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2.5. Problems 75

f’(x) has k real zeros, what can be said about the number of real zeros of

f (x)3

Show that a real polynomial of degree n cannot have more than n real

zeros, counting multiplicity.

Rolle’s result as stated above can be strengthened. If f (x) has consecutive

real zeros at a and 6, then by the Factor Theorem, we can write f(x) in

the form

f(x) = (x - a)r(x - b)‘g(z)

where g(x) is a polynomial which does not vanish at any point in the

interval [a, b]. (Justify this statement.) Show that

(x - a)(x - b)f’(x) = f(x)[r(x - b) + s(x - a)

Every zero of f’(x) b e t ween a and b is a zero of the function in the square
brackets on the right hand side. Now look at the value of this function at
x = a and x = b, and draw the conclusion that the number of zeros of
f’(x) strictly between a and b must be odd, if we adopt the convention of
counting each zero as often as its multiplicity indicates.
Assume that the following is the graph of a real polynomial. What can
be said about its degree and about the signs of its first three and last three
coefficients?

2.5 Problems

1. What is the highest multiplicity a root can have for the equation

x(x - 1)(x - 2)... (x - n + 1) = k?