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Unit 3: Polynomial and

Rational Functions

MATH 1001 Pre-Calculus Mathematics

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permission from Cengage Learning:

Swokowski, E. W., & Cole, J. A. (2012). Precalculus: Functions and graphs

(12th ed.). Belmont, CA: Brooks/Cole, Cengage Learning

Course Revision Team (2019) Course Reviser: Saeed Rahmati, PhD

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Table of Contents Unit 3: Polynomial and Rational Functions Introduction ................................................................................................................. U3‐1 Learning Outcomes ..................................................................................................... U3‐1 3.1: Polynomial Functions of Degree Greater Than 2 .......................................... U3‐2 Introduction .............................................................................................................. U3‐2 Section Learning Outcomes .................................................................................... U3‐2 Study Plan ................................................................................................................. U3‐2 Notes on Graphing Polynomials ............................................................................ U3‐3 Sample Questions from Section 3.1 ....................................................................... U3‐4 Further Practice for Section 3.1 ............................................................................... U3‐7

3.2: Properties of Division ......................................................................................... U3‐8 Introduction .............................................................................................................. U3‐8 Section Learning Outcomes .................................................................................... U3‐8 Study Plan ................................................................................................................. U3‐8 Notes on Long Division........................................................................................... U3‐9 Notes on the Remainder and Factor Theorems ................................................... U3‐9 Sample Questions from Section 3.2 ..................................................................... U3‐10 Further Practice Section 3.2 ................................................................................... U3‐11

3.5: Rational Functions ............................................................................................. U3‐12 Introduction ............................................................................................................ U3‐12 Section Learning Outcomes .................................................................................. U3‐12 Study Plan ............................................................................................................... U3‐12 Sample Questions from Section 3.5 ..................................................................... U3‐13 Further Practice Section 3.5 ................................................................................... U3‐16

3.6: Variation .............................................................................................................. U3‐17 Introduction ............................................................................................................ U3‐17 Section Learning Outcomes .................................................................................. U3‐17 Study Plan ............................................................................................................... U3‐17 Sample Questions from Section 3.6 ..................................................................... U3‐18 Further Practice Section 3.6 ................................................................................... U3‐19

MATH 1001: Pre-Calculus Mathematics U3-1

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Unit 3: Polynomial and Rational Functions This unit corresponds to Chapter 3 of the textbook. In Unit 3 you will be covering the following sections in your textbook:

 3.1: Polynomial Functions of Degree Greater Than 2

 3.2: Properties of Division

 3.5: Rational Functions

 3.6: Variation

Introduction The functions obtained by adding multiples of whole number powers of the independent variable are called polynomials (e.g., 4( ) 3 1p x x x   ) and their ratios

are called rational functions (e.g., 2

1( ) 1

tf t t 

  ). The larger class of algebraic

functions includes polynomials and rational functions as well as functions involving radicals. Polynomial functions arise in applications when we are interested in sums and products of related quantities. Examples include business applications where the total revenue is obtained by multiplying unit price times number of items sold, motion problems where distance equals rate multiplied by time and areas and volumes that are obtained by multiplying lengths. Quotients arise, for example, when computing the time needed for a trip from distance and speed.

Learning Outcomes In this unit, you will become familiar with polynomial and rational functions. In particular, you will study their graphs and their use in solving practical problems. Specific learning outcomes are listed at the beginning of each section in the unit.

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3.1: Polynomial Functions of Degree Greater Than 2

Introduction In this section we go beyond the linear and quadratic cases that we studied in Chapter 2 to investigate functions involving higher powers of the variable.

Among the practice problems from Section 2.6 in the textbook, you saw examples in which quadratic functions were used to describe areas—not surprising since the area of a rectangle is obtained by multiplying two lengths. But the volume of a box is the product of three lengths (or the cube of the side length if they are all equal), so volume problems often lead to functions involving the third power—also known as cubic functions. All three cases of linear, quadratic and cubic functions are included in the larger class of polynomials. This class of functions is the topic of the current section.

Section Learning Outcomes After completing this section you should be able to:

 Identify the long run behaviour and y‐intercepts of polynomial graphs.

 Use the factored form to locate all intercepts and to determine where the graph is above the x‐axis and where below and sketch the resulting graph.

 Recognize applied situations involving polynomial functions and use the above skills to solve problems arising from such situations.

Study Plan 1. Read section 3.1 of the textbook. Keep a pencil and paper at hand and check your

understanding by working through the examples as you go.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions Not all of the material presented in each section is required for this course and some topics are more important than others. To master the required material in this section, review the following topics and examples and the supplemental “Notes on Graphing Polynomials” and “Sample Questions from Section 3.1.”

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Section 3.1: Polynomial Functions of Degree Greater than 2

Topic Page, Example Number

Sketching the Graph:

a)   31 2

f x x

b)   31 2

f x x 

Page 184—Example 1

Sketching the Graph:

  3 2 4 4f x x x x    Page 186—Example 3

Sketching the Graph:

  4 3 24 3f x x x x   Pages 187–188—Examples 4, 5

Notes on Graphing Polynomials Perhaps the most important fact about polynomial functions is their behaviour for large values of the argument (independent variable). This behaviour is determined by the highest power term (also known as the “leading term”) because for large enough values of the variable that term will be much larger than all of the rest. For example, for 3 4100x x , the 4x term will be bigger than the 3100x term whenever

100x  . Note that, despite its name, the “leading term” does not always have to be written first, although by convention it usually is.

In general, if the degree of p is even, then the graph of p will have the same behaviour at both ends—that is, either ( ) as p x x     with the graph going up at both ends, or ( ) as p x x   which means that it goes down at both ends.

Even Degree Graphs

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On the other hand, if the degree is odd then the graph will go up at one end and down at the other—that is, either ( ) as p x x   , and ( ) as p x x   , as in the graph on the left below, or

( ) as , and ( ) as p x x p x x      .

Odd Degree Graphs

To see how the graph behaves in the middle region we can start by looking for intercepts.

The y‐intercept is easy. It is just equal to the value of ( )p x when 0x  . So we just need to evaluate (0)p .

For x‐intercepts we need to solve an equation that is not always easy. In fact, unlike the quadratic case, there is no general formula for the roots of a polynomial of degree higher than 4. But as for quadratics, the factored form of p, will help us to find solutions of the equation ( ) 0p x  .

Sample Questions from Section 3.1

Page 190—q. 2

Sketch the graph of   32f x x c   for 2c  

Solution:

  32 2f x x  

Without a graphing calculator, you can graph this function starting from the graph of 3y x and using TRANSFORMATIONS. Hence, the graph of f can be obtained from the graph of 3y x by doing the following:

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1) Stretch this graph by a factor of 2 to get the graph 32y x .

2) Reflect the graph of 32y x over the x

axis to get the graph 32y x  .

3) Shift the graph of 32y x  down 2 units to get the graph of 32 2y x  

NOTE: 3

0 2 0 2 2

0 2 2 0 1

intercept:

x y y y

y x x x

         

         30 intercept: 2 2 0 1y x x x        

Page 191—q. 24

Find all values of x such that   0f x  , and all x such that ( ) 0f x  and sketch the graph of f where

       1 4 2 6 8

f x x x x    

Solution:

  0f x  if 4 0 4x x     , 2 0 2x x    , or

6 0 6x x   

Hence the points:  4,0 ,  2,0 ,  6,0 are x intercepts for the graph of f .

The x‐intercepts divide the x‐axis into 4 intervals and we can use test values and a sign diagram to determine the sign of  f x .

Test the values 5, 0, 3, 7x   .

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1( 5) ( 1)( 7)( 11) ( ) 8

f         1(3) (7)(1)( 3) ( ) 8

f     

1(0) (4)( 2)( 6) ( ) 8

f       1(7) (11)(5)(1) ( ) 8

f    

So ( ) 0f x  when x is in the interval ( , 4) (2, 6)   and ( ) 0f x  when x is in the interval ( 4, 2) (6, )   . When ( ) 0f x  the graph of f is above the x‐axis and when

( ) 0f x  the graph is below the x‐axis.

To find the y  y–intercept of the graph of f find (0).f

1(0) (4)( 2)( 6) 6 8

f       . So the point  0, 6 is the y–intercept.

Now, using all this information, we can sketch the graph of f

Page 191—q. 28 Find all values of x such that   0f x  , and all x such that ( ) 0f x  and sketch the graph of f where

  4 2 4 212 27 or 12 27f x x x y x x       

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Solution:

 f x can be factored into:

          2 2 23 9 3 3 3f x x x x x x           0f x  if 2 3 0x   , or 3 0x   , or 3 0x   and the solutions are

3x   , 3x  , or 3x   .

So the x‐intercepts are the points      3, 0 , 3, 0 , 3, 0 ,  and  3, 0 . If 0x  then (0) 27y f   so the y‐intercept is  0, 27 .

Using test values and a sign diagram as in the previous question we get

and the graph of f is as shown.

Further Practice for Section 3.1 Check your understanding and improve your speed by working through some of the exercises on pages 190–192 of the textbook. Do enough of the odd‐numbered questions of each type to convince yourself that you can get the right answers. Note that the answers are at the back of the textbook and complete worked‐out solutions are in the Student Solutions Manual––but try to avoid looking at answers or solutions until you have made your own best effort.

As a minimum you should do questions 11, 13, 21, 25, 29, and 43 from Section 3.1 and when done, compare your solutions with those in the Student Solutions Manual.

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3.2: Properties of Division

Introduction We’ve seen that factoring can help us understand the behaviour and graphs of polynomial functions. One way to factor numbers is by trying division by potential prime factors and the same is true of polynomials. In this section we study how to divide with polynomials using a pattern very similar to the “long division” we use for numbers. When the division of numbers does not give an exact integer result we are led to consider fractions—or, in other words, rational numbers. Similarly, when the result of dividing two polynomials does not give a polynomial result we call the result a rational function.

Section Learning Outcomes After completing this section you should be able to:

 Use the “Long Division” algorithm to express a ratio of polynomials in the form of a polynomial quotient plus a ratio in which the numerator has lower degree than the denominator.

 Use the Remainder Theorem to determine the remainder on dividing a polynomial p(x) by x—a.

 Use the Factor Theorem to check whether x—a is a factor of p(x).

Study Plan 1. Read section 3.2 of the textbook. Keep a pencil and paper at hand and check your

understanding by working through the examples as you go.

2. Read the following “Study Notes and Sample Questions” to prepare for the practice exercises.

3. Follow the instructions for “Further Practice.”

Study Notes and Sample Questions Not all of the material presented in each section is required for this course and some topics are more important than others. To master the required material in this section, review the following topics and examples from Section 3.2 and the supplemental “Notes on Long Division,” “Notes on the Remainder and Factor Theorems,” and “Sample Questions from Section 3.2.”

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Section 3.2: Properties of Division

Topic Page, Example Number

Long Division for Polynomials Page 194

Division Algorithm for Polynomials

Page 194

Remainder Theorem Page 195

Factor Theorem Page 196—Examples 2, 3

Notes on Long Division You definitely will need to be able to carry out the Long Division algorithm, so read pages 194–195 of the textbook carefully.

The statement ( ) ( ) ( ) ( )f x p x q x r x   in the blue box on page 194 can be written

also as ( ) ( )( ) ( ) ( )

f x r xq x p x p x

  . So the result of the division of f(x) by p(x) is to express

the ratio as the sum of a polynomial q(x) (called the “quotient”) and another ratio in which the numerator r(x) is of lower degree than the denominator p(x).

Notes on the Remainder and Factor Theorems Note that the Remainder Theorem can be used in two ways. One way, as in Example 1 on page 195 of the textbook is to evaluate ( )f a by using the remainder after long division of ( )f x by ( )x a . The other would be to find the remainder without actually doing the division. For example, if we want to find the remainder on dividing 5( )f x x x  by x—2, we need only to evaluate 5(2) 2 2 32 2 30f      which is much quicker than actually going through the whole long division process.

In particular, if ( ) 0f a  , then the remainder on dividing ( )f x by ( )x a is 0, so ( )x a is a factor of ( )f x . This is the Factor Theorem.

A polynomial of degree n has at most n real roots and a consequence of the Factor Theorem is that if a polynomial of degree n has n specified roots, for example

1 2, , , nx r r r  , the polynomial must be of the form 1 2( ) ( )( ) ( )nf x a x r x r x r    for some constant a.

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Sample Questions from Section 3.2

Page 199—q. 8

Find the quotient and remainder if  f x is divided by  p x :

where   27 3 10f x x x   and   2 10p x x x  

Solution:

Use the long division to find the quotient and remainder.

2 2

7 10 7 3 10

(7 7 70) 10 80

x x x x x x

   

   

The quotient is 7 and the remainder is 10 80x  .

Page 199—q. 12

Use the remainder theorem to find  f c :

  4 23 12 2f x x x c    

Solution:

( 2)f  will be the value of the remainder when ( )f x is divided by

( ( 2)) ( 2)x x    3 2

4 3 2

4 3

3 2

2 7 14 2 0 3 0 12

( 2 )

2 3 ( 2 4 )

7 0 (7 14 )

14 12 ( 14 28)

x x x x x x x x

x x

x x x x

x x

       

 

  



 

    

So according to the remainder theorem ( 2) 16f   .

It is easy to check: 4 2( 2) ( 2) 3( 2) 12 16f       

MATH 1001: Pre-Calculus Mathematics U3-11

Page 199—q. 20

Find a polynomial ( )f x with leading coefficient 1 and having the degree 3 and zeros -2, 2, 3.

Solution:

The factors are ( ( 2)), ( 2),x x −−− and ( 3)x − and 1a = .

( ) ( ) ( ) ( ) ( ) ( )2 3 22 2 3 4 3 3 4 12f x x x x x x x x x= + − − = − − = − − +

Page 200—q. 48

Show that x c− is not a factor of ( )f x for any real number ( ) 4 23 2f x x x= − − −

Solution:

x c− is a factor of f if ( ) 0f c = . 4 2

4 2 ( ) 0 3 2 0

3 2 0 f c c c

c c = ⇒ − − − =

+ + =

Let 2z c= then we have a quadratic equation in z: 2 3 2 0z z+ + = which has the solutions 1, 2z = − − . But there are no solutions to 2 1c = − or 2 2c = − .

Hence ( )f c is never equal to zero. So, x c− is not a factor of ( )f x .

Further Practice Section 3.2 Check your understanding and improve your speed by working through some of the exercises on Page 199 of the textbook.

Do enough of the odd-numbered questions of each type to convince yourself that you can get the right answers. Note that the answers are at the back of the textbook and complete worked-out solutions are in the Student Solutions Manual––but try to avoid looking at answers or solutions until you have made your own best effort.

As a minimum you should do questions 1, 3, 7, 9, 11, 15, and 19 from Section 3.2. When done, compare your solutions with those in the Student Solutions Manual.

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3.5: Rational Functions

Introduction The ratios of polynomials are known as rational functions. In this section we shall investigate the graphs of such functions and look at some further applications.

Section Learning Outcomes After completing this section you should be able to:

 Find the y  intercepts for the graphs of rational functions.

 Find the horizontal and vertical asymptotes for the graphs of rational functions.

 Identify the long‐run behaviours of rational functions.

 Identify the behaviours of rational functions near their vertical asymptotes.

 Use the above skills to sketch the graphs of rational functions.

 Produce a possible formula for a rational function for which the graph is given or described.

 Solve various applied problems in which quantities are related by rational functions.

Study Plan 1. Read section 3.5 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

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Section 3.5: Rational Functions

Topic Page, Example Number

Division of a Rational Function Page 221—Illustration

Holes Pages 227—Example 4

Definition of Vertical Asymptote Page 222

Definition of Horizontal Asymptote

Pages 223, 224—Example 1

Theorem of Horizontal Asymptote

Page 224—Example 2

Finding an Equation of a Rational Function Satisfying Prescribed Conditions

Page 227

Sketching Graph of Rational Function

Page 225—Examples 3–9

Sample Questions from Section 3.5

Page 234—q. 16

Sketch the graph of 5 3( ) 3 7

xf x x 

  using the following:

(a) Find the domain of f .

(b) Find the xand y  intercepts.

(d) Find the horizontal asymptote(s).

(e) Sketch the graph of f .

Solution:

(a) Domain:   73 7 0 3f f

D x x D x x         

i.e. the domain of f consists of all real numbers except for 7 3 .

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(b) x intercept: 5 3 30 5 3 0 3 7 5 x x x x 

       

y  intercept: 5(0) 3 3(0) 3(0) 7 7

f    

So the intercepts are 3 , 0 5

    

and 30, 7

    

(c) Vertical Asymptote: The vertical asymptotes (VA) are the vertical lines x k where k is a root of the denominator in the reduced form of  f x . So the line

7 3

x  is the only VA for the graph of f.

To determine the behaviour of f near the VA, calculate the one‐sided limits:

as 7 3

x 

     

then   5 3 3 7 xf x x 

   

as 7 3

x 

     

then   5 3 3 7 xf x x 

   

(d) Horizontal Asymptote(s): The horizontal asymptotes (HA) are the horizontal lines y c where c is the value that ( )f x approaches as x becomes very large in the (+) or (‐) direction.

As x  , 5 3 5( ) 3 7 3 xf x x 

  

and so the line 5 3

y  is the HA for the

graph of f.

(e) Graph:

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Page 235—q. 48

Simplify   3 22 4 8

2 x x x

f x x

   

 and sketch the graph of f using the following:

(a) Find the domain of f .

(b) Find the xand y  intercepts.

(d) Find the horizontal asymptote(s).

(e) Sketch the graph of f .

Solution:

(a)

  3 22 4 8

2 x x xf x

x   

 

factor the numerator by grouping

      2 2 4 2

2 x x x

f x x

   



     2

24 2 4 2 2

for x x

f x x x x

     



Domain:  2fD x x 

(b) x‐intercept:   

     2

24 2( ) 0 0 4 2 0 2 2

x x f x x x x

x  

          

But 2x  , so the only valid solution is 2x   .

intercepty  : ( 4)( 2)(0) 4 2

f     

So the intercepts are ( 2, 0) and (0, 4) .

(c) Because the factor ( 2)x  “divides out” in the reduced version of f , there is a “hole” in the graph at 2x  instead of a VA and so there are no vertical asymptotes for the graph.

(d) As 2, ( )x f x x   and so ( )f x  . So there is no horizontal asymptote.

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(e) Graph of f :

Further Practice Section 3.5 Check your understanding and improve your speed by working through some of the exercises on pages 234–235 of the textbook. Do enough of the odd‐numbered questions of each type to convince yourself that you can get the right answers. Note that the answers are at the back of the textbook and complete worked‐out solutions are in the Student Solutions Manual––but try to avoid looking at answers or solutions until you have made your own best effort.)

As a minimum you should do questions 1, 3, 5 a, 5 b, 7, 9, 15, 19, 45, 51, and 53 from Section 3.5. When done, compare your solutions with those in the Student Solutions Manual.

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3.6: Variation

Introduction Certain relationships can be described by showing how one quantity changes in relation to other quantities. For example, the relationship between the area of a circle and its radius is given by the formula 2A r . So as the radius, r , changes, so does the area, A . The idea of “change” is often referred to as variation and in this section we shall construct variation formulas to model practical problems.

Section Learning Outcomes After completing this section you should be able to:

 Determine the general formula that involves direct, inverse and joint variation involving several variables and a constant.

 Determine the value of the proportionality constant in a given variational relation.

 Use the above skills to solve practical applications.

Study Plan 1. Read section 3.6 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

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Section 3.6: Variation

Topic Page, Example Number

Direct and Inverse Variation Page 237

Directly Proportional Variables Page 237—Example 1

Guidelines for Solving Direct Variation Problems

Page 238—Example 2

Joint Variation Page 240—Examples 3, 4

Sample Questions from Section 3.6

Page 241—Q12, 16 Express the statement as a formula that involves the given variables and a constant of proportionality k , and then determine the value of k from the given condition:

q. 12 r is directly proportional to the product of s and v and inversely proportional to the cube of p . If 2, 3s v= = , and 5p = , then 40r = .

Solution:

(2)(3) 250040 125 3

svr k k k p

= ⇒ = ⇒ =

So 3 2500

3 svr

p =

q. 16 y is directly proportional to the square of x and inversely proportional to the square root of z . If 5x = and 16z = then 10y = .

Solution: 2 (25) 810

4 5 xy k k k z

= ⇒ = ⇒ = So 28

5 xy

z =

Page 241—q. 18 Hooke’s Law

Hooke’s Law states that the force F required to stretch a spring x units beyond its natural length is directly proportional to x .

(a) Express F as a function of x by means of a formula that involves a constant of proportionality k

MATH 1001: Pre-Calculus Mathematics U3-19

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(b) A weight of 4 pounds stretches a certain spring from its natural length of 10 inches to a length of 10.3 inches. Find the value of k in part a)

(d) Sketch the graph of a relationship between F and x for 0x 

Solution:

(a) F kx

(b) 404 (10.3 10) 3

k k   

F    . The force is 20 lb.

(d) 40 3

F x is a simple linear relationship whose graph is a line with slope

40 3

m  and y  intercept 0b  .

Further Practice Section 3.6 Check your understanding and improve your speed by working through some of the exercises on pages 241–243 of the textbook. Do enough of the odd‐numbered questions of each type to convince yourself that you can get the right answers. Note that the answers are at the back of the textbook and complete worked‐out solutions are in the Student Solutions Manual––but try to avoid looking at answers or solutions until you have made your own best effort.

As a minimum you should do questions 5, 7, 9, 11, and 23. When done, compare your solutions with those in the Student Solutions Manual.