-Budan for linear and quadratic poly-
    nomials. (In the quadratic case, let the polynomial be t2 + bt + c and
    look at the possible arrangements of signs in the sequence (t2 + bt + c,
    2t + b, b) at various points u and v. Sketch graphs to illustrate each
    possibility.)


21. Locate intervals which contain real zeros of the following polynomials.
    Get as much information as you can about these zeros.

(a) t5 - 3t4 - t2 - 4t + 14

(b) 24t5 + 143t4 - 136t3 + 281t2 + 36t - 140

(c) 2t4 + 5t3 + t2 + 5t + 2

(d) 16t6 + 3t4 - 3t3 - 142t2 - 9t - 21

(e) 16t7 - 5t5 - 97t4 - 95t3 - 79t2 + 36.

Explorations

E.47. Proving the Fourier-Budan Theorem. The key to proving the

Fourier-Budan Theorem is the observation that deleting the first entry of

the sequences (p(u),p’(u),.. .) and (p(v), p’(v),.. .) gives the corresponding

sequences for the derivative. This suggests that we should explore the rela-

tionship between the zeros of p(t) and those of p’(t), and the tool for doing

this is Rolle’s Theorem (see Exploration E.28). The proof can be executed