Welcome to Precalculus!
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Write “I can” statements. (on the right)
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Focus and Review
Attendance
Questions from yesterday’s work
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Factoring a Polynomial
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Factoring a Polynomial
The Linear Factorization Theorem shows that you can write
any nth-degree polynomial as the product of n linear factors.
f(x) = an(x – c1)(x – c2)(x – c3) . . . (x – cn)
However, this result includes the possibility that some of the
values of ci are complex.
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Factoring a Polynomial
The following theorem says that even if you do not want to
get involved with “complex factors,” you can still write f(x) as
the product of linear and/or quadratic factors.
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Factoring a Polynomial
A quadratic factor with no real zeros is said to be prime or
irreducible over the reals.
Note that this is not the same as being irreducible over the
rationals.
For example, the quadratic x2 + 1 = (x – i)(x + i) is irreducible
over the reals (and therefore over the rationals).
On the other hand, the quadratic x2 – 2 =
is irreducible over the rationals but reducible over the reals.
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Application
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Example 11 – Using a Polynomial Model
You are designing candle making kits. Each kit contains 25
cubic inches of candle wax and a mold for making a
pyramid-shaped candle.
You want the height of the candle to be 2 inches less than
the length of each side of the candle’s square base. What
should the dimensions of your candle mold be?
Solution:
The volume of a pyramid is
where B is the area of
the base and h is the height. The area of the base is x2 and
the height is (x – 2).
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Example 11 – Solution
So, the volume of the pyramid is
25 for the volume yields the following.
cont’d
Substituting
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Example 11 – Solution
cont’d
The possible rational solutions are
Use synthetic division to test some of the possible solutions.
Note that in this case it makes sense to test only positive
x-values.
Using synthetic division, you can determine that x = 5 is a
solution.
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Example 11 – Solution
cont’d
The other two solutions that satisfy x2 + 3x + 15 = 0 are
imaginary and can be discarded.
You can conclude that the base of the candle mold should
be 5 inches by 5 inches and the height should be
5 – 2 = 3 inches.
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Application
Before concluding this section, here is an additional hint that
can help you find the real zeros of a polynomial.
When the terms of f(x) have a common monomial factor, it
should be factored out before applying the tests in this
section.
For instance, by writing
you can see that x = 0 is a zero of f and that the remaining
zeros can be obtained by analyzing the cubic factor.
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2.6
Rational Functions
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Objectives
 Find the domains of rational functions.
 Find the vertical and horizontal asymptotes of
the graphs of rational functions.
 Sketch the graphs of rational functions.
 Sketch the graphs of rational functions that
have slant asymptotes.
 Use rational functions to model and solve
real-life problems.
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Introduction
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Introduction
A rational function is a quotient of polynomial functions. It
can be written in the form
where N(x) and D(x) are polynomials and D(x) is not the
zero polynomial.
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Introduction
In general, the domain of a rational function of x includes all
real numbers except x-values that make the denominator
zero.
Much of the discussion of rational functions will focus on
their graphical behavior near the x-values excluded from
the domain.
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Example 1 – Finding the Domain of a Rational Function
Find the domain of
and discuss the behavior of f
near any excluded x-values.
Solution:
Because the denominator is zero when x = 0 the domain
of f is all real numbers except x = 0.
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Example 1 – Solution
cont’d
To determine the behavior of f near this excluded value,
evaluate f(x) to the left and right of x = 0, as indicated in the
following tables.
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Example 1 – Solution
cont’d
Note that as x approaches 0 from the left, f(x) decreases
without bound. In contrast, as x approaches 0 from the
right, f(x) increases without bound.
The graph of f is shown below.
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Vertical and Horizontal Asymptotes
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Vertical and Horizontal Asymptotes
In Example 1, the behavior of f near x = 0 is denoted as
follows.
The line x = 0 is a vertical asymptote
of the graph of f, as shown in
Figure 2.21.
Figure 2.21
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Vertical and Horizontal Asymptotes
From this figure, you can see that the graph of f also has a
horizontal asymptote—the line y = 0.
The behavior of f near y = 0 is denoted as follows.
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Vertical and Horizontal Asymptotes
Eventually (as x 
or x 
), the distance between
the horizontal asymptote and the points on the graph must
approach zero.
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Vertical and Horizontal Asymptotes
Figure 2.22 shows the vertical and horizontal asymptotes of
the graphs of three rational functions.
(a)
(b)
(c)
Figure 2.22
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Independent Practice
Section 2.5 (page 164)
# 113, 131
Section 2.6 (page 177)
# 1 – 4 (vocabulary), 7, 8, 71
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