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198 6. Symmetric Functions of the Zeros

6.3 Sums of the Powers of the Roots

While the determination of an arbitrary symmetric function of the zeros

of a polynomial might be quite difficult, it is relatively easy to determine

the sums of various powers of the roots by setting up a recursion relation.

In the exercises, we suppose that p(t) = t” + c,-ltnB1 +... + clt + co is a

manic polynomial with zeros tr ,... , t,. Define

PO = n

and

pk=t:+tt+...+ti (Ic=1,2 )... ).

As before, s, will denote the rth elementary symmetric function of the
zeros.

Exercises

1. Let a, b, c be the roots of the equation

x3 - 2x2 + x + 5 = 0.

Find the value of a4 + b4 + c4.

2. Verify that pl = s1 and that p2 = sf - 2s~.


3. It is straightforward to give a recursion relation for pk when k 1 n.
   Prove that, for r 3 0,

Pn+r + cn-1pn+r-1 + Cn-2pn+r-2 +.. * + clpr+l + cop, = 0.

4. When k < n, the recursion relation to express pk in terms of sums of
   earlier powers is a little more complicated. Verify that

(a) PI + h-1 = 0

(b) p2 + cn-IPI + 2c,+2 = 0.

5. The purpose of this exercise is to develop a conjecture concerning ~3.

(a) Observe that p3 is of degree 3 in the ti. Infer that ~~~-4, ~~-5,.. .,

cl, CO will not likely be involved in an expression of ~3.

(b) On the basis of the information obtained in Exercises 3 and 4,

argue that it is reasonable to guess an equation of the form

~3 + klc,-lpz + kzc,.-m + k3c,-3 = 0.