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78 2. Evaluation, Division, and Expansion

17. Find all odd manic quintic polynomials over Z which have at least
    two integer zeros and take the value -29670 when evaluated at 10.
    What is the value of the integer zero?


18. Let f(x) be a polynomial of degree at most n. Determine the degree
    of the polynomial

f(x) - xf’(x) + (x2/2!)f”(X) - (x3/3!)f”‘(X) + *. *

+ (-l)“(x”/n!)f(^)(z).

19. Let T be a real number, and let A be the set of polynomials f over R
    which satisfy

(9 f(O) 2 0;

(ii) if f(0) = 0, then f’(0) = 0 and f”(0) 10;

(iii) f(O)f”(O) - j’(O)2 2 rf(O)f’(O).

(a) Give an example of a nonconstant polynomial in A.

(b) Prove that, if c > 0 and f, g are in A, then cf, f + g, fg all

belong to A.

Hints

Chapter 2

1.13. If the zeros of j’(t) are known, what are the zeros of f(t - 3)?

2.11. The remainder has degree not exceeding 1. Write it in the form
u(t - Q) + v(t - b).

3.9. (a) Use induction on the degree of p. Note that, if p(t) = ~,t” +.. .,

then deg(p(t) - a,(t - c)“) < degp(t).

4.6. Translate the graph so that the inflection point is at the origin.

5.1. Let f(x) = x(x - 1)(x - 2). .a (x - n + 1) - Ic. Show that f’(z) =

(x - 1)(x - 2)(x - 3) * *. (x - n + 1) +x(x - 2)(x - 3). 1. (x - n + 1) +

x(x - 1)(x -3)=(x-n+l)+*** and consider the signs of f’(O),

f’(l), f’(2), * * ** What can be deduced about the number of distinct

zeros of f(z) and their multiplicities?

5.2. Let the three polynomials be a(x - u)(x - o), b(x - u)(x - w),

c(x - v)(x - w) and examine the sum of these evaluated at u, V,

w in turn.