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9
Objective 2
Use the greatest common factor and the Distributive Property to factor polynomials
with the grouping technique, and use these techniques to solve equations.
Objective 2 Pretest
Students complete the Objective 2 Pretest on the same day
as the Objective 1 Posttest.
Using the Results
• Score the pretest and update the class record card.
• If the majority of students do not demonstrate mastery
of the concepts, use the 5-Day Instructional Plan for
Objective 2.
Chapter 9 • Objective 2
Pretest
• If the majority of students demonstrate mastery of the
concepts, use the 3-Day Instructional Plan for Objective 2.
Name __________________________________________ Date ____________________________
Factor the polynomials using the greatest common factor (GCF) and the
distributive property.
x 2 + 3x
2.
9x 4y 3 + 12x 2y 5 – 15x 3y
3.
4c 2d – 8c 3d 2
4.
2m 2 – 10m
5.
18s 4t 3 + 9s 2t 4 – 27s 3t 5
–2x 2 – 4x = 0
7.
5a 2 + 30a = 0
6b 2 – 12b = 0
9.
0 = 9x 2 – 3x
1.
x(x + 3)
4c 2d(1 – 2cd)
3x 2y(3x 2y 2 + 4y 4 – 5x)
2m(m – 5)
9s2t 3(2s 2 + t – 3st 2)
Solve the equations.
6.
8.
126
786
6b(b – 2) = 0
b = 0, 2
5a(a + 6) = 0
a = 0, –6
3x(3x – 1) = 0
x = 0, 1
3
–3a 2 – 9a = 0
–3a(a + 3) = 0
a = 0, –3
Chapter 9 • Objective 2
Chapter 9 • Objective 2
Inside Algebra
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
10.
–2x(x + 2) = 0
x = 0, –2
Objective 2
Goals and Activities
Objective 2 Goals
The following activities, when used with the instructional plans on
pages 788 and 789, enable students to:
• Factor the polynomial 24x 2 – 16x using the greatest common
factor (GCF) and the Distributive Property to get 8x(3x − 2)
• Factor the polynomial 18x 2y 3 + 15x 3y 5 – 21x 4y 2 using the
greatest common factor (GCF) and the Distributive Property
to get 3x 2y 2(6y + 5xy 3 − 7x 2)
• Solve the equation 2x 2 – 4x = 0 to get x = 0, 2
• Solve the equation 3x 2 + 6x = 0 to get x = 0, −2
Objective 2 Activities
Concept Development Activities
★CD 1 Using
CD 2 Finding
What’s in
Common—
Part 1, page 792
Algebra Tiles,
page 790
CD 3 Finding
What’s in
Common—
Part 2, page 793
CD 4
Factoring and
Solving,
page 795
Practice Activities
PA 1 Drawing Cards,
page 796
PA 2 Solving
Equations, page 797
PA 3 Finding the
Solutions, page 798
Progress-Monitoring Activities
PM 1 Apply Skills 1,
page 799
PM 2 Apply Skills 2,
page 800
PM 3 Apply Skills 3,
page 801
Problem-Solving Activity
★
★PS 1 Using the Length and Width of a Rectangle, page 802
Ongoing Assessment
Posttest Objective 2, page 803
Pretest Objective 3, page 804
CD = Concept Development PM = Progress Monitoring PS = Problem Solving
PA = Practice Activity ★ = Includes Problem Solving
Chapter 9 • Objective 2
787
CHAPTER
9
Objective 2
Instructional Plans
5-Day Instructional Plan
Use the 5-Day Instructional Plan when pretest results indicate that students would
benefit from a slower pace. This plan is used when the majority of students need
more time or did not demonstrate mastery on the pretest.
★CD 1 Using Algebra Tiles
Day 1
CD 2 Finding What’s in Common—Part 1
PM 1 Apply Skills 1
ACCELERATE
Day 2
DIFFERENTIATE
PA 1 Drawing Cards
CD 3 Finding What’s in Common—
Part 2
PM 2 Apply Skills 2
PA 1 Drawing Cards
CD 4 Factoring and Solving
PM 2 Apply Skills 2
PA 2 Solving Equations
CD 4 Factoring and Solving
PM 3 Apply Skills 3
PA 3 Finding the Solutions
Day 3
Day 4
★PS 1 Using the Length and Width
of a Rectangle
PM 3 Apply Skills 3
Day 5
Posttest Objective 2
Pretest Objective 3
788
CD = Concept Development PM = Progress Monitoring PS = Problem Solving
PA = Practice Activity ★ = Includes Problem Solving
Chapter 9 • Objective 2
3-Day Instructional Plan
Use the 3-Day Instructional Plan when pretest results indicate that students can move
through the activities at a faster pace. This plan is ideal when the majority of students
demonstrate mastery on the pretest. This plan does not include all activities.
★CD 1 Using Algebra Tiles
CD 3 Finding What’s in Common—Part 2
Day 1
Day 2
ACCELERATE
DIFFERENTIATE
PM 2 Apply Skills 2
PM 1 Apply Skills 1
CD 4 Factoring and Solving
PA 1 Drawing Cards
PA 3 Finding the Solutions
PM 2 Apply Skills 2
PM 3 Apply Skills 3
CD 4 Factoring and Solving
★PS 1 Using the Length and Width
of a Rectangle
PA 3 Finding the Solutions
PM 3 Apply Skills 3
Day 3
Posttest Objective 2
Pretest Objective 3
CD = Concept Development PM = Progress Monitoring PS = Problem Solving
PA = Practice Activity ★ = Includes Problem Solving
Chapter 9 • Objective 2
789
Objective 2
Concept Development
Activities
★ CD
1
Using Algebra Tiles
Use with 5-Day or 3-Day Instructional Plan. In this
activity, students factor binomials using algebra tiles.
MATERIALS
5.Write 2x 2 + 6x on the board. Tell students to
think about how we factor the expression using
algebra tiles. Ask students to discuss how we find
the dimensions for the length and width. Students
should recognize that both a 2 and an x can be
factored out. 2x 2 + 6x = 2x (x + 3)
• Algebra tiles
x2
•Variation: Gizmos
Modeling the Factorization of x 2 + bx + c
DIRECTIONS
1. Review the following term with students:
factor A monomial that evenly divides a value
2.Distribute algebra tiles to students. Have each
student build a rectangle with dimensions A • B.
B
3.Write x + 3x on the board, and demonstrate how
to build a rectangle for that equation. Point out that
the area of the rectangle and the factored form of
the equation is x(x + 3).
x2
x
x
x
x
x
+3
x2
x x x
4.Have students use algebra tiles to factor several
expressions.
Sample problems:
x 2 + 4x x (x + 4)
x 2 − 3x x(x − 3)
x 2 + 6x x(x + 6)
x 2 + x x(x + 1)
★ = Includes Problem Solving
790
Chapter 9 • Objective 2
x
x
x
+3
x2
x x x
x2
x x x
x
x
2x
A
2
x
x2
6.Have students build rectangles and name the
factors for several more expressions.
Sample problems:
x 2 + 4x x(x + 4)
2x 2 + 4x 2x (x + 2)
x 2 − 5x x(x − 5)
3x 2 + 6x 3x (x + 2)
2x 2 − 6x 2x(x − 3)
7.Discuss the problems and the factors using the
Distributive Property.
x
Variation: Gizmos For this activity, use the tiles
in the Gizmo Modeling the Factorization of
x 2 + bx + c to model the factoring of these
quadratic expressions.
• Gizmos
Modeling the Factorization of x 2 + bx + c
NEXT STEPS • Differentiate
5-Day Instructional Plan:
CD 2, page 792—All students, for additional
concept development
3-Day Instructional Plan:
CD 3, page 793—All students, for additional
concept development
Chapter 9 • Objective 2
791
Objective 2
Concept Development
Activities
CD 2
Finding What’s in Common—Part 1
Use with 5-Day Instructional Plan. In this activity,
students develop an understanding of GCF.
MATERIALS
• Blank cards, 48 per group
DIRECTIONS
1. Review the following terms with students:
6.If students have difficulty writing the expression
from the cards they were dealt, go over an example.
factor A monomial that evenly divides a value
g reatest common factor (GCF) The largest
factor that a set of monomials has in common
2.Divide the class into groups of four. Give one set
of 48 cards to each group. Have each group label
the cards as follows:
•Twelve cards with the number 2
•Twelve cards with the number 3
•Twelve cards with the word red
•Twelve cards with the word blue
3.Explain the process for this activity.
• O
ne student from each group shuffles and deals
five cards to each person in the group. The
remaining cards are put aside.
• E ach student places his or her five cards
faceup on the table so all students can look
at everyone’s hand.
• S
tudents look for cards that all four group
members have in common. These cards are
placed in a new stack in the middle of the table.
• T he group members continue this process
until their hands no longer have any cards that
are common to those of all other members in
their group.
4.Tell students the cards that were placed in the
center of the table represent the greatest common
factor for their group. Have one student from each
group share the greatest common factor his or her
group found. Have each group write its GCF next to
an open parenthesis symbol on a sheet of paper; for
example, 2rb(.
5.Explain to students that the cards remaining in
their hands represent the portion of their hands
that were not common. Have groups list the cards
remaining in each group members’ hands, one at
a time. Explain to students that these should be
listed to the right of the parenthesis with a plus sign
between the entry. Have students place a closing
parenthesis symbol on the right side of the sum of
their leftover cards.
792
Chapter 9 • Objective 2
Examples:
Student 1 red, 2, red, blue, 3 or r + 2 + r + b + 3
Student 2 blue, 2, 2, 3, red or b + 2 + 2 + 3 + r
Student 3 red, red, 3, blue, 2 or r + r + 3 + b + 2
Student 4 3, blue, 2, red, 3 or 3 + b + 2 + r + 3
Common cards/GCF: red, blue, 2, and 3 or (2)(3)br
Leftover cards:
Student 1 red or r
Student 2 2
Student 3 red or r
Student 4 3
Expression: (2)(3)br (r + 2 + r + 3) or
(2)(3)br (2 + 3 + 2r)
7.Have students shuffle all the cards and repeat the
activity as time allows.
8.At the end of the activity, ask students to write
the factored sentences for 3x 2y 2 + 15xy 3 and
12a 2bc 3 + 30a 2bc. 3x 2y 2 + 15xy 3 = 3xy 2 (x + 5y);
12a 2bc 3 + 30a 2bc = 6a 2bc(2c 2 + 5)
NEXT STEPS • Differentiate
5-Day Instructional Plan:
PM 1, page 799—All students, to assess progress
Objective 2
Concept Development
Activities
CD 3
Finding What’s in Common—Part 2
Use with 5-Day or 3-Day Instructional Plan. In this
activity, students develop an understanding of GCF.
MATERIALS
•Blank cards, 52 per group, or the card sets from
CD 2 and 4 blank cards
DIRECTIONS
1. Review the following term with students:
factor A monomial that evenly divides a value
2.Divide the class into groups of four. Give each group
four blank cards and one set of the 48 cards students
made in Concept Development Activity, Finding
What’s in Common—Part 1. If students did not
complete this activity, give each group 52 blank cards
and have them label 48 of the blank cards as follows:
•Twelve cards with the number 2
•Twelve cards with the number 3
•Twelve cards with the word red
•Twelve cards with the word blue
Have the groups keep the four blank cards out
of the deck for use later in the activity.
3. Explain the process for this activity.
•One student in each group shuffles and deals
five cards to each person. The dealer puts the
remaining cards aside.
•All four group members place the card with their
abbreviated hand faceup in the center of the table
so all group members may view them.
•Students look at the abbreviated hands to find any
2s that are common to all four cards. Students
that find 2s write a 2 on a blank piece of paper,
then determine the lowest exponent of 2 that
appears in the four hands. It is written as an
exponent for the 2 they have on their paper.
•Students repeat the process for the number 3,
for the letter r, and for the letter b.
Note: This process will take a while, and it is
important you keep students organized so they
stay on track.
4.Tell students that once they complete the process,
the term they have written on their paper should be
the GCF for their group.
5.Have students complete the expression. Tell all
students to write an open parenthesis symbol to the
right of the GCF on their paper. Have each student
determine what would be left if the GCF were
removed from his or her hand. Students write these
remaining elements inside the open parenthesis.
Make sure students remember to put a plus symbol
between each student’s leftover cards. The last
person should place a closing parenthesis on the
right side of his or her term.
Example:
•Each student places his or her five cards
faceup on the table so all students can look
at everyone’s hand.
Student 1 2, 2, 3, r, r or 22 • 3 • r 2
Student 2 2, 3, 2, r, b or 22 • 3 • b • r
Student 3 b , 2, b, r, 3 or 2 • 3 • b 2 • r
Student 4 2, 3, 3, r, b or 2 • 32 • b • r
•Each student determines the most abbreviated
way to write his or her combination of cards.
Expression: 2 • 3 • r (2r + 2b + b 2 + 3b)
•Each student writes his or her abbreviated hand
on one of the four blank cards. They should write
the numbers first, followed by the variables,
using exponents for any cards that repeat within
the hand. Students should write a multiplication
symbol between each element of their hands to
avoid confusion later (e.g., r, r, 2, 3, 2 should be
written as 2 2 • 3 • r 2).
GCF: 2 • 3 • r or (2)(3)r
Chapter 9 • Objective 2
793
Objective 2
Concept Development
Activities
6.After some practice with the cards, have students
write a factored sentence for the following:
(22 • r • b 2) + (3 • 2 • r 2 • b) + (3 • 2 • r • b 2) +
(22 • r • b 2)
2rb(2b + 3r + 3b + 2b) or 2rb(7b + 3r)
7.Have students write factored sentences for some
expressions.
Sample problems:
8a 2b + 4ab 2 4ab(2a + b)
14a 3b 2c 2 + 7a 2bc 4 7a 2bc 2 (2ab + c 2)
NEXT STEPS • Differentiate
5-Day Instructional Plan:
PA 1, page 796—All students, for additional
practice
3-Day Instructional Plan:
PM 2, page 800—Students who demonstrate
understanding of the concept, to assess progress
PM 1, page 799—Students who need additional
support, to assess progress
794
Chapter 9 • Objective 2
Objective 2
Concept Development
Activities
CD 4
Factoring and Solving
Use with 5-Day or 3-Day Instructional Plan. In this
activity, students understand the Zero Product Property
and use it to solve equations.
DIRECTIONS
1.Write a • b = 0 on the board. Ask students what
numbers make it a true statement. If students have
difficulty, give them a value for a or b, and have
them determine the value of the other variable.
Examples:
factor A monomial that evenly divides a value
6.Write x 2 + 3x = 0 on the board. Elicit from students
that the factored form of the equation is x(x + 3) = 0.
Explain that by factoring the original problem, you
have created an equivalent equation that involves
the multiplication of two terms, which can easily be
solved using the Zero Product Property. Explain that
if x • (x + 3) = 0, then either x = 0 or (x + 3) = 0. Point
out that solving these two equation factors yields
two answers, x = 0 or x = –3.
7.Solve 4x 2 + 6x = 0 with the class. 2x(2x + 3) = 0;
x = 0 or x = – 3
2
If a = 4, what must b equal? 0
8.Have students solve several equations independently.
If b = 6, what must a equal? 0
1
, what must b equal? 0
If a = 16
2.Make sure students recognize that no matter what
value we pick for a or b, in order for a • b = 0 to
be true, one or both of the values must be equal
to zero. Refer to this concept as the Zero Product
Property.
3.Write 2x • y = 0 on the board. Continue discussing
the concept that either x or y must be equal to zero
for this equation to be a true statement. Write 2x = 0
and y = 0 on the board, and explain that one or both
of these equations must be true if 2x • y = 0.
4.Write a + b = 0 on the board. Have students
discuss what combinations work for the equation.
Make sure students recognize that many
combinations work.
5.Review the following term with students:
Examples:
Sample problems:
x 2 − 9x = 0 x = 0, 9
5x + x 2 = 0 x = 0, –5
2x 2 − 6x = 0 x = 0, 3
2x + 2 = 0 x = –1
NEXT STEPS • Differentiate
5-Day Instructional Plan:
PA 2, page 797—Students who are on the
accelerated path, for additional practice
PA 3, page 798—Students who are on the
differentiated path, for additional practice
3-Day Instructional Plan:
PA 3, page 798—All students, for additional
practice
(1, −1) 1 + (−1) = 0
(2, −2) 2 + (−2) = 0
(8, −8) 8 + (−8) = 0
Emphasize the fact that the Zero Product Property
works only for multiplication and that it does not
hold for addition.
Chapter 9 • Objective 2
795
Objective 2
Practice
Activities
PA 1
Drawing Cards
Use with 5-Day or 3-Day Instructional Plan. In this
activity, students find the GCF of polynomials.
MATERIALS
• Blank cards, one per student
Directions
1. Review the following term with students:
factor A monomial that evenly divides a value
2.Label one side of each card with an integer starting
from 1 and continuing to the number that represents
the student population in your class. On the other
side of each card, write a monomial that contains a
random combination of factorable integers between
1 and 60 and the variables x, y, and/or z. Any
variable used may contain an exponent from
1 to 6—for example, 8x 2y 3z 2 . Include cards such
as x 2 and 6x.
3.Have students number off from 1 to whatever your
class population is.
4.Shuffle the cards and randomly select two of them.
Have the two students whose numbers correspond
to the numbers on the cards go up to the board.
Give them their corresponding cards.
5.Tell the two students to write the monomials from
their cards on the board side-by-side with an
addition or subtraction symbol between them—
for example, 8x 2y 3z 2 + 12xy 2z 2 .
6.Have the rest of the class factor the two monomials
and write the GCF on their own papers—for the
example, 4xy 2z 2 . Have students work individually.
7.Repeat this process until all cards have been used.
If students are confident with this task, make it a
competition to see who can get the most correct.
796
Chapter 9 • Objective 2
8.Point out to students that they should have answers
for all but their own problem. Alternatively, give
them time to get back to their seats to complete that
problem as well.
NEXT STEPS • Differentiate
5-Day and 3-Day Instructional Plans:
PM 2, page 800—All students, to assess progress
Objective 2
Practice
Activities
PA 2
Solving Equations
Use with 5-Day Instructional Plan. In this activity,
students solve equations.
Directions
3.Put students in groups of four. Have each student
work independently to solve all the equations for his
or her group on a new sheet of paper. Instruct the
students in each group to compare their answers.
If any of the solutions differ, have the group work
together to determine the correct answer.
1.Write x 2 ± bx = 0 on the board. Tell students to get
out a sheet of paper and write an equation like the
one on the board, but substitute a number for b and
choose either + or – for the equation.
4.Have each group exchange their eight problems
with a different group and solve the new problems.
Continue the sharing of problems as time allows.
NEXT STEPS • Differentiate
Sample answers:
x 2 + 10x = 0
x2 − x = 0
5-Day Instructional Plan:
PM 3, page 801—All students, to assess progress
x 2 + 5x = 0
1
2.Write ax 2 ± bx = 0 on the board. Have students
write an equation like the one on the board, but
substitute numbers for a and b and choose either
+ or – for the equation.
Sample answers:
3x 2 + 2x = 0
4x 2 + 3 x = 0
2
0.5x 2 − x = 0
The students should now have two equations on
their paper.
Chapter 9 • Objective 2
797
Objective 2
Practice
Activities
PA 3
Finding the Solutions
Use with 5-Day or 3-Day Instructional Plan. In this
activity, students solve equations.
MATERIALS
Directions
1.Have students form groups of four, and give each
group 20 blank cards. Write the following equations
on the board, and tell the groups to put one
equation per card on six of the blank cards:
x 2 + 4x = 0
3x + x 2 = 0
2x 2 − 4x = 0
x 2 + 2x = 0
x 2 − 3x = 0
2.Tell students to write the following solutions, one
solution per card, on the remaining 14 cards:
0, 0, 0, 0, 0, 0, 1, −1, 2, −2, 3, −3, 4, −4
3.Explain the game rules to students.
•Each group shuffles their equation cards and their
solution cards separately and deals one equation
card to each student in the group. The two
extra equation cards are set to the side, and the
solution cards are placed in a stack facedown.
•Each student solves the equation he or she was
dealt on a sheet of paper.
•After all students in a group are finished solving
their equations, the dealer takes the top card from
the stack of solution cards and places it next to
the stack, facing up. The student keeps the card if
it is a solution to his or her equation.
798
Chapter 9 • Objective 2
•Play continues clockwise.
•After all solution cards have been selected, each
student must show the group that his or her
solution cards are correct by solving the original
equation to prove it is a true statement.
• Blank cards, 20 per group
5x − 5x 2 = 0
•The next player takes the card facing up or the
card on the top of the stack of cards that are
facedown. He or she keeps the card if it is a
solution to his or her equation.
4.Have students separate the cards into equation
cards and solution cards, shuffle all the equation
cards, and deal the cards again.
NEXT STEPS • Differentiate
5-Day and 3-Day Instructional Plans:
PM 3, page 801—All students, to assess progress
Progress-Monitoring
Activities
PM 1
Apply Skills 1
Use with 5-Day or 3-Day Instructional Plan.
MATERIALS
• Interactive Text, page 342
DIRECTIONS
1.Have students turn to Interactive Text, page 342,
Apply Skills 1.
3.Monitor student work, and provide feedback as
necessary.
Watch for:
•Do students recognize when they have not found
all the factors that can be pulled out?
Name __________________________________________ Date __________________________
A p p ly S k i l l S 1
Factor the polynomials using the greatest common factor (GCF) and the
Distributive Property.
Example:
2x 2 + 6x =
2x(x + 3)
1.
x 2 + 6x =
2.
x 2 – 4x =
3.
x 2 + 5x =
4.
7x – x 2 =
5. x 2 + x =
6.
x 2 – 11x =
7. 3x 2 + x =
8.
2x 2 – 4x =
x(x + 6)
x(x + 5)
x(x + 1)
x(3x + 1)
x(x – 4)
x(7 – x)
x(x – 11)
2x(x – 2)
6x 2 + 8x =
10. 3x 2 + 9x =
11.
m 2n 2 + mn 3 =
12.
m 3n 2 + m 2n 3 =
13.
2m 2n 2 + 8mn 2 =
14.
5m 4n – 40m 5n 3 =
15.
–8m 2n 5 + 32m 2n 3 =
16. 20a 3b 5 + 50a 5b 3 =
9.
2x(3x + 4)
mn 2(m + n)
2mn 2(m + 4)
–8m 2n 3(n 2 – 4)
17. 12a 2b + 18a 3b 3 =
6a 2b(2 + 3ab 2)
19.
342
2a 5b 3 + 22a 5b 2 =
2a 5b 2(b + 11)
Chapter 9
•
Objective 2 • PM 1
3x(x + 3)
m 2n 2(m + n)
5m4n(1 – 8mn 2)
10a 3b 3(2b 2 + 5a 2)
18.
–8a 4b – 48a 2b 2 =
–8a 2b(a 2 + 6b)
20. 14a 3b – 35a 2b 3 =
7a 2b(2a – 5b 2)
Inside Algebra
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
2.Remind students of the key term: greatest common
factor (GCF).
progress moNitoriNg
Objective 2
•Do students compare each factor separately?
NEXT STEPS • Differentiate
5-Day Instructional Plan:
PA 1, page 796—Students who demonstrate
understanding of the concept, for additional
practice
CD 3, page 793—Students who need additional
concept development
3-Day Instructional Plan:
PA 1, page 796—All students, for additional
practice
Chapter 9 • Objective 2
799
Progress-Monitoring
Activities
PM 2
A p p ly S k i l l S 2
Factor the polynomials using the greatest common factor (GCF) and the
Distributive Property.
Example:
6x 2 + 36x =
6x(x + 6)
Apply Skills 2
1.
–2x 2 + 6x =
2.
5x 2 – 15x =
3.
–3x 2 + 12x =
4.
5x 2 + 25x =
5. 5m 2n 3 – 35m 5n =
6.
11mn 2 – 33m 4n =
7. –5m 4n – 20m 5n 3 =
8.
–8m 2n + 24m 2n2 =
Use with 5-Day or 3-Day Instructional Plan.
• Interactive Text, page 343
•Do students find the GCF using number trees,
algebra tiles, or another method?
NEXT STEPS • Differentiate
5-Day and 3-Day Instructional Plans:
CD 4, page 795—All students, for additional
concept development
800
Chapter 9 • Objective 2
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
1.Have students turn to Interactive Text, page 343,
Apply Skills 2.
Watch for:
•Do students correctly factor out negatives?
–3x(x – 4)
–5m 4n(1 + 4mn 2)
5x(x – 3)
5x(x + 5)
11mn(n – 3m 3)
–8m 2n(1 – 3n)
9.
24a 3b 5 + 48ab 3 =
10. 16ab 4 + 32ab 2 =
11.
–9a 4b – 45a 3b 4 =
12.
13a 3b 2 – 26ab =
13.
7x 4y 3 + 14x 2y 5 – 28x 3y 2 =
14.
6x 4y 2 – 24x 3y + 12xy 5 =
15.
–15x 2y – 5xy 4 – 30xy 4 =
16. 10x 4y 3 + 16x 2 – 24y 2 =
DIRECTIONS
3.Monitor student work, and provide feedback as
necessary.
–2x(x – 3)
5m 2n(n 2 – 7m 3)
MATERIALS
2.Remind students of the key term: greatest common
factor (GCF).
progress moNitoriNg
Name __________________________________________ Date __________________________
Objective 2
24ab 3(a 2b 2 + 2)
–9a 3b(a + 5b 3)
7x 2y 2(x 2y + 2y 3 – 4x)
–5xy(3x + 7y 3)
17. –72x 4y + 16x 3y 3 – 20x 2y 2 =
–4x 2y(18x 2 – 4xy 2 + 5y)
19.
14ab 2 + 35a 3b 2 – 7a 3b + 70ab 4 =
7ab(2b + 5a 2b – a 2 + 10b 3)
Inside Algebra
16ab 2(b 2 + 2)
13ab(a 2b – 2)
6xy(x 3y – 4x 2 + 2y 4)
2(5x 4y 3 + 8x 2 – 12y 2)
18.
12a 5b 3 – 3a 4b 3 + 27a 3b 4 + 9a 3b 4 =
3a 3b 3(4a 2 – a + 12b)
20. –32a 4b 3 + 16a 3b 3 – 4a 3b + 24a 2b 2 =
–4a 2b(8a 2b 2 – 4ab 2 + a – 6b)
Chapter 9 • Objective 2 • PM 2
343
progress moNitoriNg
Objective 2
Progress-Monitoring
Activities
PM 3
Apply Skills 3
Use with 5-Day or 3-Day Instructional Plan.
MATERIALS
• Interactive Text, pages 344–345
DIRECTIONS
1.Have students turn to Interactive Text, pages
344–345, Apply Skills 3.
A p p ly S k i l l S 3
Factor the polynomials using the greatest common factor (GCF) and
the Distributive Property.
Example:
3x 2 + 6x =
2x(x – 7)
3.
–3x 2 + 12x =
5x(x + 11)
5. 36x + 9x 2 =
2x(x + 5)
2.
2x 2 – 14x =
4.
55x + 5x 2 =
5x(11 + x)
9x(4 + x)
Example:
3x
3
3x 2 – 6x = 0
3x(x – 2) = 0
either 3x = 0 or x – 2 = 0
9x(x + 4)
= 03 or x – 2 = 0
+2 +2
6. x 2 + 5x = 0
7. x 2 – 3x = 0
8. 4x 2 + 8x = 0
9. 3x 2 – 12x = 0
x(x + 5) = 0
x = 0 or x + 5 = 0
x = –5
x = 0, –5
344
x(x – 3) = 0
x = 0 or x – 3 = 0
x=3
x = 0, 3
Chapter 9
•
3x(x – 4) = 0
3x = 0 or x – 4 = 0
x = 0 or x = 4
x = 0, 4
Objective 2 • PM 3
Inside Algebra
Name __________________________________________ Date __________________________
A p p ly S k i l l S 3
10. 9x 2 + 27x = 0
11. x 2 + 8x = 0
12. 2x 2 + x = 0
13. 10x 2 – 5x = 0
9x(x + 3) = 0
9x = 0 or x + 3 = 0
x = 0 or x = –3
x = 0, –3
NEXT STEPS • Differentiate
5-Day and 3-Day Instructional Plans:
PS 1, page 802—Students who are on the
accelerated path, to develop problem-solving skills
x(2x + 1) = 0
x = 0 or 2x + 1 = 0
2x = –1
Objective 2 Posttest, page 803—Students who are
on the differentiated path
x = –1
2
x = 0, –1
2
14. 12x 2 + 4x = 0
4x(3x + 1) = 0
4x = 0 or 3x + 1 = 0
x = 0 or 3x = –1
x = –1
3
x = 0, –1
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
(continued )
Inside Algebra
3
x(x + 8) = 0
x = 0 or x + 8 = 0
x = –8
x = 0, –8
5x(2x – 1) = 0
5x = 0 or 2x – 1 = 0
x = 0 or 2x = 1
x=1
2
x = 0, 1
2
15. 27x 2 – 18x = 0
9x(3x – 2) = 0
9x = 0 or 3x – 2 = 0
x = 0 or 3x = 2
x=2
3
x = 0, 2
3
Chapter 9 • Objective 2 • PM 3
progress moNitoriNg
•Do students see that the product of two numbers
will be zero if and only if at least one of those
numbers is zero?
–3x(x – 4)
x = 0 or x = 2
x = 0, 2
4x(x + 2) = 0
4x = 0 or x + 2 = 0
x = 0 or x = –2
x = 0, –2
Watch for:
•Do students remember the Zero Product Property
of multiplication?
3x(x + 2)
Factor the polynomials and solve them using the Zero Product Property.
3.Monitor student work, and provide feedback as
necessary.
1.
2x 2 + 10x =
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
2.Remind students of the key term: greatest common
factor (GCF).
Name __________________________________________ Date __________________________
345
Chapter 9 • Objective 2
801
Objective 2
Problem-Solving
Activity
Using the Length and Width
of a Rectangle
★ PS
1
Use with 5-Day or 3-Day Instructional Plan. In this activity,
students solve geometric problems involving area.
DIRECTIONS
6.Point out that the x to the left of the parentheses
in x(x + 3) is the GCF of x 2 and 3x .
1.Draw a 6 by 3 rectangle on the board. Do not
include the measurements.
Tell students to find the length, width, and total area
in square units. Make sure students know the
formula is area = length • width. Invite students to
count the square units if they want to check their
work. Length = 6 units, width = 3 units, area =
18 square units
2.Draw the rectangle at right on
the board. Point out that the
areas of the interior spaces are
labeled. Ask students to think
about what the dimensions are.
x2
x
3
4.Tell students that if the width of
the left rectangle is x units, then
the width of the right rectangle x
must also be x units because
both rectangles have the same
width. Remind students of the area formula. Point
out that because the area of the right part of the
rectangle is 3x and the width is x units, the length
must be 3. Label the right part of the rectangle with
the length.
802
Chapter 9 • Objective 2
7.Have students find the lengths and widths of more
rectangles with given areas of the interior spaces.
Remind students that because the left rectangle
in each problem has square units, they can assume
it is a square.
Sample problems:
4x2
14x
3x
x
3.Tell students to think of the
left part of the rectangle as a
square. Remind students that
x
if it is a square, its dimensions
must be equal. Therefore, both
the length and the width of the left part of the
rectangle are x units long. Label the left part of the
rectangle with these measurements.
★ = Includes Problem Solving
5.Point out that the area of the rectangle is x 2 + 3x
because we add the small areas together to find the
large area. Tell students we can double-check this
area by multiplying the length, x + 3, by the width,
x . Have students substitute the length and width
into the area formula to find the total area
of the rectangle. x(x + 3) = x 2 + 3x
x2
5x
length = 2x + 7
width = 2x
length = x + 5
width = x
9x2
15x
length = 3x + 5
width = 3x
8.Tell students to find the dimensions of more
rectangles without the visual drawings.
Sample problems:
x 2 + 7x length = x + 7, width = x
x 2 + 40x length = x + 40, width = x
2x 2 + 8x length = x + 4, width = 2x
NEXT STEPS • Differentiate
5-Day and 3-Day Instructional Plans:
Objective 2 Posttest, page 803—All students
CHAPTER
9
Objective 2
Ongoing Assessment
Objective 2 Posttest
Discuss with students the key concepts in Objective 2.
Following the discussion, administer the Objective 2
Posttest to all students.
Using the Results
• Score the posttest and update the class record card.
• Provide reinforcement for students who do not
demonstrate mastery of the concepts through individual
or small-group reteaching of key concepts.
4x 2 + 2x
2.
30a 3b 5 + 20ab 3
3.
–12m 2n 3 – 4m 3n
4.
18cd 4 – 9c 3d 2 + 36c 2d 3
5.
14x 5y 3 – 7x 3y 4 + 21x 4y 2
7.
12x 2 + 6x = 0
1.
2x(2x + 1)
–4m 2n(3n 2 + m)
10ab 3(3a 2b 2 + 2)
9cd 2(2d 2 – c 2 + 4cd)
Chapter 9 • Objective 2
Factor the polynomials using the greatest common factor (GCF) and the
distributive property.
Posttest
Name __________________________________________ Date ____________________________
7x 3y 2(2x 2y – y 2 + 3x)
Solve the equations.
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
6.
–4x 2 + 28x = 0
–4x(x – 7) = 0
x = 0, 7
8.
5b 2 + 15b = 0
10.
12x 2 – 6x = 0
5b(b + 3) = 0
b = 0, –3
6x(2x + 1) = 0
x = 0, –1
2
9.
0 = 8a 2 – 24a
8a(a – 3) = 0
a = 0, 3
6x(2x – 1) = 0
x = 0, 1
2
Inside Algebra
Chapter 9 • Objective 2
127
Chapter 9 • Objective 2
803