2.2. POWERS, LOGS, AND EXPONENTIALS 49
5.
5.0
-5.0
5.0-5.0
6.
5.0
-5.0
5.0-5.0
Problem 2.20 Suppose that the largest number of points that any horizontal line intersects
the graph of a polynomial f .x/ in is m. Prove that the degree of f .x/ is at least m.
Problem 2.21 Give an example of a polynomial of degree 8 that has the property that the
maximum number of times it intersects any horizontal line is two.
Problem 2.22 Verify that the polynomial f .x/ D x
3
6x
2
C 11x 6 has the property that
f .1/ D f .2/ D f .3/ D 0. Use this information and the root-factor theorem to factor f .x/ .
Problem 2.23 Use the root factor theorem and some cleverness to factor
g.x/ D x
4
625
Problem 2.24 If
h.x/ D .x 2/.x
2
C 1/.x
2
C x C 1/
then how many roots does h.x/ have and what are they?
2.2 POWERS, LOGS, AND EXPONENTIALS
is section deals with a very important category of functions: logarithms and exponential func-
tions, as well as the related algebra of powers. We link logs and exponentials together because
each can undo what the other does, like square and square root. As with square root, there is
also a concern with negative numbers when computing logs. Logarithms don’t exist at zero,
while square roots do, so there is a difference. We begin with the algebra of powers and their
corresponding roots.
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