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204 6. Symmetric Functions of the Zeros

1.11. (a) Express p and r in terms of the roots to get two equations for

(a + b) and (c + d), w h ere a, b, c, d are the zeros and ab = cd. If

r2 = p2s, the equation is quasi-reciprocal (Exercise 1.4.17).

(c), (d) Consider the substitution x = -p/2 - y.

3.1. a3 and a4 can be expressed in terms of lower powers of a.

3.9. Use Exercise 6 to determine the polynomial of degree n with zeros zi.

4.1. (a) Let the zeros be r - s, r, r + s.

4.3. (a) A double root is a zero of the derivative.

(b) Use b k+3 = bk+2 + bk+’ + bk, etc. and induction.

4.4. The reciprocals of terms in harmonic progression are in arithmetic

progression.

4.8. Note that (u - 1) + (V - 2) = -(w - 3). Cube this equation.

4.9. Let yk = zk - 1. What polynomial has zeros yk?

4.11. Let 21 = r + s, v = rs, w = p + q, t = pq and express the coefficients

of the given quartic and the zeros of the required quartic in terms of

u, v, w, %.

4.12. Look at values assumed by t4 - 14t3 + 16t2 - 84t (which can be easily

factored).

4.13. What can be said about the sum of the roots of the equation?

4.14. Where ri are the roots, look at C(rf +ri”). What is the lower bound

for the values of each summand?

4.15. Look at f(x)/f’(x).

4.16. The purported roots have the form v3(4v3-32r)-l, where u = cose for

suitable 0. One strategy is to first determine the cubic equation whose

roots are the reciprocals of these. The reciprocals can be simplified to

an expression in v -2. The cubic equation whose roots are v should be

found, so that the elementary functions in the vs2 can be determined.

4.17. a, b, c, d are the zeros of the quartic t4 - wt3 - zt2 - yt - x.

4.18. u, V, w are the zeros of a polynomial t3+pt+q. Express the symmetric

functions of x, y, z in terms of p and q.

4.19. Consider the cubic polynomial (t + a)(t + b)(t + c) - x(t + b)(t + c) -
y(t + a)(t + c) - z(t + a)(t + b). What are its zeros? What values oft
will isolate the coefficients x, y, z?