Unknown

74 2. Evaluation, Division, and Expansion

9. Discuss the possible graphs of the general quartic ax4 + bx3 + cx2 +
   dx + e. You may find it helpful to make a translation of coordinates
   which will eliminate the cubic term.


10. (a) Sketch the graph of the polynomial

6x5 - 15x4 - 10x3 + 30x2 + 10.

How many real zeros does this polynomial have?

(b) For each nonnegative integer m, find the set of values k for which

the polynomial

6x5 - 15x4 - 10x3 + 30x2 + k

has exactly m real zeros (i) not counting multiplicity; (ii) count-

ing multiplicity.

11. Prove that every polynomial p(t) of odd degree with real coefficients
    has at least one real root. Deduce that p(t) = (t - r)q(t) for some real
    r and polynomial q(t) over R.

Explorations

E.27. Consider the graphs of the polynomials you have drawn already. In
how many points can such a graph be intersected by a line with equation
of the form y = k? of the form y = mx + b? Make a conjecture concerning
this number and the degree of a polynomial. Investigate further using poly-
nomials of higher degree than 4; be sure to sketch the derivative as well
and to relate the values of the derivative to the behaviour of the graph of
the polynomial.

E.28. Rolle’s Theorem. The task of finding the zeros of a polynomial
becomes more difficult as the degree of the polynomial increases. Accord-
ingly, it is often helpful to be able to relate the zeros of a polynomial to the
roots of its derivative, whose degree is lower. The technique we are about
to discuss was initiated by the mathematician Michel Rolle in a book called
l+aite’ d’algkbre published in 1690.
Suppose that a and b are two consecutive zeros of a real polynomial f(x);
that is, f(a) = f(b) = 0 and f d oes not vanish between a and b. Sketch
some possible graphs for f ( x ) on the interval [a, b], and argue that f must
have at least one maximum or minimum in the interior of the interval.
Deduce Rolle’s Theorem, that between any two zeros of f(x) there is at
least one real zero of f’(x).
Suppose that u and v are two consecutive zeros of f’(x). What can be said
about the number of real zeros of f(x) between u and v? If the derivative