192 5. Approximation and Location of Zeros4.29. Multiply up the denominators and evaluate the two sides at x = ei.
4.31. If Re z > a, then 1% + a,-rl 2 Re (z + a,-~) > a + Re(a,-1).
4.32. q(t) can be factored.
4.33. Use induction. Multiply the left side by n - 1.
4.34. It is easy to find a polynomial of degree n - 1 over Z whose zeros are
prescribed rationals. If such a polynomial is perturbed by adding a
sufficiently small multiple oft”, the zeros will not move much.
4.35. Express ej(x) + bg(x) in terms of (1 + ix)“’ and (1 - ix)“‘. What can
be said about the absolute value of the ratio of these two quantities
when x is a zero?