Unknown

4.6. Consequences of the Fundamental Theorem 147

(a) If si is a nonreal zero of p(t), prove that the polynomial

t2 - (2Re si)t + Isi12

is a polynomial irreducible over R which divides p(t).

(b) Prove that p(t) can be written as a product of real irreducible

linear and quadratic polynomials.

(c) Deduce that a polynomial over R is irreducible if and only if it

is linear or quadratic of the form at2 + bt + c with b2 - 4ac < 0.

4. Let p(t) be a polynomial over C of positive degree n. Prove that p(t)
   assumes every complex value at least once and assumes all but finitely
   many complex values n times.


5. Show that every real polynomial of odd degree has an odd number of
   real zeros, counting multiplicity, and deduce that such a polynomial
   has at least one real zero.


6. Let r1, r2,... , r, be all the roots (each repeated as often as its multi-
   plicity indicates) of the complex polynomial ant” + a,-lP-’ +... +
   alt + a~. Show that

rl + r2 +... + rn = -an-Jan

rlr2r3 .. .r, = (-l)“ao/an,

and that the sum of all possible products of k of the roots is equal to

(--l)“-k,,-k/a,.

7. Suppose that a complex polynomial p(t) can be factored in two ways:

p(t) = fi(t - ai) = fi(t - bj).

i=l j=l

Show that m = n and that the ai’s are the same as the bj’s in some

order.

8. Let as, bo, al, bl,... ,a,, 6, be 2(n + 1) complex numbers, with the
   ei distinct. Show that there is at most one polynomial p(t) of degree
   not exceeding n for which p(ai) = bi (0 5 i 5 n).


9. Suppose that a polynomial p(t) over C has the following properties:

(i) the multiplicity of 1 as a zero of p(t) is even (possibly 0);

(ii) If r is a zero of p(t), then l/r is a zero with the same multiplicity

as r.

Prove that p(t) is a reciprocal polynomial (see Exercises 1.4.13 and

1.4.16).