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Hints 191

4.13. What is the sign of the value of the polynomial at 0 and l? What

about the discriminant? What about the product of the zeros? Use

(u - l)(c - 1) 1 0 to show b 2 5.

4.14. Use induction. Show that xzm+z > -l/(m + 1) by plugging z =

-l/(m + 1) into both equations.

4.16. Let z = r(cos0 + isine) and write the equation as a real system in r

and 0. Let 0 < 0 < ?r. What is the sign of sin60?

4.17. Differentiate. Factor out the highest power of z and z+ 1. Differenti-

ate some more. The strategy is to use Holle’s Theorem to relate the

positive roots of the equation to the roots of an equation of the form

~(x)P(z + 1)” = 0, where the degree of u(x) can be identified.

4.18. First look at the case in which all zeros are real. For the situation in

which there are two nonreal zeros u f iv and one real zero r, use the

relationship between zeros and coefficients to obtain a system for r,

u and u2 + v2. Eliminate r and u and get a cubic equation for u2 + v2

which involves a.

4.19. Make a linear change of variable and deal with a polynomial whose

zeros are -u, -v, v where u 1 v 1 0. Identify the endpoints of the

interval which should contain a zero of the derivative.

4.20. What quadratic over Z has (l/2)(&- 1) as zero? Deal with the case

nl > 2 first. If 121 = 1, multiply the polynomial by^1 - z.

4.21. What is the sum of the zeros? If the zeros are on a straight line, what

is the relationship between this line and the origin?

4.22. The polynomial has a minimum. Its derivative vanishes there.

4.23. What does the property imply concerning the zeros of p(t)? Look at
p(t) + k for all real values of k.

4.24. Let q’(t) = p(t) and q(0) = 0. Show that q(1) < q(-1) and that q(t)
is increasing at t = 0. Sketch the graph of q(t).

4.25. Look at n - kp’(r)/p(z).

4.26. What roots of unity will satisfy the equation? Suppose IzI = 1. Note
that bzP = &‘+‘J + (b - e). Take absolute values.

4.27. If there is a root with modulus unequal to 1, there is a root z with
modulus exceeding 1. For such a root, Z”(Z - u) = (1 - UZ). Let
z = r(cos 0 + i sin 0) and look at the square of the absolute values of
both sides.

4.28. Use induction.