Unknown

6.3. Sums of the Powers of the Roots 199

(c) Assuming for the moment that (b) is valid, what must the coef-

ficients ki be? Try the substitutions (l,O, 0, 0,.. .), (1, l,O, 0,.. .)

and (1, l,l,O,.. .) for (tl, t2,t3, t4,.. .) to obtain three linear

equations which must be satisfied by the ki. Solve these equa-

tions to obtain kl = k2 = 1 and k3 = 3.

6. On the basis of Exercise 3, 4 and 5, it is reasonable to conjecture
   that,fork=1,2,3 ,..., n-l,

pk + Cn-@k-l + cn-2pk-2 +.. * + cn-k+lpl + h&k = 0.

This result can actually be established by induction on the degree n

of p(t) (i.e. the number of the ti).

(a) Verify that th e result holds for n = 2 and n = 3.

(b) Suppose that the result has been established for polynomials of

degree up to n - 1. Let c: and pi be obtained respectively from

ci to pi by setting t, = 0. Verify that the manic polynomial with

zeros tr, t2,... , t,-1 is

P-l + C;J”-2 +. f * + c&t2 + c;t + c;.

Use the induction hypothesis to establish that

pk + cn-@k-l +... + c,-k+lpl + h-k

(considered as a polynomial in the ti) vanishes for t, = 0. De-

duce that it is divisible by t,, and therefore, because of its sym-

metry is actually divisible by tit;!... t,. Conclude that, because

its degree as a polynomial in the ti is equal to k < n, it must in

fact vanish.

7. Let t2 + clt + cc have zeros tl and tz. Write down recursion relations
   for pk = tf +tf (1 < k 5 5) and verify them directly. Determine these
   pk in terms of cl and cc only.
   Carry this out for the polynomials t2 - 3t + 2 and t2 + t + 1, and verify
   your answers directly.


8. Find the sum of the fifth powers of the zeros of t3 + 7t2 - 6t - 1.


9. Let ~1, zz,... , Z, be complex numbers for which Z! + ~5 +... + Z: = 0
   for 1 5 k 2 n. Must each Zi vanish?

Explorations

E.52. Use the recursion relations for the pk to obtain expressions for each
of them which involve only the coefficients ci and none of the other pi.