Chapter 5: Polynomial and Rational Functions Section 5.1 ......Graphing Polynomial Functions Using Transformations In Section 3.5 we learned the methods of graph transformations. We

128 Chapter 5 Polynomial and Rational Functions

Copyright © 2016 Pearson Education, Inc.

Chapter 5: Polynomial and Rational Functions Section 5.1: Polynomial Functions and Models

Definition: A Polynomial Function in one variable is a function of the form

_____________________________________________

1 1 0, , , , n na a a a− …where are constants, called the _________________ of the polynomial,

0n ≥ is an ______________, and x 0,na ≠is the ______________. If it is called the

____________________, and n is the ______________ of the polynomial. The domain of a polynomial function is ____________________________.

• The leading term is ________

• The term _____ is called the constant term.

A polynomial is in standard form if the terms are listed in _______________________. The graph of a polynomial is a smooth unbroken curve (by “smooth” we mean that the graph does not have any sharp corners as turning points).

Example 1*: Identify Polynomial Functions and Their Degree Determine which of the following are polynomial functions. For those that are, state the degree. For those that are not, explain why not. Write each polynomial in standard form, and then identify the leading term and constant term.

2(a) ( ) 3 5 10f x x x= + − 2

3(b) ( ) 3 5H x x= −

3(c) ( ) 2 3 8f x x x−= + − (d) G( ) 5x = −

2(e) ( ) 3 (2 1)g x s s= −

Section 5.1 Polynomial Functions and Models 129


Definition: A Power function of degree n is a monomial function in the form _________ where a is a real number, 0a ≠ and 0n > is an integer. Notice that we could also define a polynomial as a sum of power functions.

Exploration 1: The Graphs of Power Functions Use a graphing utility to label each graph. Then answer the questions below: Label the graphs of 2 4, ,y x y x= = Label the graphs of 3 5, ,y x y x= =

and 6y x= on the curves below. and 7y x= on the curves below.

Compare the graphs of even powered functions.

• Do they share the same shape?

• Do they share any points?

• Are they symmetric to the y-axis, x –axis, or the origin?

• What is the domain of even powered functions?

• What is the range of even powered functions?

• What happens to the graph as n (the exponent) increases and → ±∞ verses → 0?

Compare the graphs of odd powered functions.

• Do they share the same shape?

• Do they share any points?

• Are they symmetric to the y-axis, x –axis, or the origin?

• What is the domain of odd powered functions?

• What is the range of odd powered functions?

• What happens to the graph as n (the exponent) increases and → ±∞ verses → 0?



Graphing Polynomial Functions Using Transformations In Section 3.5 we learned the methods of graph transformations. We can use these methods along with what we’ve just learned about power functions to graph polynomial functions.

Example 2*: Graph Polynomial Functions Using Transformations 1st: Use words to describe the graphs of the functions below as a transformation of the power functions 4 5 or y x y x= = .

( ) 4(a)* 2( 1)f x x= − + ( ) ( )51(b) 2 1

2g x x= − +

2nd: Graph each of the following polynomials using graph transformations of their corresponding power function.

( ) 4(a)* 2( 1)f x x= − + ( ) ( )51(b) 2 1

2g x x= − +



The graph to the right shows a polynomial function with four x-intercepts. At the x-intercept, the graph must either _______ the x-axis or _______ the x-axis. Consequently, between consecutive x-intercepts the graph is either _______ the x-axis or ________ the x-axis. If a polynomial function f is factored completely, it is easy to locate the x – intercepts of the graph by solving the equation ( ) 0f x = using the

_______________ property.

For example, if ( ) ( )( )21 5f x x x= − + , we can see that the zeros of the function are x = 1

and x = -5. Based on this, the following statements are equivalent, where r is the point that crosses or touches the x-axis:

1. r is a ____________ of a polynomial function f 2. r is an ___________ of the graph of f. 3. _____ is a factor of f. 4. r is a solution to the equation ______.

Example 3: The Zeros of a Polynomial Function Find the zeros of each of the following polynomials.

(a) ( ) 2 2 3f x x x= + −

( ) ( ) ( )2 3(b) 5 1 2f x x x= + −



Example 4*: The Zeros of a Polynomial Function Find a polynomial of degree 3 whose zeros are 4,− 2,− and 3. Use a graphing utility to verify your results.

Definition: If ____________ is a factor of a polynomial of f and ___________ is not a factor of f, then r is called a zero of multiplicity m of f. In other words, if r is a zero of a polynomial and the exponent of the factor that produced the root is m then we say that r has multiplicity m.

1− ( )21x +Notice in Example 3b the zero was produced by the factor and the zero 2 was

( )32x −produced by the factor . Since the exponents of these terms are 2 and 3 respectively,

1−we would say that has multiplicity 2, and that the zero 2 has multiplicity 3.

Example 5*: Multiplicity, Turning Points, and End Behavior For the polynomial 3 2( ) ( 3) ( 2)f x x x x= − − + (a) Find the x – and y – intercepts of the graph of f.

(b) Using a graphing utility, graph the polynomial.

(c) For each x – intercept, determine whether it is of odd or even multiplicity.

Example 6: Graphing a Polynomial Using Its x – Intercepts

2( ) ( 2) ( 1)f x x x= − +Consider the following polynomial: (a) Find the x- and y – intercepts of the graph of f.

(b) Use the x – intercepts to find the intervals on which the graph is above the x – axis and the intervals on which the graph of f is below the x – axis.



(c) Locate other points on the graph, and connect all the points plotted with a smooth, continuous curve.

MultiplicityExploration 2: Using the graph in Example 5b record below what the difference in even and odd multiplicities produces in the graph at the zeros.

Fact: If r is a zero of even multiplicity,

( )f xNumerically: The sign of _____________ from one side to the other of r. Graphically: The graph of f _________________ the x – axis at r.

( )f xNumerically: The sign of _____________ from one side to the other of r. Graphically: The graph of f _________________ the x – axis at r.

Exploration 3: Turning Points

1. Graph 3 3 3 21 2 3, - , and 3 4Y x Y x x Y x x= = = + + .

(a) How many turning points do you see for each graph? (b) How does the number of turning points relate to the degree?

2. Graph 4 4 3 4 21 2 3

4, , and 2

3Y x Y x x Y x x= = − = − .

(a) How many turning points do you see for each graph? (b) How does the number of turning points relate to the degree?



Turning Points Theorem: A turning point on a graph is a point where the function changes from increasing to decreasing or vice versa. These points are the local maximums or local minimums of the function. An example of a turning point is the vertex of a parabola. In the parabola, 2y x= , the degree of the polynomial is _____, and there is _____ turning point.

In general:

If is a polynomial function of degree , then the graph of has at most _____ turning points. If the graph of a polynomial function has –1 n turning points, the degree of is at least_____.

Example 7: Turning Points State the degree and number of turning points for ( ) ( )( )2 2 1f x x x= − − + .

End Behavior Theorem: x, either positive or negative, the graph of a For large values ofpolynomial function will resemble the graph of its leading term when the polynomial is written in standard form. Note that this leading term will be a power function. In Calculus, we’d say, “the value of becomes larger or smaller as becomes larger or smaller,” and in the case of ( ) 3f x x= , we’d write ( ) f x as x→ −∞ → −∞ and ( ) f x as x→ ∞ → ∞ . In

addition, another notation involves using limits to convey this idea, i.e. lim ( )x

f x→∞

= ∞ or

lim ( )x

f x→−∞

= −∞ .

Example 8: Find the Zeros of a Polynomial and State its End Behavior (a) For the polynomial ( ) ( )( )( ) ( )2

5 1 2 4f x x x x x= + + − − , list all zeros and multiplicity.

(b) Discuss the general appearance of the graph of the function in (a) using the idea of “end

behavior.”



Example 9: Write a Polynomial Function from its Graph Write a polynomial function whose graph is shown (use the smallest degree possible). Verify your results with a graphing utility.

Example 10: Write a Polynomial Function from its Graph Write a polynomial function whose graph is shown (use the smallest degree possible). Verify your results with a graphing utility.



Steps to Analyze the Graph of a Polynomial Function

Step 1: Determine the ____________________ of the graph of the function.

Step 2: Find the __________________ of the graph of the function.

Step 3: Determine the ________ of the function and their multiplicity. Use this information to determine whether the graph __________or __________ the x – axis at each x – intercept.

Step 4: Determine the maximum number of ___________ on the graph of the function.

Step 5: Use the information in Steps 1 – 4 to draw a complete graph of the function.

Example 11*: Analyze the Graph of a Polynomial Function Analyze the polynomial function ( ) ( ) ( )( )2

4 1 2f x x x x= + + − .

Example 12: Analyze the Graph of a Polynomial Function Analyze the polynomial function ( ) ( )( )2 3 4g x x x x= − − + .



Using a Graphing Utility to Analyze the Graph of a Polynomial Function

Step 1: Determine the ____________________ of the graph of the function.

Step 2: Graph the function using a graphing utility.

Step 3: Use a graphing utility to approximate the __________________________.

Step 4: Use a graphing utility to create a TABLE to find points on the graph around each ____________

Step 5: Approximate the _________________ on the graph.

Step 6: Use the information in Steps 1 through 5 to draw a complete graph of the function by hand.

Step 7: Find the ____________ and ___________ of the function.

Step 8: Use the graph to determine where the function is _____________ and where it is ___________.

Example 13: Use a Graphing Utility to Analyze the Graph of a Polynomial Function. Analyze the graph of the polynomial function 3 2( ) 1.23 4.124 2.321f x x x x= − + −



In Chapter 4, we learned how to find the line of best fit and quadratic function of best fit. We can also find the cubic function of best fit. Below are two scatterplots of data that follow a cubic relation.

Example 14: Build Cubic Models from Data Use the table to answer the following questions. (a) Draw a scatter diagram of the data using x as the

independent variable and y as the dependent variable. Comment on the type of relation that may exist between the two variables x and y.

(b) Using a graphing utility, find the cubic function of best fit ( )y f x= that models the

relation between x and y.

(c) Graph the cubic function of best fit on your scatter diagram.

(d) Use the function found in part (b) to predict y if 13x = .

x y 0 3.5 1 5.6 2 4 3 3 4 2.5 5 5.12 6 12.5 7 20 8 43 9 67

Section 5.2 Properties of Rational Functions 139


Chapter 5: Polynomial and Rational Functions Section 5.2: Properties of Rational Functions

Definition: A rational function is a function of the form ( ) R x = where p and q are

polynomial functions and q is not the ___________________. The domain of a rational function is _______________________ except those for which the denominator q is ____.

WARNING: The domain of a rational function must be found before writing the function in its lowest terms.

Example 1*: Find the Domain of a Rational Function Find the domain of the following rational functions

( ) ( )

( ) ( ) ( )

2

2

2 2

2

4 6(a)* (b)*

4 8 12

5 9 4(c)* (d) (e)

2 3 2

x xR x R xx x x

x x xR x R x R xx x

− += =+ + +

− − −= = =+ +

Example 2: Graph the following functions

1yx

= 2

1yx

=



In Section 3.4 we created our library of functions and in Section 3.5 we transformed those basic functions. We can do the same types of transformations with the basic rational

functions 1yx

= and 2

1yx

= .

Example 3*: Graph Transformations of a Rational Function

Graph the rational function ( )( )2

12

3R x

x= +

− using transformations.

Long-run behavior: As we learned in Section 51, the long-run behavior of a function concerns what happens as the inputs get extreme ( → ±∞). As we saw before with polynomials, the leading terms of ( ) and ( ) determine the long run behavior of ( ).r x

Exploration 1: Long-Run Behavior and Vertical Asymptotes

1. Use a graphing utility to graph ( ) 3 1

1

xf xx

+=+

.

(a) What is happening to the function’s values as the x values move towards positive infinity? How can you use mathematical notation to write this?

(b) What is happening to the function’s values as the x values move towards negative infinity? How can you use mathematical notation to write this?

(c) Look at the equation for ( ). Can you explain why the function’s values get

closer and closer to the values found above as the x values approach positive and negative infinity?



Horizontal and Vertical Asymptotes Let R denote a function. If, as _________ or as ________, the values of ( )R x approach some fixed number L, then the line _______ is a horizontal asymptote of the graph of R. [Refer to (a) and (b) below]. If, as x approaches some number c, the values ________________________, then the line __________ is a vertical asymptote of the graph of R. The graph of R never intersects a vertical asymptote. [Refer to (c) and (d)].

Can the graph of a function intersect a horizontal asymptote?

(d) What is the equation for the horizontal asymptote of ( )f x ?

(e) What is/are the equation(s) of the vertical asymptote(s) of ( )f x ?

2. Use a graphing utility to graph ( )2

3

4

1

xh xx

=−

.

(a) What is happening to the function’s values as the x values move towards positive infinity? How can you use mathematical notation to write this?

(b) What is happening to the function’s values as the x values move towards negative infinity? How can you use mathematical notation to write this?

(c) What is the equation for the horizontal asymptote of ( )h x ?



(d) Look at the equation for ( )h x . Can you explain why the function’s values get closer and closer to the values found above as the x values approach positive and negative infinity?

(e) What is/are the equation(s) of the vertical asymptote(s) of ( )h x ?

3. Use a graphing utility to graph the functions below and fill in the table. Function Leading

Term in the Numerator

Leading Term in the Denominator

Equation of the Horizontal Asymptote

Equation(s) of the Vertical Asymptote(s)

( ) 3 1

1

xf xx

+=+

( ) 2

2

2 2

xq xx

+=−

( )2

2

2 3 2

3 2

x xs xx x

+ −=+ +

( )2

3

3 2 1

1

x xg xx

− − +=−

4. Some patterns/relationships exist in the table above that will help us determine the

horizontal and vertical asymptotes of a rational function. What patterns do you notice?

Locating Vertical Asymptotes Theorem: A rational function( )

( )( )

p xR xq x

= , in lowest

terms, will have a vertical asymptote at ______ if r is a ___________________ of the denominator q.

What do we mean by lowest terms in the theorem above?

Multiplicity and Vertical Asymptotes: If the multiplicity of the zero that gives rise to the vertical asymptote is odd, the graph approaches +∞ on one side of the asymptote and −∞ on the other side. If the multiplicity is even, the graph approaches either +∞ or −∞ on both sides of the vertical asymptote. We will use this for graphing rational functions in the next section.



Example 4*: Find Vertical Asymptotes of a Rational Function Find the vertical asymptotes, if any, of the graph of each rational function.

( )25

3

xR xx

=+

( ) ( )( )3

2 2

xH xx x

−=+ −

( ) 2

1

5 4

xF xx x

−=+ +

( )2

2

3 2

4

x xG xx+ +=

−

Finding a Horizontal Asymptote or Oblique Asymptote of a Rational Function

Consider the rational function 1

1 1 01

1 1 0

...( )( )

( ) ...

n nn n

m mm m

a x a x a x ap xR xq x b x b x b x b

−−

−−

+ + + += =+ + + +

in which the

degree of the numerator is n and the degree of the denominator is m. 1. If n m< , the line ____________ is a horizontal asymptote. 2. If n m= , the line ____________ is a horizontal asymptote. (That is, the horizontal

asymptote is y equals “the ratio of the leading coefficients.”) 3. If 1n m= + , the line ____________ is an oblique asymptote (sometimes called a

slant asymptote since it is a slanted line), which is the quotient found using long division.

4. If 2n m≥ + , there are ___________________________. The end behavior of the

graph will resemble the power function n mn

m

ay xb

−= .

Note: a rational function will never have both a horizontal asymptote and an oblique asymptote, but may have neither.

Example 5: Find Horizontal Asymptotes of a Rational Function Create a rational function that satisfies the following, and then compare your examples with your classmates. (a) Has a horizontal asymptote at 5y = .

(b) Has a horizontal asymptote at 0y = .

(c) Does not have a horizontal asymptote.



Example 6*: Find Horizontal or Oblique Asymptotes of a Rational Function Find the horizontal asymptote, if any, of each of the following graphs:

( ) 2

3

2 5

xR xx x

+=+ +

2

2

2 7 1( )

2

x xR xx+ −=

+

6 3

6 5 4 3

5 4 3( )

2 5 8 7 2

x xR xx x x x

− +=− + − +

2 4 1

( )2

x xR xx+ +=

−

Example 7*: Find Horizontal or Oblique Asymptotes of a Rational Function

Find the oblique asymptote of 2 4 1

( )2

x xR xx+ +=

−.

Example 8: Find Horizontal or Oblique Asymptotes of a Rational Function

Find the horizontal or oblique asymptote, if any, of 22 3 2

( )1

x xR xx− +=

−.

Section 5.3 The Graph of a Rational Function 145


Chapter 5: Polynomial and Rational Functions Section 5.3: The Graph of a Rational Function

Steps for Analyzing the Graph of a Rational Function

Step 1: Factor the __________________________ of R. Find the _____________.

Step 2: Write R in ____________ (cancel out anything, if you can). If you can cancel out any terms these are the holes of the function.

Step 3: Locate the ____________ of the graph. Determine the behavior of the graph of R at each x-intercept, based on multiplicity.

Step 4: Determine the ___________________ based on the factors of the denominator of R in lowest terms. Graph each vertical asymptote using a dashed line.

Determine the behavior of the graph of R on either side of the vertical asymptote based on the following:

• If the multiplicity of a factor is odd, the graph will approach __________ infinities at the vertical asymptote.

• If the multiplicity of a factor is even, the graph will approach the ________ infinity at the vertical asymptote .

Example of opposite infinities: Example of same infinities:

Step 5: Determine the __________________ or _________________, if one exists. Determine points, if any, at which the graph of R ____________ this asymptote. Graph the asymptote using a dashed line. Plot any points at which the graph of R intersects the asymptote.

Step 6: Use the ______________ of the numerator and denominator of R to divide the x – axis into intervals. Determine where the graph of R is __________ or ___________ the x – axis by choosing a number in each interval and evaluation R there. Plot the points found.

Step 7: Use the results in Steps 1 through 6 to graph R.



Example 1*: Analyze the Graph of a Rational Function

Analyze the graph of the rational function 2

2

4( )

3 4

xR xx x

−=+ −

.

Example 2*: Analyze the Graph of a Rational Function with a Hole

Analyze the graph of the rational function 2

2

9( )

9 18

xR xx x

−=+ +

.



Example 3: Analyze the Graph of a Rational Function

Analyze the graph of the rational function 2 3 2

( )x xR x

x+ += .


Analyze the graph of the rational function ( )2

2

12

1

x xR xx− +=

−.




Analyze the graph of the rational function ( )2

2

12

1

x xR xx− +=

−.


Analyze the graph of the rational function ( ) ( ) ( )2

2

1 2

( 4)

x xR x

x x− +

=−

.



Example 7: Constructing a Rational Function from its Graph Find a formula for the rational function graphed below.

Example 8: Solve Applied Problems Involving Rational Functions The concentration C of a certain drug in a patient’s blood stream t minutes after injection is

given by 2

2(1 20 )( )

30

tC tt

+=+

.

(a) Find the horizontal asymptote of ( )C t . Interpret this horizontal asymptote in the context

of the problem.

(b) Using your graphing utility, graph ( )C C t= .

(c) Determine the time at which the concentration is highest.



Chapter 5: Polynomial and Rational Functions Section 5.4: Polynomial and Rational Inequalities

There are two ways we can solve polynomial and rational inequalities: graphically and algebraically. Let’s use our knowledge from solving quadratic inequalities in Chapter 2 as well as the information we’ve learned in the previous sections of Chapter 3 to solve these types of inequalities. First, let’s make sure we understand what certain notation means. Exploration 1: Function Inequalities What does ( ) 0f x > mean? What would these solutions look like on the graph of ( )f x ? What does ( ) 0f x ≥ mean? What would these solutions look like on the graph of ( )f x and how do they differ from the solutions to ( ) 0f x > ? What does ( ) 0f x < mean? What would these solutions look like on the graph of ( )f x ? What does ( ) 0f x ≤ mean? What would these solutions look like on the graph of ( )f x and how do they differ from the solutions to ( ) 0f x < ?

Example 1*: Solve a Polynomial Inequality Using a Graph Solve 2( 3) ( 1)( 4) 0x x x+ − − ≤ graphically.

Section 5.4 Polynomial and Rational Inequalities 151


Example 2*: Solve a Rational Inequality Using a Graph

Solve 4

22

xx

− ≥+

graphically.

Steps for Solving Polynomial and Rational Inequalities Algebraically

Step 1: Write the inequality so that a polynomial or rational expression is on the ________ side and the zero is on the right side in one of the following forms:

_________ _________ _________ _________ For rational expressions, be sure the left side is written as a _____________________ then

find the domain of f. Step 2: Determine the real numbers at which the expression f ___________, and if the

expression is rational, the real numbers at which the expression f is ____________. Step 3: Use the numbers found in Step 2 to separate the real number line into intervals. Step 4: Select a number in each interval and evaluate f at the number.

(a) If the value of f is positive, then __________________ for all numbers x in that interval.

(b) If the value of f is negative, then __________________ for all numbers x in that interval.

Now that we have steps for solving polynomial and rational inequalities algebraically, let’s go back and solve the inequalities in Examples 1 and 2 using algebra. There is space for you to do this on the following page.



Example 3*: Solve a Polynomial Inequality Algebraically Solve 2( 3) ( 1)( 4) 0x x x+ − − ≤ algebraically. Example 4*: Solve a Rational Inequality Algebraically

Solve 4

22

xx

− ≥+

algebraically.

Section 5.5 The Real Zeros of a Polynomial Function 153


Chapter 5: Polynomial and Rational Functions Section 5.5: The Real Zeros of a Polynomial Function

Exploration 1: Remainder Theorem Perform the following operations and state the quotient, divisor, and remainder. Then rewrite the dividend as a sum: For example, 14 4 3 R2÷ = since:

34 14

12

2

−

The divisor is 4, 3 is the quotient, and 2 is the remainder. We can check our division by rewriting our dividend as a sum: 14 4 3 2= × + . You will always have dividend = divisor quotient + remainder× if you’ve performed your division correctly. (a) 8 3÷ (b) 22 5÷

Just like numbers, we can also divide polynomials. Look at the following polynomial division problems, where each polynomial, ( )f x is divided by a linear function of the form,

( )g x x c= − . Write down any pattern(s) you see between the divisor, remainder, and ( )f c .

2( ) 2 4

( ) 1

f x x xg x x

= + −= −

2( ) 3 4

( ) 4

f x x xg x x

= − −= −

2( ) 2 3 1

( ) 2

f x x xg x x

= − += +

2

2

3 1 2 4

( )

3 4

(3 3)

1

xx x x

x x

xx

+− + −

− −

−− −

−

2

2

1 4 3 4

( 4 )

4

( 4)

0

xx x x

x x

xx

+− − −

− −

−− −

2

2

2 7 2 2 3 1

(2 4 )

7 1

( 7 14)

15

xx x x

x x

xx

−+ − +

− +

− +− − −

2(1) 1 2(1) 4 1f = + − = − (4) 0f = ( 2) 15f − =



Remainder Theorem: Let f be a polynomial function. If ( )f x is divided by x c− , then the remainder is _______.

Example 1*: Use the Remainder and Factor Theorems Find the remainder if 3 2( ) 3 2 1f x x x x= + + − is divided by (a) 2x + (b) 1x −

Look back at the second polynomial division problem in the Exploration 1. Why was the remainder 0 and (4) 0f = ?

Factor Theorem: Let f be a polynomial function. Then x c− is a factor of ( )f x if and only if ___________________. The Factor Theorem actually consists of two separate statements:

1. If ( ) 0f c = , then __________________________.

2. If x c− is a factor of ( )f x , then ______________.

Example 2*: Use the Remainder and Factor Theorems Use the factor theorem to determine whether the function 3 2( ) 2 4 3f x x x x= − − + + has the factor: (a) 1x + (b) 1x −

Fact: A polynomial function cannot have more real zeros than its degree.



We can determine the number of positive and negative real zeros of a polynomial function ( )f x without even having to solve ( ) 0f x = ! Descartes, a famous mathematician, came up

with this theorem cleverly named Descartes’ Rule of Signs. It is formally defined as follows: Descartes’ Rule of Signs Theorem: Let f denote a polynomial function written in standard form. The number of positive real zeros of f either equals the number of variations of the sign of the nonzero coefficients of ____ or else equals that number less an even integer. The number of negative real zeros of f either equals the number of variations of the sign of the nonzero coefficients of ____ or else equals that number less an even integer.

Example 3*: Use Descartes’ Rule of Signs Discuss the real zeros of ( ) 3 22 5 6f x x x x= + − − .

Rational Zeros Theorem: Let f be a polynomial function of degree 1 or higher of the form

1 11 1 0 0( ) ... , 0, 0n n

n n nf x a x a x a x a a a−−= ≠+ + + ≠+

where each coefficient is an _____________. If pq

, in lowest terms, is a rational zero of f,

then p must be a factor of _______, and q must be a factor of _______. Example 4*: List the Potential Rational Zeros of a Polynomial Function List the potential rational zeros of ( ) 3 23 8 7 12f x x x x= + − − .



Steps for Finding the Real Zeros of a Polynomial Function Step 1: Use the degree of the polynomial to determine the ________________________.

Step 2: Use Descartes’ Rule of Signs to determine the possible number of

________________________________.

Step 3:

(a) If the polynomial has integer coefficients, use the _________________ to

identify those rational numbers that potentially could be zeros.

(b) Use substitution, synthetic division, or long division to test each potential

rational zero. Each time that a zero (and thus a factor) is found, repeat Step 3 on

the _____________.

In attempting to find zeros, remember to use (if possible) the factoring techniques that you already know (special products, factoring by grouping, and so on).

Example 5*: Find the Real Zeros of a Polynomial Function Find the real zeros of ( ) 3 23 8 7 12f x x x x= + − − (note, we already found the potential

rational zeros of ( )f x in Example 4).



Example 6*: Find the Real Zeros of a Polynomial Function Find the real zeros of ( ) 4 3 22 13 29 27 9f x x x x x= + + + + . Note: This is the same thing as

solving the equation 4 3 22 13 29 27 9 0x x x x+ + + + = . Example 7*: Solve a Polynomial Equation Find the real solutions of the equation 4 3 22 4 8 0x x x x− + − − = . Theorem: Every polynomial function (with real coefficients) can be uniquely factored into a product of ___________ factors and_______________________________ factors. Theorem: A polynomial function with real coefficients of odd degree has at least ___________ real zero.

Bounds on Zeros Theorem: Let f denote a polynomial function whose leading coefficient is positive.

• If 0M > is a real number and if the ____________ in the process of ___________

of f by x M− contains only numbers that are ____________ or ___________, then M is an upper bound to the zeros of f.

• If 0m < is a real number and if the ____________ in the process of ____________ of f by x m− contains numbers that alternate _____________ or ______________, then m is a lower bound to the zeros of f.



Example 8: Use the Theorem for Bounds on Zeros Find the bound to the zeros of each polynomial (a) 4 3 2( ) 2 4 1g x x x x x= − − + − (b) 4 3 2( ) 4 2 2f x x x x x= − + + −

Intermediate Value Theorem: Let f denote a polynomial function. If ______ and if ____ and _____ are of opposite sign, then there is at least one real zero of f between _________.

Why does the Intermediate Value Theorem work? Example 9: Use the Intermediate Value Theorem Show that ( ) 3 1f x x x= − + has a zero between 2 and 1.− −



Approximate the Real Zeros of a Polynomial Function Step 1: Find two consecutive integers a and 1a + such that f has a ______ between them. Step 2: Divide the interval [ ], 1a a + into ____ subintervals.

Step 3: Evaluate f at each endpoint of the subintervals until the ______________________ applies; this interval then contains a zero.

Step 4: Now divide the new interval into 10 equal subintervals and repeat Step ___. Step 5: Continue with Steps 3 and 4 until the desired accuracy is achieved.

Example 10: Approximate the Zeros of a Polynomial Function Find the zero of ( ) 3 1f x x x= − + correct to two decimal places.



Chapter 5: Polynomial and Rational Functions Section 5.6: Complex Zeros; Fundamental Theorem of Algebra

Exploration 1: Complex Zeroes In Section 1.3, we found complex zeros of quadratic equations. In this section, we will use those concepts to find complex zeros of any polynomial.

1. Find the zeros of the functions below in both the real and complex number system:

(a) ( ) 2 4f x x= + (b) 2( ) 9f x x= + (c) 2( ) ( 1)( 1)f x x x x= − + +

2. Make a conjecture about the degree of the polynomial and the number of zeros.

3. Recall complex conjugates: If z a bi= + then its conjugate is z a bi= − . Based on your

solutions to (1b) and (1c) make a conjecture about complex zeros of polynomials and complex conjugates.

To expand our knowledge about complex zeros from quadratics to any polynomial, we need to first learn some important definitions and theorems. Let’s start by defining a complex polynomial, complex variable, and complex zero.

Definitions A variable in the complex number system is referred to as a complex variable.

A complex polynomial function f of degree n is a function of the form:

__________________________________________________________

where _______________ are complex numbers, 0na ≠ , n is a nonnegative integer,

and x is a complex variable. As before, _____ is called the leading coefficient of f.

A complex number r is called a complex zero of f if _____________.

Section 5.6 Complex Zeros; Fundamental Theorem of Algebra 161


Now that we have defined a complex polynomial we can learn about the Fundamental Theorem of Algebra! Fundamental Theorem of Algebra: Every complex polynomial function ( )f x of degree

1n ≥ has at least one ___________________. OK and now for the big one… Theorem: Every complex polynomial function ( )f x of degree 1n ≥ can be factored into n linear factors (not necessarily the same factors) of the form:

_________________________________________ where ___________________________________ are complex numbers.

This means, that a polynomial of n degree has exactly n zeros (zeros can be real or complex some of which may repeat). Look back at Exploration 1. Is this what you found? Example 1: Number of Zeros of a Polynomial How many zeros does the polynomial 5 3( ) 3 2 4 1f x x x x= + − + have?

Now that we know how many zeros a polynomial has, let’s talk about how to find all of the zeros. In order to do so, we need to introduce some more theorems and corollaries. Look back at your answer to (3) in Exploration 1. What did you find about the complex zeros of a polynomial and complex conjugates? Hopefully, you saw that if a polynomial had a complex zero, its conjugate was also a zero of that polynomial. This leads us to the following theorem:

Conjugate Pairs Theorem: Let ( )f x be a polynomial function whose coefficients are real numbers. If r a bi= + is a zero of f, the complex conjugate _________ is also a zero of f. Corollary: A polynomial function f of odd degree with real coefficients has at least one _____ zero. Why is this corollary true?



Example 2*: Use the Conjugate Pairs Theorem A polynomial of degree 5 whose coefficients are real numbers has the zeros 1, 5i, and 1+i. Find the remaining two zeros. Example 3*: Find a Polynomial Function with Specified Zeros Find a polynomial f of degree 4 whose coefficients are real numbers and that has the zeros 1, 1, 4 i− + . Theorem: Every polynomial function with real coefficients can be uniquely factored over the real numbers into a product of _____________ and/or irreducible ________________.

To find the complex zeros of a polynomial function, we follow the same steps as those for finding the real zeros from Section 5.5.

Steps for Finding the Zeros of a Polynomial Function Step 1: Use the degree of the polynomial to determine the maximum number of zeros.

Step 2: Use Descartes’ Rule of Signs to determine the possible number of positive and

negative zeros.

Step 3:

(a) If the polynomial has integer coefficients, use the Rational Zeros Theorem to

identify those rational numbers that potentially could be zeros.

(b) Use substitution, synthetic division, or long division to test each potential

rational zero. Each time that a zero (and thus a factor) is found, repeat Step 3 on

the depressed equation.

In attempting to find zeros, remember to use (if possible) the factoring techniques that you already know (special products, factoring by grouping, and so on).

Section 5.6 Complex Zeros; Fundamental Theorem of Algebra 163


Example 4*: Find the Complex Zeros of a Polynomial Function. Find the complex zeros of the polynomial function 4 3 2( ) 3 5 25 45 18f x x x x x= + + + − . Write f in factored form. Example 5: Find the Complex Zeros of a Polynomial Function. Find the complex zeros of the polynomial function 3 2( ) 2 3 8 5f x x x x= + + − . Write f in factored form.

164 Chapter 6 Exponential and Logarithmic Functions


Chapter 6: Exponential and Logarithmic Equations Section 6.1: Composite Functions

Exploration 1*: Form a Composite Function

1. Suppose you have a job that pays $10 per hour. Write a function, g that can be used to determine your gross pay (your pay before taxes are taken out) per hour, h, that you worked.

( )g h =

2. Now let’s write a formula for how much money you’ll actually take home of that

paycheck. Let’s assume your employer withholds 20% of your gross pay for taxes. Write a function, n, that determines your net pay based off of your gross income, g.

( )n g =

3. How much money would you net if you worked for 20 hours?

4. Instead of having to use two different functions to find out your net pay, as you most likely did in (3), let’s combine our functions from (1) and (2) and write them as one function. This is called composing functions. Write a function that relates the number of hours worked, h, to your net pay, n.

Definition: Given two functions f and g, the composite function, denoted by __________ (read as “f composed with g”) is defined by________________________.

Note: f g does not mean f multiplied by g(x). It means input the function g into the

function f: ( )( ) ( ) ( )f g f g x f x g x= ≠

The domain of f g is the set of all numbers x in the domain of g such that ( )g x is in the

domain of f. In other words, f g is defined whenever both ( )g x and ( )( )f g x are defined.

Section 6.1 Composite Functions 165


Example 1*: Form a Composite Function; Evaluate a Composite Function Suppose that ( ) 22 3f x x= + and ( ) 34 1g x x= + . Find:

(a) ( )(1) (b) ( )(1) (c) ( )( 2) (d) ( )( 1) f g g f f f g g− − Example 2: Evaluate a Composite Function If f (x) and g(x) are polynomial functions, use the table of values for f (x) and g(x) to complete the table of values for ( )( )f g x .

Example 3: Find the Domain of a Composite Function Suppose that ( ) 22 3f x x= + and ( ) 34 1g x x= + . Find the following and their domains:

(a) f g (b) g f

x ( )g x -2 4 -1 1 0 0 1 1 2 4

x ( )f x 0 3 1 4 2 5 3 6 4 7

x ( )( )f g x -2 -1 0 1 3



Example 4*: Find the Domain of a Composite Function

Find the domain of ( )( )f g x if ( ) ( )1 4 and

4 2f x g x

x x= =

+ −.

Example 5: Find a Composite Function and Its Domain

Suppose that ( ) 1f xx

= and ( ) 1g x x= − . Find the following and their domains:

(a) f g (b) f f

Example 6*: Showing Two Composite Functions Are Equal

If ( ) ( ) 12 and ,

2f x x g x x= = show that ( )( ) ( )( )f g x g f x x= = for every x in the

domain of and .f g g f

Section 6.1 Composite Functions 167


Example 7: Showing Two Composite Functions Are Equal

If ( ) ( ) ( )11 and 2 1,

2f x x g x x= − = + show that ( )( ) ( )( )f g x g f x x= = for every x in

the domain of and .f g g f Example 8*: Find the Components of a Composite Function

Find functions f and g such that ( ) ( )4if 2 3f g H H x x= = +

Example 9: Find the Components of a Composite Function

Find functions f and g such that ( ) 2

1 if

2 3f g H H x

x= =

−



Chapter 6: Exponential and Logarithmic Functions Section 6.2: One-to-One Functions; Inverse Functions

Definition: A function is one – to – one if any two different inputs in the domain correspond to _________________________________________________. That is, if 1x

and 2x are two different inputs of a function f, then f is one – to – one if ____________.

Example 1*: Determine Whether a Function is One – to – One Determine whether the following functions are one – to – one. Explain why or why not.

(a) (b) {(1,5), (2,8), (3,11), (4,14)}

The Horizontal Line Test Theorem: If every horizontal line intersects the graph of a function f in at most ________________, then f is one – to – one.

Why does this test work? You may want to refer to the definition of one – to – one functions.

Dan

John

Joe

Andy

Saturn

Pontiac

Honda

Student Car

Section 6.2 One – to – One Functions; Inverse Functions 169


Example 2: Determine whether a Function is One – to – One Using the Horizontal Line Test For each function, use the graph to determine whether the function is one – to – one.

Theorem: A function that is increasing on an interval I is a one – to – one function on I. A function that is decreasing on an interval I is a one – to – one function on I.

Why is this theorem true?

Exploration 1: Inverse Functions – Reverse the Process You might have experienced converting between degrees Fahrenheit and degrees Celsius when measuring a temperature. The standard formula for determining temperature in degrees

Fahrenheit, when given the temperature in degrees Celsius, is9

325

F C= + . We can use this

formula to define a function named g, namely ( ) 932

5F g C C= = + , where C is the number

of degrees Celsius and ( )g C is a number of degrees Fahrenheit. The function g defines a

process for converting degrees Celsius to degrees Fahrenheit. 1. What is the value of ( )100g ? What does it represent?

2. Solve the equation ( ) 112g C = and describe the meaning of your answer.

3. What happens if you want to input degree Fahrenheit and output degree Celsius?

Reverse the process of the formula 9

325

F C= + by solving for C.



Definition: Suppose that f is a one – to – one function. Then, to each x in the domain of f, there is _______________________ y in the range (because f is a function); and to each y in the range of f there is exactly one x in the domain (because f is one – to – one). The correspondence from the range of f back to the ______________ of f is called the inverse

function of f. We use the symbol 1f − to denote the inverse of f. Note: 1 1ff

− ≠

In other words, two functions are said to be inverses of each other if they are the reverse process of each other. Notice in the exploration, the formula found in part (c) was the reverse process of g. Instead of inputting Celsius and outputting Fahrenheit, the new function inputs Fahrenheit and outputs Celsius.

Example 3*: Determine the Inverse of a Function Find the inverse of the following function. Let the domain of the function represent certain students, and let the range represent the make of that student’s car. State the domain and the range of the inverse function. Example 4*: Determine the Inverse of a Function Find the inverse of the following one – to – one function. Then state the domain and range of the function and its inverse.

{(1,5), (2,8), (3,11), (4,14)} Domain and Range of Inverse Functions: Since the inverse function, 1f − , is a reverse mapping of the function f :

Domain of f = _____________ of 1f − and Range of f = _______________ of 1f −

Dan

John

Joe

Michelle

Saturn

Pontiac

Honda

Chrysler

Student Car

Section 6.2 One – to – One Functions; Inverse Functions 171


Fact: What f does, 1f − undoes and vice versa. Therefore,

( )( )1 ____f f x− = where x is in the domain of f

( )( )1 ____f f x− = where x is in the domain of 1f −

We can use this fact to verify if two functions are inverses of each other. Example 5*: Determine the Inverse of a Function ; Verifying Inverse Functions

Verify that the inverse of ( ) ( )3 1 32 is 2g x x g x x−= + = − by showing that ( )( )1g g x x− =

for all x in the domain of g and that ( )( )1 g g x x− = for all x in the domain of -1.g

Exploration 2: Graphs of Inverse Functions

1. Using a graphing utility, graph the following functions on the same screen 3 3, , and y x y x y x= = =

2. What do you notice about the graphs of 3 3, its inverse ,y x y x= = and the line y x= ?

3. Repeat this experiment by graphing the following functions on the same screen: 1

, 2 3, and ( 3)2

y x y x y x= = + = −

4. What do you notice about the graphs of 1

2 3, its inverse ( 3),2

y x y x= + = − and the

line y x= ?



Theorem: The graph of a one – to – one function f and the graph of its inverse 1f − are symmetric with respect to the line ______________.

Example 6*: Obtain the Graph of the Inverse Function The graph shown is that of a one – to – one function. Draw the graph of its inverse.

Procedure for Finding the Inverse of a One – to – One Function Step 1: In ( )y f x= , interchange the variables x and y to obtain ____________. This

equation defines the inverse function 1f − implicitly. Step 2: If possible, solve the implicit equation for y in terms of x to obtain the explicit form

of 1f − : ____________________. Step 3: Check the result by showing that _______________ and ________________. Example 7*: Find the Inverse Function from an Equation I Find the inverse of ( ) 4 2.f x x= +

Example 8*: Find the Inverse Function from an Equation II

The function 2 1

( ) , -11

xf x xx

−= ≠+

is one – to – one. Find its inverse and state the domain

and range of both f and its inverse function.

Documents

Chapter 5: Polynomial and Rational Functions Section 5.1 ......Graphing Polynomial Functions Using Transformations In Section 3.5 we learned the methods of graph transformations. We