9-2
9-2
Multiplying and Factoring
1. Plan
Objectives
1
2
To multiply a polynomial by
a monomial
To factor a monomial from a
polynomial
Examples
1
2
3
Multiplying a Monomial and
a Trinomial
Finding the Greatest Common
Factor
Factoring Out a Monomial
Math Background
What You’ll Learn
Check Skills You’ll Need
• To multiply a polynomial by
Multiply.
a monomial
GO for Help
1. 3(302) 906
• To factor a monomial from
a polynomial
Lesson 2-4
2. 41(7) 287
3. 9(504) 4536
Simplify each expression.
. . . And Why
4. 4(6 + 5x) 24 ± 20x
5. -8(2y + 1) –16y – 8
6. (5v - 1)5 25v – 5
To factor a monomial out of a
polynomial, as in Example 3
7. 7(p - 2) 7p – 14
8. (6 - x)9 54 – 9x
9. -2(4x - 1)
–8x ± 2
1
1 1 Distributing a Monomial
Part
The process of applying the
distributive property with
polynomials combines the
distributive property for real
numbers with the rules for
multiplying powers. Factoring
reverses this process.
In Chapter 2 you used the Distributive Property to multiply a number by a sum
or difference.
More Math Background: p. 492C
2x(3x + 1) = 2x(3x) + 2x(1)
5(a + 3) = 5a + 15 (x - 2)(3) = 3x - 6
-2(2y + 7) = -4y - 14
You can also use the Distributive Property
or an area model to multiply polynomials.
Consider the product 2x(3x + 1).
See p. 492E for a list of the
resources that support this lesson.
1
x2
x2
x2
x
x2
x2
x2
x
You can use the Distributive Property for multiplying powers with the same base
when multiplying by a monomial.
4
5
A
A
B
B
E
D
C
D
E
E
Test-Taking Tip
PowerPoint
When you distribute a
negative expression
such as -4y2, be sure
to apply the negative
sign to each term
within the
parentheses.
Bell Ringer Practice
Check Skills You’ll Need
For intervention, direct students to:
The Distributive Property
EXAMPLE
Multiplying a Monomial and a Trinomial
E
D
C
1
E
D
C
B
E
D
C
B
A
D
C
B
A
3
C
B
A
1
2
Lesson 2-4: Examples 1, 3
Extra Skills and Word
Problem Practice, Ch. 2
Multiple Choice Simplify -4y 2 A 5y 4 - 3y 2 + 2 B .
-20y6 + 12y 4 - 8y 2
-20y6 - 12y 4 - 8y 2
-20y8 + 12y 4 - 8y 2
-20y6 + 12y 4 + 8y 2
-4y 2 A 5y4 - 3y2 + 2 B
= -4y 2 A 5y4 B - 4y 2 A -3y2 B - 4y 2(2)
Use the Distributive Property.
= -20y 2 + 4 + 12y 2 + 2 - 8y 2
Multiply the coefficients and add the
exponents of powers with the same base.
= -20y6 + 12y 4 - 8y 2
Simplify.
So A is the correct answer.
Quick Check
500
1 Simplify each product.
a. 4b A 5b2 + b2 + 6 B
20b3 ± 4b2 ± 24b
b. -7h A 3h2 - 8h - 1 B
–21h3 ± 56h2 ± 7h
c. 2x A x2 - 6x + 5B
2x3 – 12x2 ± 10x
Chapter 9 Polynomials and Factoring
Special Needs
Below Level
L1
Some students have difficulty using variables. In
Example 2 have them find the GCF of the terms of a
parallel expression such as 4(5)3 + 12(5)2 - 8(5). Point
out the similarity of this example to Example 2.
500
⫹
2x
= 6x2 + 2x
Lesson Planning and
Resources
3x
learning style: visual
L2
To help students keep track of which terms they have
already multiplied, suggest that they arrange their
work carefully.
learning style: verbal
2
1
2. Teach
Factoring a Monomial From a Polynomial
Factoring a polynomial reverses the multiplication process. To factor a monomial
from a polynomial, first find the greatest common factor (GCF) of its terms.
Vocabulary Tip
The greatest common
factor (GCF) is the
greatest factor that
divides evenly into each
term of an expression. See
Skills Handbook page 755.
2
1
Finding the Greatest Common Factor
EXAMPLE
Guided Instruction
List the prime factors of each term. Identify the factors common to all terms.
4x 3 = 2 ? 2 ? x ? x ? x
12x 2 = 2 ? 2 ? 3 ? x ? x
PowerPoint
8x = 2 ? 2 ? 2 ? x
Additional Example
The GCF is 2 ? 2 ? x or 4x.
1 Simplify -2g2(3g 3 + 6g - 5).
2 Find the GCF of the terms of each polynomial.
a. 5v 5 + 10v 3 5v3
b. 3t 2 - 18 3
c. 4b 3 - 2b 2 - 6b 2b
–6g5 – 12g3 ± 10g2
3
To factor a polynomial completely, you must factor until there are no common
factors other than 1.
3
Factor 3x 3 - 12x 2 + 15x.
Step 1 Find the GCF.
Step 2 Factor out the GCF.
3x 3 = 3 ? x ? x ? x
15x = 3 ? 5 ? x
The GCF is 3 ? x or 3x.
Quick Check
c.
- 24m
6m(m2 – 2m – 4)
6m 3
12m 2
For more exercises, see Extra Skill and Word Problem Practice.
Practiceand
andProblem
ProblemSolving
Solving
Practice
A
Practice by Example
Example 1
GO for
Help
(page 500)
Simplify each product. 5–12. See back of book.
1. 8m(m + 6) 8m2 ± 48m 2. (x + 10)3x 3x2 ± 30x 3. 9k(7k + 4) 63k2 ± 36k
4. -5a(a - 1) –5a2 ± 5a 5. 2x 2(9 + x)
7.
10.
Example 2
(page 501)
2x A 6x 3
-
-7q 2 A 6q 5
x2
+ 5x B
- 2q - 7 B
6. -p 2(p - 11)
8.
4y 2 A 9y 3
+
8y 2
11.
-3g7 A g 4
-
6g 2
- 11 B
9. -5c 3 A 9c 2
+ 5B
12. -4x 6 A 10x 3
- 8c - 5 B
+ 3x 2 - 7 B
Find the GCF of the terms of each polynomial.
13. 15w + 21 3
14. 6a 2 - 8a 2a
15. 36v + 24 12
16. x 3 + 7x 2 - 5x x
17. 5b 3 + 15b - 30 5
18. 9x 3 - 6x2 + 12x 3x
Lesson 9-2 Multiplying and Factoring
Advanced Learners
Additional Examples
2 Find the GCF of the terms of
2x 4 + 10x 2 - 6x. 2x
3 Factor 4x 3 - 8x 2 + 12x.
3 Use the GCF to factor each polynomial.
b. 5d 3 + 10d
a. 8x 2 - 12x
4x(2x – 3)
5d(d 2 ± 2)
EXERCISES
Alternative Method
PowerPoint
3x 3 - 12x 2 + 15x
= 3x A x 2 B + 3x(-4x) + 3x(5)
= 3x A x 2 - 4x + 5 B
12x 2 = 2 ? 2 ? 3 ? x ? x
EXAMPLE
Help students learn to factor
expressions mentally by scanning.
They should first scan the
coefficients and find their GCF.
Next, they should scan for the
least power of the variable.
Factoring Out a Monomial
EXAMPLE
Error Prevention
Remind students that the
negative sign is part of the
monomial. It is -4y 2 that must be
distributed.
Find the GCF of the terms of 4x3 + 12x2 - 8x.
Quick Check
EXAMPLE
4x(x 2 – 2x ± 3)
Resources
• Daily Notetaking Guide 9-2 L3
• Daily Notetaking Guide 9-2—
L1
Adapted Instruction
Closure
Ask students how to find the GCF
of a polynomial. First, list the
prime factors of each coefficient.
Then identify the factors
common to all coefficients.
Finally, write the product of these
common factors with the least
power of the variable used in any
term.
501
English Language Learners ELL
L4
Help students understand that factoring and
multiplying polynomials are each applications of the
distributive property.
learning style: verbal
In Exercise 43, be sure students understand the
meaning of diagonal and vertex and vertices by
having them identify these terms on the diagram.
learning style: visual
501
3. Practice
Example 3
(page 501)
Assignment Guide
C Challenge
B
Apply Your Skills
13-24, 33-42
43, 44
Test Prep
Mixed Review
19. 6x - 4 2(3x – 2)
2t 2
-
10t 4 2t2(1
–
20.
Connection to History
Exercise 33 European castles
M
2x
4x
34. 9m 12 - 36m7 + 81m 5
35. 24x 3 - 96x 2 + 48x
36. 16n 3 + 48n 2 - 80n
37. 5x 4 + 4x 3 + 3x 2
38. 13ab 3 + 39a 2b 4
39. 7g 2k 3 - 35g 5k 2
40. Critical Thinking The GCF of two numbers p and q is 5. What is the GCF of p 2
and q 2? Explain your answer. 25; 52 ≠ 25.
41. a. Factor n 2 - n. n(n – 1)
b. Writing Suppose n is an integer. Is n 2 - n always, sometimes, or never even?
Justify your answer. Always; the product of two consecutive integers
is always even, since one of the integers is even.
42. A triangular number is a number you can represent with a triangular
arrangement of objects. A triangular number can also be written as a product of
two factors, as in the table.
a. Find the values of a, b, c, and d, and then write an expression in factored
form for the nth triangular number. 1; 2; 3; 4; n
2 (n ± 1)
Math Tip
Exercise 44 Remind students that
“in terms of s” means that s is the
variable in the formula.
1
2
3
4
1
3
6
10
a
2 (a ⫹ 1)
b
2 (b ⫹ 1)
c
2 (c ⫹ 1)
d
2 (d ⫹ 1)
Triangular
Number
GO
L4
Practice
Name
L1
Class
Date
Practice 9-2
nline
Homework Help
Visit: PHSchool.com
Web Code: ate-0902
L2
Adapted Practice
C
Challenge
L3
Multiplying and Factoring
Simplify each product.
1. 4(a - 3)
4. 4x3(x - 3)
7. -x2(-2x2 + 3x - 2)
10. a2(2a + 4)
2. -5(x - 2)
5. -5x2(x2 + 2x + 1)
8. 4d2(d2 - 3d - 7)
11. 4(x2 - 3) + x(x + 1)
3. -3x2(x2 + 3x)
6. 3x(x2 - 5x - 3)
9. 5m3(m + 6)
12. 4x(5x - 6)
Find the GCF of the terms of each polynomial.
13. 8x - 4
14. 15x + 45x2
15. x2 + 3x
16. 4c3 - 8c2 + 8
17. 12x - 36
18. 12n3 + 4n2
19. 14x3 + 7x2
20. 8x3 - 12x
21. 9 - 27x3
22. 25x3 - 15x2
23. 11x2 - 33x
24. 4n4 + 6n3 - 8n2
25. 8d3 + 4d2 + 12d
26. 6x2 + 12x - 21
27. 8g2 + 16g - 8
28. 8x + 10
29. 12n3 - 8n
30. 14d - 2
31. 6h2 - 8h
32. 3z4 - 15z3 - 9z2
33. 3y3 - 8y2 - 9y
34. x3 - 5x2
35. 8x3 - 12x2 + 4x
36. 7x3 + 21x4
37. 6a3 - 12a2 + 14a
38. 6x4 + 12x2
39. 3n4 - 6n2 + 9n
40. 2w3 + 6w2 - 4w
41. 12c3 - 30c2
42. 2x2 + 8x - 14
43. 4x3 + 12x2 + 16x
44. 16m3 - 8m2 + 12m
45. 4a3 - 20a2 - 8a
46. 18c4 - 9c2 + 7c
47. 6y4 + 9y3 - 27y2
48. 6c2 - 3c
29. 12c A -5c 2 + 3c - 4 B
Factor each polynomial. 34–39. See margin.
usually consisted of two
concentric square or triangular
shaped systems of steep walls and
round towers surrounding an
inner courtyard. Outside the
outer wall, a deep dry ditch or a
moat served as defense, garbage
dump, and sewer. Moats also
contained eels and fish for food.
Reteaching
28. -7p2 A -2p3 + 5p B
30. y(y + 3) - 5y(y - 2) 31. x 2(x + 1) - x A x 2 - 1 B 32. 4t A 3t 2 - 4t B - t(7t )
–4y2 ± 13y
x2 ± x
12t 3 – 23t2
33. Building Models Suppose you are building a model of the square castle shown
GPS at the left. The moat of the model castle is made of blue paper.
a. Find the area of the moat using the diagram with the photo. A ≠ 16πx2 – 4x2
b. Write your answer in factored form. A ≠ 4x2(4π – 1)
To check students’ understanding
of key skills and concepts, go over
Exercises 10, 22, 32, 33, 41.
Enrichment
+ 4v v(v ± 4)
5t 2)
27. -3a A 4a 2 - 5a + 9 B
Homework Quick Check
L3
5(2x3 – 5x2 ± 4)
21. 10x 3 - 25x 2 + 20
Simplify. Write in standard form. 27–29. See margin.
45-50
51-67
GPS Guided Problem Solving
v2
23. 15n 3 - 3n2 + 12n
24. 6p6 + 24p5 + 18p 3
2
3n(5n – n ± 4)
6p3(p3 ± 4p2 ± 3)
25. Error Analysis Kevin said that -2x(4x - 3) = -8x2 - 6x. Karla said that
-2x(4x - 3) = -8x 2 + 6x. Who is correct? Explain. Karla; Kevin multiplied
–2x by 3 instead of –3.
26. Open-Ended Write a polynomial that has a common factor in each term.
Factor your polynomial. See margin.
22.
1 A B 1-12, 25-32
2 A B
Factor each polynomial.
Factored
Form
b. Use the expression you wrote to find the 100th triangular number. 5050
43. a. Geometry How many sides does the polygon have? How many of its
diagonals come from one vertex? 6; 3
b. Suppose a polygon has n sides. How many diagonals
will it have from one vertex? n – 3
c. The number of diagonals from all the vertices is
3
n
1 2
2 (n 2 3) . Multiply the two factors. 2 n – 2 n
d. For a polygon with 8 sides, what is the total number of
diagonals that can be drawn from the vertices? 20
© Pearson Education, Inc. All rights reserved.
Factor each polynomial.
49. A circular pond will be placed on a square piece of land. The length of a
side of the square is 2x. The radius of the pond is x. The part of the square
not covered by the pond will be planted with flowers. What is the area of
the region that will be planted with flowers? Write your answer in
factored form.
50. A square poster of length 3x is to have a square painting centered on it.
The length of the painting is 2x. The area of the poster not covered by
the painting will be painted black. What is the area of the poster that
will be painted black?
51.
The formula for the surface area of a sphere is A = 4pr 2. A square
sticker of side x is placed on a ball of radius 3x. What is the surface
area of the sphere not covered by the sticker? Write your answer in
factored form.
502
502
Chapter 9 Polynomials and Factoring
pages 501–503
Exercises
28. 14p5 – 35p3
36. 16n(n2 ± 3n – 5)
26. Answers may vary.
Sample: 8x3 ± 12x2 ±
24x; 4x(2x2 ± 3x ± 6)
29. –60c3 ± 36c2 – 48c
37. x2(5x2 ± 4x ± 3)
34. 9m5(m7 – 4m2 ± 9)
38. 13ab3(1 ± 3ab)
27. –12a 3 ± 15a2 – 27a
35. 24x(x2 – 4x ± 2)
39. 7g2k2(k – 5g3)
4s
s
4s
48 in.
4s
44. Manufacturing The diagram shows a cube of metal with a cylinder cut out of
it. The formula for the volume of a cylinder is V = pr 2h, where r is the radius
and h is the height.
a. Write a formula for the volume of the cube in terms of s. V ≠ 64s3
b. Write a formula for the volume of the cylinder in terms of s. V ≠ 48πs2
c. Write a formula in terms of s for the volume V of the metal left after the
cylinder has been removed. V ≠ 64s3 – 48πs2
d. Factor your formula from part (c). V ≠ 16s2(4s – 3π)
e. Find V in cubic inches for s = 15 in. about 182,071 in.3
Alternative Assessment
48. Which of the following represents an odd number for any integer n? G
F. n + 1
G. 2n + 1
H. 3n
J. 3n + 1
49. Factor 6g 8 - 3g 4 + 9g 2 completely. B
A. g 2 A 6g 4 - 3g 2 + 9 B
C. g 2 A 6g 6 - 3g 2 + 9g B
B. 3g 2 A 2g 6 - g 2 + 3 B
D. 3g 2 A 2g 6 - g 2 + 3g B
50. How do you know if you’ve factored out the GCF of a polynomial?
Illustrate your explanation by using the GCF to factor 10x 4 + 6x 3 + 2x 2.
See margin.
Mixed Review
Lesson 9-1
Lesson 8-1
Organize students into groups of
three. Using the examples in this
lesson, have each group write a
similar polynomial and its factors
on cards. For example, students
should write 6x2 - 4x on one
card, the factor 2x on one card,
and the factor (3x - 2) on a third
card. Collect all the cards from the
class and shuffle them. Randomly
redistribute the cards to the class.
Have students find their matching
cards. Tell students not to be
alarmed if there is more than
one card with the same factor
on it. Each factor will have a
polynomial to match it.
Simplify. Write each answer in standard form. 52–54. See margin.
51. A x2 + 3 B - A 4x2 - 7 B –3x2 ± 10
52. A m3 + 8m + 6 B + A -5m2 + 4m B
53. A g 2 + 6g - 2 B + A 4g 2 - 7g + 2 B
54. A 3r 2 - 8r + 7 B - A 2r 2 + 8r - 9 B
55. A t 4 - t 3 + 1 B + A t 3 + 5t 2 - 10 B
56. A 3b 3 - b 2 B - A 5b2 + 12 B
t 4 ± 5t 2 – 9
Simplify each expression.
57. 5-1 15
2
61. n-3m2 m3
n
Lesson 7-3
1. Simplify
-2x2(-3x2 + 2x + 8).
6x4 – 4x3 – 16x2
4. Factor 10y3 + 5y2 - 15y.
5y(2y2 ± y – 3)
47. Let n represent the number of different pairs of integers whose
product is n. For example, -1 3 10, -2 3 5, 1 3 (-10), and 2 3 (-5)
give -10. So -10 = 4. What does -24 equal? C
A. 4
B. 6
C. 8
D. 10
GO for
Help
Lesson Quiz
3. Factor 3x3 + 9x2.
3x2(x ± 3)
45. x A 6x 2 - 4x - 2 B equals which of the following expressions? B
A. 6x 3 - 4x - 2
B. 6x 3 - 4x 2 - 2x
D. 7x 3 - 5x 2 - 3x
C. 6x 3 - 4x 2 - 2
46. Simplify A p 2 - 3 B - A 5 - p + 2p 2 B - A 4p + 5 - 2p 2 B . What is the
coefficient of p 2? F
F. 1
G. 2
H. 4
J. 5
Short Response
PowerPoint
2. Find the GCF of
16b4 - 4b3 + 8b2. 4b2
Standardized
Test Prep
Test Prep
Multiple Choice
4. Assess & Reteach
3
62. v3w-5 v 5
w
59. (-2)-3 –18
4 4c3
63. 23
c
60. 80 1
28
64. ab 5 8a 5
c
b c
Solve by elimination.
65. 7x + 6y = 33
2x - 6y = - 6 (3, 2)
lesson quiz, PHSchool.com, Web Code: ata-0902
50. [2] 2x2(5x2 ± 3x ± 1); the
terms 5x2, 3x, and 1
have no common
factor other than 1.
Resources
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 545
• Test-Taking Strategies, p. 540
• Test-Taking Strategies with
Transparencies
3b3 – 6b2 – 12
1
58. 5-2 25
Test Prep
Exercise 48 Remind students that
66. 8x + 4y = 28
3x - 2y = 21 (5, –3)
67. 4x + 2y = 16 (1, 6)
11x - 3y = - 7
Lesson 9-2 Multiplying and Factoring
[1] incorrect factoring OR
incorrect explanation
503
any odd number is one more than
an even number. 2n always
represents an even number for
any integer n.
52. m3 – 5m2 ± 12m ± 6
53. 5g2 – g
54. r 2 – 16r ± 16
503