102 3. THE ARITHMETIC, GEOMETRY, AND CALCULUS OF POLYNOMIALS
while the odd ones have a sign chart of
.1/ CC C .0/ CC C .C1/:
˙
3.2.1 MULTIPLICITY OF ROOTS
What makes it possible for a polynomial of degree n to have less than n roots? Part of this is we
can add constants to a polynomial so that it intersects the x-axis as little as possible. Another
factor is that roots can be repeated. Look at the polynomial shown in Figure 3.4:
p.x/ D
1
4
.x C2/.x 2/
2
D
1
4
x
3
1
2
x
2
x C2
e fact that the root at x D 2 is squared makes the graph bounce off of the x axis instead of
passing through. is leads to a definition.
Definition 3.1 If .x a/ divides a polynomial p.x/, then the highest power k so that .x a/
k
di-
vides p.x/ is the multiplicity of the root x D a in p.x/.
3.2. QUALITATIVE PROPERTIES OF POLYNOMIALS 103
4.0
-4.0
4.0-4.0
x D 2
x D 2
Figure 3.4: p.x/ D
1
4
.x C2/.x 2/
2
e multiplicity of the root x D 2 in Figure 3.4 is two.
Knowledge Box 3.5
e effect of root multiplicity
If p.x/ is a polynomial, and x D a is a root of p.x/, then
• If the root at x D a is of odd multiplicity, the graph of p.x/ passes
through the x-axis at x D a.
• If the root at x D a is of even multiplicity, the graph of p.x/
touches the x-axis and then bounces back on the same side it was on
before at x D a.
104 3. THE ARITHMETIC, GEOMETRY, AND CALCULUS OF POLYNOMIALS
Notice that, since the root at x D 2 in Figure 3.4 has multiplicity two, there is a local minimum
and a critical value at x D 2. is effect is fairly general and can be summarized as in Knowledge
Box 3.6.
Knowledge Box 3.6
Roots with multiplicity and derivatives
If p.x/ is a polynomial, and x D a is a root of p.x/ with multi-
plicity k > 1, then x D a is also a root of f
0
.x/ with multiplicity
k 1.
Example 3.21 Verify that if
f .x/ D .x 2/
2
.x C2/ D x
3
2x
2
4x C 8;
then f
0
.2/ D 0.
Solution:
Compute
f
0
.x/ D 3x
2
4x 4
en f
0
.2/ D 12 8 4 D 0, and 2 is a root of the derivative.
˙
e examples in this section show us how we can construct polynomials with particular prop-
erties quite easily by putting in the roots we want with the correct multiplicities. is sort of
information is made even more useful by the fact that polynomials can be used to approximate
other functions on bounded intervals.
PROBLEMS
Problem 3.22 Show that the number of horizontal tangents between two roots – that do not
have another root between them – must be odd.
Problem3.23 Construct a polynomial that has more than one horizontal tangent between two
roots that do not have another root between them.
3.2. QUALITATIVE PROPERTIES OF POLYNOMIALS 105
Problem 3.24 Prove that a non-constant polynomial can have zero inflection values; do this
by giving an example of one.
Problem 3.25 Show that a polynomial of degree n can have at most n 2 inflection values.
Problem 3.26 Give an example of a non-constant polynomial with two roots that never takes
on negative values. Explain why your example is correct.
Problem 3.27 Suppose a polynomial has three distinct roots, r
1
, r
2
, r
3
. What can you deduce
from this fact alone about the degree of the polynomial?
Problem 3.28 Suppose we have two polynomials p .x/ and q.x/ and that all roots of p.x/ are
roots of q.x/ but at least one root of q.x/ is not a root of p.x/. If p.x/ has two roots, provide
three examples. In the first, p.x/ should have a degree less than q.x/; in the second, p.x/ and
q.x/ have equal degree; in the third, p.x/ should have a higher degree than q.x/.
Problem 3.29 Deduce as much as you can from the graph above. e grids are of length one
unit. In particular, what is the minimum degree and what can you deduce about the roots and
their multiplicity?
106 3. THE ARITHMETIC, GEOMETRY, AND CALCULUS OF POLYNOMIALS
Problem 3.30 Deduce as much as you can from the graph above. e grids are of length one
unit. What is the minimum possible degree and what can you deduce about the roots and their
multiplicity?
Problem 3.31 Deduce as much as you can from the graph above. e grids are of length one
unit. In particular what is the minimum degree and what is possible to deduce about the roots
and their multiplicity?
Problem3.32 What is the smallest possible degree for a polynomial that is an answer to Ques-
tion 3.26.
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