41
C H A P T E R 2
e Library of Functions
is chapter continues our review of background material needed to study calculus. We start with
polynomials and then review log, exponential, and trigonometric functions. If you have already
had classes in these topics, you may be able to skim through this chapter, but even someone who
has passed courses in these topics will benefit from reviewing the knowledge boxes.
2.1 POLYNOMIALS
We’ve already learned about lines, where the highest power of x is one, and quadratics, where
the highest power of x is two. Polynomials are functions that let this highest power be any
positive number.
Knowledge Box 2.1
A polynomial is a function that is a sum of constant multiples of powers of
a variable.
Example 2.1 e following are examples of polynomials.
y D 3
f .x/ D 3x C 4
g.x/ D x
2
C 2x 1
h.x/ D x
3
C 4x 1
k.x/ D x
3
C 3x
2
C 3x C 1
q.x/ D x
6
2
˙
42 2. THE LIBRARY OF FUNCTIONS
Polynomials are very well-behaved functions. ey have a domain of .1; 1/. e range of
a polynomial is a bit trickier. As we’ve seen, a quadratic may have a range that doesn’t include
everything. We need some additional terminology before we go on.
Definition 2.1 e standard form of a polynomial is when it is written as a sum of constant multiples
of a variable with the powers in descending order from left to right.
e polynomials in Example 2.1 are in standard form. A example of a polynomial not in standard
form is:
f .x/ D .x 1/.x 2/.x
2
C 4/
In standard form this polynomial would be:
f .x/ D x
4
3x
3
C 6x
2
12x C 8
Definition 2.2 e degree of a polynomial is the highest power of x that appears in the polynomial
when it is in standard form.
e polynomials in Example 2.1 have, from top to bottom, degrees as follows: 0, 1, 2, 3, 3, 4,
and 6. e degree of a polynomial may not be obvious, as we see in the next example.
Example 2.2 What is the degree of
f .x/ D .x 1/.x 2/.x 3/‹
Solution:
To find the degree of this polynomial we have to multiply it out first.
.x 1/.x 2/.x 3/ D .x 1/.x
2
5x C 6/
D x.x
2
5x C 6/ 1.x
2
5x C 6/
D x
3
5x
2
C 6x x
2
C 5x 6
D x
3
6x
2
C 11x 6
Now, we can see that the degree of this polynomial is 3.
˙
Actually, we don’t have to multiply it out. It’s possible to see that it will have a degree of three
by imagining how the multiplication would come out. In fact:
Knowledge Box 2.2
If we multiply several polynomials, the degree of the product is the sum
of the degrees of the polynomials we multiplied.
2.1. POLYNOMIALS 43
Definition 2.3 e coefficients of a polynomial are the constants multiplied by the powers of x.
Example 2.3 For the polynomial f .x/ D x
3
C 4x C 3 the coefficients are 1, 4, and 3. We can
be more specific: the coefficient of x
3
is 1; the coefficient of x is 4; and the constant coefficient
is 3. Since there is no x
2
term, we can say that the coefficient of x
2
is 0.
˙
Now that we know what degrees and coefficients are, we can say a little about the range of
polynomial functions. ese results are given here without explanation. We will revisit them in
the second book, Fast Start Integral Calculus.
Knowledge Box 2.3
e range of a polynomial of odd degree is .1; 1/.
Knowledge Box 2.4
e range of a polynomial of positive even degree is:
• .C; 1/ for some constant C if the coefficient of its highest power is
positive, and
• .1; D/ for some constant D if the coefficient of its highest power
is negative.
Normally “odd” and “even” cover all the possibilities, but zero is quite peculiar for an even num-
ber. f .x/ D cx
0
is just a long way of saying f .x/ D c, which has a very small range. Let’s
complete our Knowledge Box collection of possible ranges of polynomials with the following.
Knowledge Box 2.5
e range of a polynomial of degree zero is a single constant c; the function
has the form f .x/ D c.
44 2. THE LIBRARY OF FUNCTIONS
Example 2.4
e picture on the left is the graph of a second-degree polynomial, while the one on the right
is the graph of a third-degree polynomial.
1.5
-1.5
1.5-1.5
An even degree polynomial
1.5
-1.5
1.5-1.5
An odd-degree polynomial
Check these against the Knowledge Box rules for polynomials of even and odd degree and see
if they have the predicted ranges.
˙
e next definition comes up a lot when we are trying to solve problems. We have already seen
that quadratic equations can have 0, 1, or 2 roots and, in Example 1.53, found a formula for
those roots. We didn’t formally define what a root was then, so here it is now.
Definition 2.4 A root of a polynomial function f .x/ is any number r for which f .r/ D 0.
If you want to think about roots geometrically, think of them as places where the graph
of a function crosses the x-axis. When we are solving problems, we can sometimes write a
polynomial so that the places where the polynomial is zero are the solutions to the problem.
is will come up frequently in Chapter
4 when we are doing optimization.
Figure 2.1 shows the odd-degree polynomial from Example 2.4 with dots where its roots are.
e ranges of polynomials that we mentioned before arise from the way that a polynomial with
positive degree heads toward infinity. is geometric behavior influences the roots as well. Poly-
nomials can only change which direction they are going (up or down) a number of times that
is one less than their degree. Notice a second-degree polynomial changes from down to up or
from up to down exactly once. All this has the following implications for roots of polynomials.
Knowledge Box 2.6
A polynomial of odd-degree n has from one to n roots. A polynomial of
positive even degree n has from zero to n roots.
2.1. POLYNOMIALS 45
1.5
-1.5
1.5-1.5
Figure 2.1: Roots of an odd-degree polynomial.
Polynomials have another interesting property: they form a closed set relative to addition,
multiplication, and multiplication by constants. Adding or multiplying two polynomials or
multiplying a polynomial by a constant results in another polynomial.
Knowledge Box 2.7
Polynomials obey the following rules:
• A constant multiple of a polynomial is a polynomial.
• e sum of two polynomials is a polynomial.
• e product of two polynomials is a polynomial.
Next we note that something that was true of quadratics is also true for polynomials.
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