Download Chapter Three - Polynomials and Rational Functions

Math 100
Section 3.1
Polynomial Functions and Their Graphs
A polynomial is a function of the form:
P(x)  an x n  an 1 x n 1    a1 x  a0
where an, an-1, ... a0 are constant numbers known as coefficients with an  0. The
number n is the highest power of x to appear and is known as the degree of the
polynomial. If this formal definition of polynomial seems confusing, an example
may clarify the definitions:
Example: P(x) = 6x3 – 7x2 + x – 17 is an example of a polynomial of degree 3,
because 3 is the highest power of x that appears. The coefficients of the
polynomial are the numbers 6, -7, 1 and –17. The number 6 is known as the leading
coefficient, because it is the coefficient of the highest degree x term, x3. The
number –17, which is not multiplied by an x-term at all, is known as the constant
coefficient.
Exercise 1 For each polynomial, state the degree, list the coefficients and the
leading coefficient:
Leading
Polynomial
Degree
Coefficients
coefficient
P(x) = 4x2 – 12
F(x) = -10x9 + 3x4 – 12x - 1
Y = x3 – 2x + 4
You have already studied polynomials of degree 0, 1 and 2.
Exercise 2 Given a examples of polynomials of degree 1 and 2, and state the
more familiar names for such polynomials.
Degree 1:___________________________________
Degree 2:___________________________________
Math 100 Chapter 3
67
Exercise 3 Plot the graph of each polynomial given below, using techniques you
have already learned:
a) p(x) = 2
b) p(x) = - ½ x + 2
a) p(x) = x2 – 6x + 5 In this case, find the vertex
by completing the square, determine whether
there is any vertical reflection, and find where
the graph crosses the x and y axes.
It is important to notice the differences between the x-intercepts and the yintercept. Here are some properties to note:
y-intercept



There is always exactly one
It is equal to P(0)
It is equal to the constant
coefficient
x-intercepts




Math 100 Chapter 3
There are at most n (for degree n),
but there might not be any
They are the solutions to P(x) = 0
If c is a zero, then (x – c) is a factor
in P(x).
They are often called the zeros of
P(x)
68
The end behaviour of a polynomial P(x) tells us what happens to the value of
P(x) as x gets far away from zero (in both directions). Thus the end behaviour
occurs for values of x at the ends of the x-axis
x-
x 
ends
So when we refer to the end-behaviour of a function, we are discussing the values
f(x) takes on when x is a number that is quite far away from zero.
 As x gets far away from zero, going right, we say x 
 As x gets far away from zero, going left, we say x -
Exercise 4 Each graph below represents a function f(x). Describe what f(x)
seems to be is doing when x  and x -
a)
b)
c)
Math 100 Chapter 3
As x ,
f(x)______
_________
As x ,
f(x)______
_________
As x -,
f(x)______
_________
_________
__
As x ,
f(x)______
_________
As x -,
f(x)______
_________
_________
__
As x ,
f(x)______
_________
As x -,
f(x)______
_________
_________
__
d)
As x -,
f(x)______
_________
_________
__
69
The end behaviour of a polynomial depends on the parity (+ or -) of the leading
coefficient and on whether the degree of the polynomial is odd or even. (Even
Degrees: 2, 4, 6, 8, 10, . . . .; Odd Degrees: 1, 3, 5, 7, 9, 11, . . .). There are four
types of end behaviour, that have here been numbered 1, 2, 3, and 4. (This
numbering system was made up for these notes and is not standard.)
End Behaviour of Polynomials
P(x) has odd degree
P(x) has even degree
Leading coefficient
positive
Type 1
y  as x 
y - as x -
Leading coefficient
negative
Type 2
y - as x 
y  as x -
leading coefficient
positive
Type 3
y  as x 
y  as x -
Leading coefficient
negative
Type 4
y - as x 
y - as x -
Exercise 5 For each polynomial, state the degree, the leading coefficient, and
what type of end behaviour the polynomial has:
a) P(x) = x4 + 11 x2 + 18
Degree____ Leading coefficient______ End behaviour(circle one): 1 2 3 4
b) P(x) = 2 x3 + 14 x – 12
Degree____ Leading coefficient______ End behaviour(circle one): 1 2 3 4
c) P(x) = -x5 + 11 x4 - 33 x3 - 11 x2 + 154 x – 120
Degree____ Leading coefficient______ End behaviour(circle one): 1 2 3 4
d) P(x) = -2 x2 + 4 x
Degree_____ Leading coefficient_____ End behaviour(circle one): 1 2 3 4
Math 100 Chapter 3
70
Exercise 6 Match each of the graphs on the following pages to a polynomial on this
list. Each polynomial is given in its factored and expanded form:
1. P(x) = (x2 + 9) (x2 + 2)
P(x) = x4 + 11 x2 + 18
_____________
2. P(x) = (x + 2) (x - 4)
P(x) = x2 - 2 x - 8
_____________
3. P(x) = x(2 – x)
P(x) = -x2 + 2x
_____________
4. P(x) = (4 - x) (x + 2)
P(x) = -x2 + 2 x + 8
_____________
5. P(x) = (x - 1) (x - 2) (2 x + 8) (x + 1) _____________
P(x) = 2 x4 + 4 x3 - 18 x2 - 4 x + 16
6. P(x) = (4 – x)(x2 + 2)
P(x) = -x3 + 4x2 – 2x + 8
_____________
7. P(x) = (x - 3) (x + 5) (1 - x)
P(x) = -x3 - x2 + 17 x - 15
_____________
8.
P(x) = -(x + 3)2(x – 2)2
_____________
4
3
2
P(x) = - x - 2x + 11x + 12x - 36
9.
P(x) = (x + 3)2(x – 2)2
P(x) = x4 + 2x3 – 11x2 –12x + 36
_____________
10. P(x) = (x - 1) (2 - x) (2 x + 8) (x + 1) ____________
P(x) = -2 x4 - 4 x3 + 18 x2 + 4 x - 16
11. P(x) = (4 - x)(x + 2)(x - 1)(x - 5)(x - 3) ___________
P(x) = -x5 + 11 x4 - 33 x3 - 11 x2 + 154x – 120
12.
P(x) = (x - 3)(x + 5)(x - 1)
P(x) = x3 + x2 - 17x + 15
_____________
13. P(x) = (2x + 6)(x - 1)(x - 2) _______________
P(x) = 2x3 - 14x + 12
Math 100 Chapter 3
71
Circle the correct answer:
A
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Circle the correct answer:
B
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Circle the correct answer:
C
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Circle the correct answer:
D
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Math 100 Chapter 3
72
Circle the correct answer:
E
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Circle the correct answer:
F
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Circle the correct answer:
G
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Circle the correct answer:
H
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Math 100 Chapter 3
73
Circle the correct answer:
I
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Circle the correct answer:
J
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Circle the correct answer:
K
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Circle the correct answer:
L
 P(x) has (odd/even) degree
 The leading coefficient is (positive/negative)
Fill in the blanks (estimate) :
 P(x) = 0 when___________________________
 P(x) < 0 when ___________________________
 P(x) > 0 when ___________________________
 The y-intercept is ______________________
The polynomial that best matches is:_______
Math 100 Chapter 3
74
Exercise 7 Sketching Graphs of Polynomials
1. P(x) = (2 x + 6) (1 - x) (x - 4)
P(x) = -2 x3 + 4x2 + 22x – 24
a) For which x’s does P(x) = 0 ? (These are the zeros/x-intercepts of P(x) )
What is P(0) equal to?
b) Plot the zeros (from part a) and
the y-intercept (from part b)
c) Choose a value of x between and
to the left and right of the zeros.
Determine the value of the
function at these points. Include
the y-intercept in the table.
x
P(x)
d) Plot the points in the table
e) Determine the end behaviour:
P(x) has (odd/even) degree, and the leading coefficient is (positive/negative)
Therefore P(x) has type ( 1 2 3 4 ) end behaviour
f) Sketch a smooth curve that passes through the points and exhibits the required end behaviour
g) Solve the inequality -2 x3 + 4x2 + 22x > 24
2. P(x) = (x-2)2(x-4)(x+2)
Math 100 Chapter 3
75
P(x) = x4 – 6x3 + 4x2 + 24x - 32
a)
When does P(x) = 0 ? (These are the zeros/x-intercepts of P(x) )
b) What does P(0) equal?
d) Plot the zeros and y-intercept
e) Choose a value of x between
and to the left and right of
the zeros. Determine the
value of the function at these
points. Include the yintercept in the table.
X
P(x)
f) Plot the points in the table
g) Determine the end behaviour:
P(x) has (odd/even) degree.
The leading coefficient is (positive/negative).
P(x) has type ( 1 2 3 4 ) end behaviour
h) Sketch a smooth curve that passes through the points and exhibits the required end behaviour
i) Solve the inequality x4 + 24x – 32  6x3 - 4x2
Exercise 8 Polynomial Inequalities.
1. Solve 7x3 + 7x2  42x by following the steps given.
Math 100 Chapter 3
76
a) Get 0 on one side. (The other side will be a polynomial. Call it P(x) ):
b) Factor the polynomial side:
c) Set the polynomial side equal to 0, and solve to find its x-intercepts (zeros)
d) Use the zeros to divide the x-axis into intervals:
e) Substituted a number from each interval into the polynomial. If the value is
positive, then the polynomial will be positive for all numbers in the interval. If
the value is negative, then the polynomial will be negative for all numbers in the
interval.
Interval
Test
number (x)
from
interval
Evaluate P(x) at
the test number
Is P(x)
+ or –
at x ?
Is this interval
part of the
solution?
f) State the solution:
2. Solve 2x3 + 6x2 < 2x + 6 by following the steps given:
a) Get 0 on one side. (The other side will be a polynomial. Call it P(x) ):
Math 100 Chapter 3
77
b) Factor the polynomial side (hint: grouping):
c) Set the polynomial side equal to 0, and solve to find its x-intercepts (zeros)
d) Use the zeros to divide the x-axis into intervals:
e) Substituted a number from each interval into the polynomial. If the value is
positive, then the polynomial will be positive for all numbers in the interval. If
the value is negative, then the polynomial will be negative for all numbers in the
interval.
Interval
Test
number (x)
from
interval
Evaluate P(x) at
the test number
Is P(x)
+ or –
at x ?
Is this interval
part of the
solution?
f) State the solution:
Objectives and Suggested Exercises for Section 3.1
Suggested Exercises are on pages 230 and 231
1. To sketch the graph of a polynomial (#9, 11, 13)
Math 100 Chapter 3
78
2. To match a polynomial function with a given graph (#27, 29)
More Objectives and Suggested Exercises for Section 1.7
Suggested Exercises are on pages 78 and 79
1.
To solve a polynomial inequality (#21, 23, 25, 31)
Math 100
Section 3.5
Rational Functions
Math 100 Chapter 3
79
A rational function is a function that is the ratio of two polynomials. Rational
functions are characterized by the fact that, because division by zero is
undefined, any x-values that cause the denominator of a rational function to equal
zero are not allowed in their domains. (This contrasts with polynomial functions
which are always defined for all the real numbers.)
Exercise 1 Evaluate each of the rational functions at the indicated values of x.
10 x  2
a) For f(x) 
find:
x 3
f(5)
f(2)
f(0.2)
b) For the function f(x) 
f(3)
x 2  x  20
find:
x2  9
f( 10)
f(-5)
f(3)
f(-3)
f(4)
f(1)
c) Based on the results of a) and b), explain how to find:
i) The values of x for which a rational function is undefined.
Math 100 Chapter 3
80
ii) The zeros of a rational function. That is, values of x that make f(x) = 0
It is a common mistake to get the answers to the two questions asked above mixed
0
1
up. If you ever feel uncertain, use your calculator to evaluate
and then . You
1
0
will see that the results are very different. Thus it is very important to be able to
distinguish between the numerator and the denominator of a rational function.
(Remember that the denominator is downstairs).
Exercise 2 For each function, clearly state the largest natural domain. If any x
values are excluded from the domain, explain why they are excluded.
10 x  2
a) f(x) 
x 3
x 2  x  20
b) f(x) 
x2  9
x8
:
x4
a) State the largest natural domain of f(x)
Exercise 3 For the function f(x) 
b) The graph of f(x) is given below. What happens when x gets close to the
number that is not in the domain of f(x)?
Math 100 Chapter 3
81
Evaluate: f(4.2)
f(4.01)
f(3.8)
f(3.99)
c) Find the zero of f(x) (solve f(x) = 0). Compare your result to the graph.
d) Find the y-intercept of f(x) (evaluate f(0) ) Compare your result to the graph.
e) What is the end behaviour of
f(x)? Justify this algebraically
and graphically.
Math 100 Chapter 3
82
In the last exercise we saw that, just as for polynomials, rational functions can
have x-intercepts and y-intercepts. They can also have an additional
characteristic, which polynomials lack, namely asymptotes. There are two types of
asymptotes, vertical and horizontal.
The vertical asymptotes of a rational function f(x) 
p(x)
q(x)
are
the solutions (if any) to the equation q(x) = 0
Vertical asymptotes generally occur at the “holes” in the domain, that is, the values
of x for which the function is undefined. The graph of a rational function is
discontinuous at a vertical asymptote. That is, the graph is “broken” at that point.
To draw the graph, you would have to lift your pencil from the paper at the vertical
asymptote. (Polynomials, in contrast, have continuous graphs, with no breaks.)
The end-behaviour of rational functions can often be described with a horizontal
asymptote. While the y-values of polynomials always increase in magnitude without
bound towards  as x gets far away from zero, the y-values of rational functions
often get closer and closer to some finite number.
We can find the horizontal asymptote of a rational function by making use of the
y = c is a horizontal asymptote of a rational function f(x) if f(x)
gets closer and closer to c as x or x-
fact that a polynomial is dominated by its leading term when x is far away from
zero.
Exercise 4
a) If p(x) = 6x3 – 2x2 +x – 3 show that p(100)  6(100)3.
b) If q(x) = -2x3 + 5x – 10 show that q(100)  -2(100)3.
c) Conclude that f(x) 
p(x)
6x 3

 -3 for values of x far away from 0.
q(x)  2x 3
Exercise 5 For each of the following functions, find (when they exist)
Math 100 Chapter 3
83
x-intercepts, y-intercepts, vertical asymptotes and horizontal asymptotes. Match
each function with one of the graphs on page 131 .
1. f(x) 
x 1
x 2  2x  3
Intercepts
x
y
Asymptotes
vertical
horizontal
y
Asymptotes
vertical
horizontal
y
Asymptotes
vertical
horizontal
Matching graph:
2. f(x)  3 
2
x
Intercepts
x
Matching graph:
3. f(x) 
x2  x  2
2x 2  1
Intercepts
x
Matching graph:
4. f(x) 
1
2
x 1
Math 100 Chapter 3
Intercepts
Asymptotes
84
x
y
vertical
horizontal
y
Asymptotes
vertical
horizontal
y
Asymptotes
vertical
horizontal
Matching graph:
5. f(x) 
3x
x 2  5x  4
Intercepts
x
Matching graph:
6. f(x) 
2x  8
x
Intercepts
x
Matching graph:
a)
Math 100 Chapter 3
b)
85
c)
d)
e)
f)
g)
h)
Rational
Math 100 Chapter 3
e)
Inequalities
86
Like polynomial inequalities, rational inequalities can be easily solved if we know
what the graph of the function looks like. When the graph of f(x) is above the xaxis, then f(x) > 0, and when the graph is below the x-axis, then f(x) < 0. Where
the graph crosses the x-axis, f(x) = 0.
Exercise 6 In each case the graph of a function f(x) is given. Solve the given
inequality by studying the graph.
a) f(x)  0
b) f(x) < 0
c) f(x)  0
d) Recall that the graphs of polynomials are continuous (with no breaks) and the
only way that the value of the polynomial function can change from positive to
negative is to cross the x-axis at a zero. Rational functions can also switch from
being positive to negative at a zero. Where else, besides zeros, can the graph of a
rational function switch from positive to negative?
___________________________________
In the following exercises, we solve rational inequalities without the graph.
Exercise 7: Solve the inequality
Math 100 Chapter 3
2
1
by following the steps given:

x 2 x 2
87
a) Get zero on one side of the inequality.
b) Express the non-zero side as a single fraction. Call it f(x).
c) Factor the numerator and denominator of f(x), if possible.
d) Find any x - intercepts (zero's of f(x) )
e) Find any vertical asymptotes ("holes" in the domain of f(x) )
d) Use the numbers from d) and e) to divide the number line into intervals. Pick one test number on
each interval, and evaluate the non-zero side of the inequality at these test numbers.
Interval
Test
Number
Is f(x) positive or negative on this interval?
Is this interval part
of the solution?
e) State the solution:
Exercise 8: Solve the inequality x  3 
3x
. (Remembering the steps is part of knowing how to
x8
solve it.)
Math 100 Chapter 3
88
Solution for Exercise 2: The solution is given in 4 ways. You can give your solution in any of these
ways. Give your solution in whatever ways that help you to understand.
 Interval notation: (- , -4 ]  [ 6, 8 )
 Inequality notation: x  -4 , 6  x < 8.
 In words: the inequality
is true if x is either less than
-4, between 6
and
8, or equal to 6
or 4.
 With a graph:
Objectives and Suggested Exercises from Section 3.5
Math 100 Chapter 3
89
Suggested Exercises are on page 274
Objectives:
1. To find the x-intercepts (also known as zeros, they occur where f(x)=0) and the
y-intercept ( f(0) ) from the formula for a rational function. (# 1, 3)
2. To determine the x and y intercepts and vertical and horizontal asymptotes
from the graph of a rational function (# 5)
3. To determine the vertical and horizontal asymptotes from the formula of
a rational function. (#7, 9, 11, 15)
More Objectives and Suggested Exercises for Section 1.7
Suggested Exercises are on page 79
1.
To solve a rational inequality (#33 to 45 odd)
Math 100 Chapter 3
90