Looking Ahead to Chapter 10

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Looking Ahead to Chapter 10
Focus
In Chapter 10, you will learn about polynomials, including how to add, subtract,
multiply, and divide polynomials. You will also learn about polynomial and
rational functions.
Chapter Warm-up
Answer these questions to help you review skills that you will need in Chapter 10.
Solve each two-step equation.
x x
1. 5x(cid:3)3(cid:4)2x(cid:5)12 2. 4x(cid:4)7(cid:2)3x 3. (cid:3)3(cid:4)
5 10
x(cid:4)(cid:5)5 x(cid:4)1 x(cid:4)(cid:5)30
Simplify each expression.
4. (32)5 5. x7(cid:2)x(cid:2)5(x2)(cid:2)3 6. (xy5)(cid:2)1
y(cid:2)2
1 1
310
x4 xy7
Read the problem scenario below.
You have just planted a flower bed in a 3-foot wide by 5-foot long rectangular section of your yard.
After planting some flowers, you decide that you would like to add to the width of your garden so that
there is more space to plant flowers. Let x represent the amount added to the length of your garden.
7. Represent the area of your expanded garden with an expression in two ways by using the
10 distributive property.
5(3(cid:3)x) (cid:4)15(cid:3)5x
8. What is the area of your garden if you expand the width by 4 feet?
5(3(cid:3)4) (cid:4)5(7) (cid:4)35 square feet
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Key Terms ng, I
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polynomial ■ p. 458 divisor ■ p. 472 polynomial expression ■ p. 488 Le
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term ■ p. 458 dividend ■ p. 472 greatest common factor ■ p. 488 gi
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coefficient ■ p. 458 remainder ■ p. 472 factoring by grouping ■ p. 489 ar
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degree of a term ■ p. 458 FOIL pattern ■ p. 476 algebraic fraction ■ p. 491 9
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degree of the polynomial ■ p. 458 square of a binomial sum ■ p. 477 rational expression ■ p. 491 2
©
monomial ■ p. 459 square of a binomial excluded or restricted
binomial ■ p. 459 difference ■ p. 479 value ■ p. 491
trinomial ■ p. 459 factoring out a common rational equation ■ p. 492
standard form ■ p. 459 factor ■ p. 483 extraneous solutions ■ p. 494
Vertical Line Test ■ p. 461 difference of two Fundamental Theorem of
combine like terms ■ p. 464 squares ■ p. 487 Algebra ■ p. 497
distributive property ■ p. 464 perfect square trinomial ■ p. 487 synthetic division ■ p. 500
area model ■ p. 468
454 Chapter 10 ■ Polynomial Functions
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C H A P T E R
10
Polynomial Functions
10
Stained glass windows have been used as decoration in homes, religious buildings,
and other buildings since the 7th century. In Lesson 10.4, you will find the area of a
stained glass window.
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ne Polynomials and Polynomial Factoring Polynomials ■ p. 481
Car Functions ■ p. 457 10.6 More Factoring
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00 10.2 Play Ball! Factoring Polynomial
2
© Adding and Subtracting Expressions ■ p. 487
Polynomials ■ p. 463
10.7 Fractions Again
10.3 Se Habla Español Rational Expressions ■ p. 491
Multiplying and Dividing
10.8 The Long and Short of It
Polynomials ■ p. 467
Polynomial Division ■ p. 497
10.4 Making Stained Glass
Multiplying Binomials ■ p. 475
Chapter 10 ■ Polynomial Functions 455
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456 Chapter 10 ■ Polynomial Functions
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10.1 Water Balloons
Polynomials and Polynomial Functions
Learning By Doing Lesson Map
Get Ready
Objectives Lesson Overview
In this lesson, you will: Within the context of this lesson, students will be
■ Identify terms and coefficients of polynomials. asked to:
■ Classify polynomials by the number of terms. ■ Identify if an expression is a polynomial.
■ Classify polynomials by degree. ■ Identify terms and coefficients of polynomials.
■ Write polynomials in standard form. ■ Classify polynomials by the number of terms as
■ Use the Vertical Line Test to determine whether well as by the highest degree term.
equations are functions. ■ Write polynomials in standard form.
■ Apply the Vertical Line Test to graphs of equations
Key Terms to determine whether the graphed equations are
functions.
■ polynomial ■ monomial
Essential Questions
■ term ■ binomial
The following key questions are addressed in this
■ coefficient ■ trinomial
section:
■ degree of a term ■ standard form
1. What is a polynomial?
■ degree of the polynomial ■ Vertical Line Test
2. What is vertical motion?
10
Sunshine State Standards
3. How can you determine the coefficient of
a term?
MA.912.A.2.3
4. How can you determine the degree of a
Describe the concept of a function, use function nota-
term? How can you determine the degree
tion, determine whether a given relation is a function,
c. of a polynomial?
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ng, I and link equations to functions. 5. How can you determine whether a graphed
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ar equation is a function?
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e 6. How can you write a function in standard form?
egi drawing a diagram, making a chart, guessing-and-
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9 tion, working backwards, and creating a table. What is a trinomial?
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© MA.912.A.10.2
Decide whether a solution is reasonable in the context
of the original situation.
Lesson 10.1 ■ Polynomials and Polynomial Functions 457 A
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Show The Way
Warm Up
Place the following questions or an applicable subset of these questions on the board before students
enter class. Students should begin working as soon as they are seated.
The vertical motion model for a ball hit by a baseball bat is y = –16t2 + 40t + 2.5,
where y is the height in feet and t is the time in seconds. Answer each question.
Recall that the general equation for a vertical motion model is y = –16t2 + vt + h,
where v is the initial velocity in feet per second and h is the initial height in feet.
1. What is velocity? Velocity is the speed and direction in which an object is moving.
2. What is the initial velocity? 40 feet per second.
3. What does the constant 2.5 represent in the equation? The initial height from where the
object was thrown was 2.5 feet off of the ground.
4. What is the height of the ball 1 second after the ball was hit? 26.5 feet
5. What shape will the graph of this equation look like? A parabola that opens down.
6. What is the vertex point of the parabola? The vertex point is at (1.25, 27.5).
7. What is the meaning of the vertex point for the parabola? This means that the ball is at the highest
point of 27.5 feet, 1.25 seconds after the ball is hit.
8. What is the axis of symmetry for this parabola? The axis of symmetry of this parabola is x= 1.25.
9. When would the ball hit the ground? The ball will hit the ground about 2.56 seconds after
it is hit. (Assuming no one else catches the ball before it hits the ground.)
Motivator
10
Begin the lesson with the motivator to get students thinking about the topic of the upcoming problem.
This lesson is about the path of a tossed water balloon. The motivating questions are about tossing
water balloons.
Ask the students the following questions to get them interested in the lesson.
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■ What did the path of the balloon look like when you threw it? ar
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SCENARIO On a calm day, you and a friend are tossing
Problem 1 water balloons in a field trying to hit a boulder in the field.
The balloons travel in a path that is in the shape of a parabola.
Students will investigate vertical motion
for a thrown water balloon. Problem 1 Ready, Set, Launch
Grouping A. On your first throw, the balloon leaves your hand 3 feet above
the ground at a velocity of 20 feet per second. Use what you
Ask for a student volunteer to read the
learned in Chapter 8 about vertical motion models to write an
Scenario and Problem 1 aloud. Have a equation that gives the height of the balloon in terms of time.
student restate the problem. Pose the
y= –16t2+ 20t+ 3
Guiding Questions below to verify student
understanding. Complete part (A) of
Problem 1 together as a whole class. B. What is the height of the balloon after one second? Show your
work and use a complete sentence to explain your answer.
When the students understand the
situation, have them work together y= –16(1)2+ 20(1) + 3 = –16 + 20 + 3 = 7
in small groups to complete parts (B) The balloon is at a height of seven feet after one second.
through (F) of Problem 1. Then call the
C. What is the height of the balloon after two seconds? Show your
class back together to discuss and present work and use a complete sentence in your answer.
their work for parts (B) through (F) of
y= –16(2)2+ 20(2) + 3 = –64 + 40 + 3 = –21
Problem 1.
The balloon is on the ground after two seconds.
D. On your second throw, the balloon leaves your hand at ground
Take Note
level at a velocity of 30 feet per second. Write an equation that
gives the height of the balloon in terms of time.
The vertical motion modelis y= –16t2+ 30t
y(cid:2)(cid:3)16t2(cid:4)vt(cid:4)h, where tis the time in
seconds that the object has been moving,
vis the initial velocity (speed) in feet per E. What is the height of the balloon after one second? Show your
second of the object, his the initial height work and use a complete sentence in your answer.
in feet of the object, and yis the height in
y= –16(1)2+ 30(1) = –16 + 30 = 14
feet of the object at time t.
The balloon is at a height of 14 feet after one second. 10
Guiding Questions F. What is the height of the balloon after two seconds? Show your
work and use a complete sentence in your answer.
■ What information is given in this
problem? y= –16(2)2+ 30(2) = –64 + 60 = –4
The balloon is on the ground after two seconds.
c. ■ What is the meaning of velocity?
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09 what is the meaning of the variable t? ■ What will a positive value for the height represent in this
20 What is the meaning of the variable y?
© situation?
What is the meaning of the variable v?
What is the meaning of the variable h? ■ Is it possible for the water balloon to have a negative height?
What would a negative height represent?
■ Will the water balloon travel in a hori-
zontal line across the field? What will ■ What would a height of zero represent?
the path of the balloon look like?
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Investigate Problem 1
Investigate Problem 1
1. How are your models the same? Use a complete sentence in
Students will compare two polynomials your answer.
used to model vertical motion. Sample Answer: They are both quadratic equations.
Take Note How are your models different? Use a complete sentence in
your answer.
Whole numbers are the numbers 0, 1, 2, 3, Sample Answer: The y-intercepts of the graphs of the
and so on. equations are different and the numbers multiplied by
tare different.
Grouping 2. Just the Math: Polynomials The expressions
(cid:3)16t2(cid:4)20t(cid:4)3and (cid:3)16t2(cid:4)30tthat model the heights of
Ask for a student volunteer to read
the balloons are polynomials. Each expression is a polynomial
Question 1 aloud. Have a student
because it is a sum of products of the form axk, where ais a
restate the problem. Have students
real number and kis a whole number. Each product is a term
work together in small groups to complete and the number being multiplied by a power is a coefficient.
Question 1. Then call the class back For each of the polynomials above, name the terms and
together to have the students discuss and coefficients. Use complete sentences in your answer.
explaintheir work for Question 1. The terms of –16t2+ 20t+ 3 are –16t2, 20t, and 3.
The coefficients of –16t2+ 20t+ 3 are –16, 20, and 3.
Just the Math
The terms of –16t2+ 30tare –16t2and 30t.
The coefficients of –16t2+ 30tare –16 and 30.
Students will be formally introduced to
terminology for polynomials in Questions 2
3. For each of your polynomial models, what is the greatest
and 3. It is important for students to
exponent? Use a complete sentence in your answer.
recognize that terms in a polynomial are
The greatest exponent in each model is two.
separated by addition. This is true because
a polynomial such as x2(cid:3)2can be written
The degree of a termin a polynomial is the exponent of the
as x2(cid:4) ((cid:3)2). So, x2and –2 are each terms
term. The greatest exponent in a polynomial determines the
10 of this polynomial. degree of the polynomial. For instance, in the polynomial
4x(cid:4)3, the greatest exponent is 1, so the degree of the poly-
Grouping
nomial is 1. What is the degree of each of your polynomial
Ask for a student volunteer to read models? Use a complete sentence in your answer.
Question 2 aloud. Have a student restate Each polynomial model has degree two.
the problem. Pose the Guiding
Questions below to verify student 4. What kind of expression is a polynomial of degree 0? Give an c.
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understanding. Complete Questions 2 example and use a complete sentence to explain your reasoning. g, I
and 3 together as a whole class. Sample Answer: 4x0= 4; A polynomial of degree 0 is a nin
constant expression. ar
Then have students work together in small Le
groups to complete Questions 4 and 5. What kind of expression is a polynomial of degree 1? Give an gie
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Guiding Questions Sample Answer: 2x+ 1; A polynomial of degree 1 is a linear Car
■ What is a polynomial? What are expression. 009
2
the important characteristics of a ©
polynomial?
■ What is a term? What is a coefficient? ■ How can you find the degree of a term? How can you find the
degree of a polynomial?
■ In the polynomial (cid:3)3x2(cid:4)4x(cid:4) ((cid:3)5) list
the terms? What is the coefficient of x2?
What is the coefficient of x?
■ What is a degree of a term? What is a
degree of a polynomial?
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Investigate Problem 1
Investigate Problem 1
What kind of expression is a polynomial of degree 2? Give an
Students will classify polynomials example and use a complete sentence to explain your reasoning.
according to the number of terms in Sample Answer: x2; A polynomial of degree 2 is a quadratic
each polynomial. expression.
Call the class back together to have the A polynomial of degree 3 is a called a cubic polynomial. Write an
example of a cubic polynomial.
students discuss and present their work
for Question 4. Sample Answer: x3
Grouping 5. For each of your polynomial models in parts (A) and (D), find the
number of terms in the model. Use a complete sentence in your
Ask for a student volunteer to read
answer.
Question 5 aloud. Have a student restate
The model for the first throw has three terms and the model
the problem. Pose the Guiding Questions
for the second throw has two terms.
below to verify student understanding.
Have students work together in small Polynomials with only one term are monomials. Polynomials
groups to complete Questions 5 and 6. with exactly two terms are binomials. Polynomials with exactly
three terms are trinomials. Classify each polynomial model in
Guiding Questions parts (A) and (D) by its number of terms. Use a complete sen-
tence in your answer.
■ What is a term for a polynomial?
The polynomial model for the first throw is a trinomial and
■ How can you find the number of terms? the polynomial model for the second throw is a binomial.
■ What is a monomial? What is a
6. Give an example of a monomial of degree 3.
binomial? What is a trinomial?
Sample Answer: 4x3
■ Why do you think that we only have
Give an example of a trinomial of degree 5.
special names for polynomials with
Sample Answer: x5+ 2x– 3
1, 2, or 3 terms?
7. Just the Math: Standard Form of a Polynomial
Call the class back together to have the Later in this chapter, we will be adding, subtracting, multiplying,
students discuss and present their work and dividing polynomials. To make this process easier, it is 10
for Questions 5 and 6. helpful to write polynomials in standard form. A polynomial is
written in standard formby writing the terms in descending
Just the Math order, starting with the term with the greatest degree and ending
with the term with the least degree. Write each polynomial in
The concept of the standard form of a
standard form.
polynomial will be introduced in Question 7.
c. 6(cid:4)5x 7(cid:3)x2
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g, I Grouping 5x+6 –x2+ 7
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ear Ask for a student volunteer to read 4(cid:4)3x(cid:4)4x2 4(cid:4)3x2(cid:4)9x(cid:3)x3
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Car below to verify student understanding. 5(cid:3)6x4 x6(cid:3)4x3(cid:4)16
09 Have students work together in small –6x4+ 5 x6– 4x3+ 16
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2 groups to complete Questions 7 and 8.
©
Guiding Questions
represent the motion of the space shuttle, but each had the
■ What does it mean for something to have polynomial represented in a different way?
a standard or to be standardized?
■ How can you write a polynomial in standard form?
■ Why is it important for us to be able
to write polynomials in a standard form? Notes Only the basic approach to standard form is presented
What would happen if several people in this problem. Students will not yet be asked to write a poly-
were considering a polynomial to nomial that has more than one variable in standard form.
Lesson 10.1 ■ Polynomials and Polynomial Functions 459
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Investigate Problem 1
Investigate Problem 1
8. Determine whether each algebraic expression is a polynomial.
Students will complete a summary question If the expression is a polynomial, classify it by its degree and
to review the words polynomials, terms, number of terms. If the expression is not a polynomial, use a
complete sentence to explain why it is not a polynomial.
and degrees.
4x2(cid:2)3x(cid:3)1 3x(cid:3)2(cid:2)4x(cid:3)1
Call the class back together to have the
polynomial of degree 2; Not a polynomial because the
students discuss and present their work trinomial power on the first term is not a
for Questions 7 and 8. whole number.
Key Formative Assessments 4x6(cid:2)1 10(cid:3)x4
■ What are the requirements for an expres- polynomial of degree 6; polynomial of degree 4;
binomial binomial
sion to be considered a polynomial?
■ How can you determine whether an 2(cid:2)x(cid:2)3x(cid:3)4 25
expression is a polynomial?
Not a polynomial because polynomial of degree 0;
■ How can you determine where a term a polynomial does not monomial
contain square roots.
begins and where the term ends?
■ How can you determine the number
of terms in a polynomial? Problem 2 The Balloon’s Path
■ What are the requirements for a
It is now your friend’s turn to throw water balloons at the boulder.
polynomial to be in standard form?
A. In your friend’s first throw, the balloon leaves her hand 2 feet
above the ground at a velocity of 36 feet per second. Write an
Problem 2 equation that gives the height of the balloon in terms of time.
y= –16t2+ 36t+ 2
Grouping
B. Complete the table of values that shows the height of the balloon
Ask for a student volunteer to read Prob- in terms of time.
10 lem 2 aloud. Have a student restate the
problem. Pose the Guiding Questions Quantity Name Time Height
below to verify student understanding.
Unit seconds feet
Have students work together in small
Expression t –16t2+ 36t+ 2
groups to complete parts (A) through (C)
of Problem 2. 0.0 2
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Guiding Questions 0.5 16 g, I
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■ What information is given in this 1.0 22 arni
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■ How is this problem similar to Problem 1? 2.5 –8 negi
How is this problem different from 3.0 –34 Car
Problem 1? 09
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■ How can you write an equation to model ©
the motion for the balloon thrown by assigned as part of a homework assignment. At the beginning of
your friend? the next class session, call the students together to discuss and
present parts (A) through (C) of Problem 2 and then continue with
If the students finish Problem 1 during the
the problem as suggested on the next few pages.
class period and do not have enough time
to complete Problem 2 during class,
parts (A) through (C) of Problem 2 can be
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Problem 2 The Balloon’s Path
Problem 2
C. Create a graph of the model to see the path of the balloon on the
Students will graph a parabolic function grid below. First, choose your bounds and intervals. Be sure to
representing the height of a water balloon label your graph clearly.
as a function of the time after the balloon
Variable quantity Lower bound Upper bound Interval
was thrown.
Time 0 3.75 0.25
Call the class back together to have the
Height 0 30 2
students discuss and present their work
for parts (A) through (C) of Problem 2. y Water Balloon
30
Key Formative Assessments 28
■ What two quantities were compared for 26
24
your graph in part (C)?
22
■ What is the unit for time? What is the
20
unit for height?
et) 18
■ What is the shape of the graph? ht (fe 16
■ Does this shape make sense for this eig 14
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problem situation? Why or why not?
10
■ Why does the parabola open downward 8
rather than upward?
6
■ How manyy-intercepts are there? 4
2
■ What is the meaning of the y-intercept
0 x
for your graph? 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.00
■ How many x-intercepts are there? Time (seconds)
■ Why is there only one x-intercept? 10
Investigate Problem 2
■ What is the meaning of the x-intercept?
1. Is the equation that you wrote in part (A) a function? How do you
Investigate Problem 2 know? Use a complete sentence in your answer.
Sample Answer: Yes. For every input, there is only one output.
nc. Grouping
ng, I Ask for a student volunteer to read 2. Just the Math: Vertical Line Test You can use a graph
arni Questions 1 and 2 aloud. Have a student to determine whether an equation is a function. The Vertical
Le Line Teststates that an equation is a function if you can pass a
gie restate the problem. Pose the Guiding vertical line through any part of the graph of the equation and the
ne Questions below to verify student under- line intersects the graph, at most, one time. Consider your graph
Car standing. Complete Questions 1 and 2 in part (C). Does your graph pass the Vertical Line Test?
09 together as a whole class. Yes.
0
2
© Guiding Questions
■ What is a function? What does it mean ■ What is the Vertical Line Test and how can you use it?
for every input value to have exactly one Does the vertical line have to pass through just one point
output value? at a selected x-value, or does it have to pass through exactly
one point at each x-value?
Lesson 10.1 ■ Polynomials and Polynomial Functions 461