Download Multiplying Polynomial Expressions

LESSON
18.2
Name
Multiplying Polynomial
Expressions
Class
Date
18.2 Multiplying Polynomial
Expressions
Essential Question: How do you multiply binomials and polynomials?
Resource
Locker
Common Core Math Standards
The student is expected to:
COMMON
CORE
Explore
A-APR.A.1
Understand that polynomials form a system analogous to the integers,
namely, they are closed under the operations of ... multiplication… Also
A-SSE.A.1, A-CED.A.1
Using algebra tiles to model the product of two binomials is very similar to using algebra tiles to model the product of
a monomial and a polynomial.
Rules
Mathematical Practices
COMMON
CORE
MP.4 Modeling
Language Objective
Explain to a partner what FOIL means and how you use the FOIL method
to multiply two binomials.
1.
The first factor goes on the left side of the grid, and the second
factor goes on the top.
2.
Fill in the grid with tiles that have the same height as tiles on the left
and the same length as tiles on the top.
3.
Follow the key. The product of two tiles of the same color is positive;
the product of two tiles of different colors is negative.
Use algebra tiles to model (x + 1)(x - 2). Then write the product. First fill in the factors and mat.
ENGAGE
View the Engage section online. Discuss what types
of herbs and vegetables one might plant in a backyard
garden. Then preview the Lesson Performance Task.
Key
= x2
x+1
= -x2
© Houghton Mifflin Harcourt Publishing Company
PREVIEW: LESSON
PERFORMANCE TASK
x-2
×
Essential Question: How do you
multiply binomials and polynomials?
Use the Distributive Property to multiply binomials
and polynomials. Use the FOIL method to multiply
binomials.
Modeling Binomial Multiplication
=x
= -x
= -1
=1
Now remove any zero pairs.
×
The product (x + 1) (x - 2) in simplest form is 1 x 2 - 1 x - 2 .
Module 18
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Lesson 2
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HARDCOVER PAGES 669676
Quest
Essential
COMMON
CORE
A1_MNLESE368187_U7M18L2 855
Turn to these pages to
find this lesson in the
hardcover student
edition.
and the second
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Module 18
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Lesson 18.2
L2 855
7_U7M18
SE36818
A1_MNLE
(x - 2) in
is
simplest form
.
2
1 x- 2
1 x -
Lesson 2
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6:04 PM
20/08/14 6:21 PM
Reflect
1.
EXPLORE
Discussion Why can zero pairs be removed from the product?
A zero pair is two tiles with a sum of zero. This means that they are adding zero to the
Modeling Binomial Multiplication
product, so they can be removed without changing the product.
2.
Discussion Is it possible for more than one pair of tiles to form a zero pair?
Yes. There is no limit to the number of zero pairs.
Explain 1
INTEGRATE TECHNOLOGY
Students have the option of completing the algebra
tiles activity either in the book or online.
Multiplying Binomials Using the Distributive
Property
To multiply a binomial by a binomial, the Distributive Property must be applied more than once.
Example 1

QUESTIONING STRATEGIES
Multiply by using the Distributive Property.
What do the algebra tiles in the grid
represent? The tiles in the grid represent the
product of the factors shown above and to the left
of the grid.
(x + 5)(x + 2)
(x + 5)(x + 2) = x(x + 2) + 5(x + 2)
Distribute.
= x(x + 2) + 5(x + 2)
Redistribute and simplify.
= x(x) + x(2) + 5(x) + 5(2)
Why can you remove pairs of like tiles? A
positive (yellow) shape matched with a
negative (red) shape of equal size will result in a
zero pair, so a total of 0 is being erased.
= x + 2x + 5x + 10
2
= x 2 + 7x + 10

(2x + 4)(x + 3)
(2x + 4)(x + 3) = 2x(x + 3) + 4 (x + 3)
Distribute.
= 2x(x + 3) + 4 (x + 3)
© Houghton Mifflin Harcourt Publishing Company
= 2x(x) + 2x (3) + 4 (x) +
4 (3)
= 2 x 2 + 6 x + 4 x + 12
= 2 x + 10 x + 12
2
Your Turn
3.
EXPLAIN 1
Redistribute and simplify.
(x + 1) (x - 2)
(x + 1) (x - 2) = x(x - 2) + 1(x - 2)
Multiplying Binomials Using the
Distributive Property
QUESTIONING STRATEGIES
Is the Commutative Property of Multiplication
true for the multiplication of two binomials?
Explain. Yes, the product will be the same
regardless of the order in which the binomials are
multiplied and regardless which binomial is
distributed across the other.
= x(x) + x(-2) + 1(x) + 1(-2)
= x 2 - 2x + x - 2
= x2 - x - 2
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Lesson 2
PROFESSIONAL DEVELOPMENT
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Learning Progressions
In this lesson, students use what they have learned about adding and subtracting
polynomials and multiplying polynomials by monomials to find products of larger
polynomials. Some key understandings for students are that the FOIL method can
be used to multiply any two binomials, and the Distributive Property can be used
to multiply any size or degree of polynomial. Students make connections between
operations with numbers and operations with polynomials. As students progress,
they will extend this understanding to multiplying and dividing a variety of
polynomial expressions.
AVOID COMMON ERRORS
3/24/14 7:13 PM
Students sometimes overlook the subtraction sign in
binomials of the form ax − b. It may be helpful to
have students rewrite the binomial as ax + (−b) so
that it is clear to them that the constant in the
binomial is negative. Highlighting or circling negative
constants and coefficients is also a good idea so that
students pay attention to them as they multiply
binomials.
Multiplying Polynomial Expressions
856
Explain 2
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Technology
MP.5 Have students use graphing to check
Multiplying Binomials Using FOIL
Another way to use the Distributive Property is the FOIL method. The FOIL method uses the Distributive Property
to multiply terms of binomials in this order: First terms, Outer terms, Inner terms, and Last terms.
Example 2

their products. For example, use a graphing
calculator to display the graphs of
y 1 = (x + 3)(x + 2) and y 2 = x 2 + 5x + 6 in the
same viewing window. Because the graphs coincide,
they confirm that the product of x + 3 and x + 2 is
x 2 + 5x + 6.
Multiply by using the FOIL method.
(x 2 + 3) (x + 2)
Use the FOIL method.
(x 2 + 3) (x + 2) = (x 2 + 3) (x + 2)
= (x 2 + 3) (x + 2)
F
Multiply the first terms. Result: x 3
O
Multiply the outer terms. Result: 2x 2
I
Multiply the inner terms. Result: 3x
L
Multiply the last terms. Result: 6
= (x 2 + 3) (x + 2)
= (x 2 + 3) (x + 2)
Add the result.
(x 2 + 3)(x + 2) = x 3 + 2x 2 + 3x + 6
EXPLAIN 2

(3x 2 - 2x) (x + 5)
Use the FOIL method.
(3x 2 - 2x) (x + 5) = (3x 2 - 2x) (x + 5)
Multiplying Binomials Using FOIL
= (3x 2 - 2x) (x + 5)
= (3x 2 - 2x) (x + 5)
QUESTIONING STRATEGIES
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Point out that the sign of the last term in the
product will be determined by the signs from the last
term in each binomial. It will be positive if the signs
of the last terms are the same, and it will be negative
if the signs are different.
Multiply the first terms. Result: 3x 3
O
Multiply the outer terms. Result: 15x 2
I
Multiply the inner terms. Result:
L
Multiply the last terms. Result:
-2x 2
-10x
Add the result.
(3x 2 - 2x)(x + 5) = 3 x 3 + 13 x 2 - 10 x
Reflect
© Houghton Mifflin Harcourt Publishing Company
What terms from FOIL can often be
combined? Explain. The product of the inner
terms and the product of the outer terms can often
be combined because they are like terms.
= (3x 2 - 2x) (x + 5)
F
4.
The FOIL method finds the sum of four partial products. Why does the result from part B only
have three terms?
When there are like terms to combine, the result will have fewer than four terms.
5.
Can the FOIL method be used for numeric expressions? Give an example.
Sample answer: The FOIL method can be used for numeric expressions. For example, when
multiplying 55 × 47, you can rewrite as (50 + 5)(40 + 7) and use FOIL:
(50 × 40 ) + (50 × 7) + (5 × 40) + (5 + 7) = 2000 + 350 + 200 + 35
= 2585
Your Turn
6.
(x 2 + 3) (x + 6)
(x 2 + 3)(x + 6)
= x 3 + 6x 2 + 3x + 18
AVOID COMMON ERRORS
Students need to remember that any like terms that
result from a FOIL expansion need to be combined.
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Lesson 2
COLLABORATIVE LEARNING
A1_MNLESE368187_U7M18L2.indd 857
Small Group Activity
Show students that they can also multiply two polynomials vertically or by using a
table. Then have students work in small groups to multiply two polynomials, such
as (x + 3)(x 2 − 5x + 2). Each student in the group should choose a different
method, such as multiplying horizontally, multiplying vertically, or using a table.
Have students discuss the ways in which the methods are alike and the ways in
which they differ.
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Explain 3
Multiplying Polynomials
EXPLAIN 3
To multiply polynomials with more than two terms, the Distributive Property must be used several times.
Example 3

Multiply the polynomials.
Multiplying Polynomials
(x + 2) (x 2 - 5x + 4)
(x + 2) (x 2 - 5x + 4) = x (x 2 - 5x + 4) + 2(x 2 - 5x + 4)
Distribute.
= x( x - 5x + 4) + 2(x - 5x + 4)
2
2
QUESTIONING STRATEGIES
Redistribute.
= x(x 2) + x(-5x) + x(4) + 2(x 2) + 2(-5x) + 2(4)
Why can’t you use the FOIL method for
multiplying a binomial by a trinomial or a
larger polynomial? Since one of the factors has
more than two terms, you would miss some terms if
you used the FOIL method. Terms between the first
and last term of the polynomial would not be
included in the multiplication.
Simplify.
= x 3 - 5x 2 + 4x + 2x 2 - 10x + 8
= x 3 - 3x 2 - 6x + 8

(3x - 4) (-2x 2 + 5x - 6)
(3x - 4) (-2x 2 + 5x - 6) = 3x (-2x 2 + 5x - 6) - 4 (-2x 2 + 5x - 6)
= 3x (-2x + 5x - 6) 2
4 (-2x + 5x - 6)
2
( ) ( ) (
= 3x (-2x 2) + 3x 5x + 3x -6 - 4 -2x
Simplify.
= -6 x
3
+ 15 x
2
- 18 x +
= -6 x
3
+ 23 x
2
- 38 x + 24
8 x
2
2
Distribute.
Redistribute.
)- 4 ( 5x )- 4 -6
AVOID COMMON ERRORS
- 20 x + 24
It may be helpful for students to insert placeholders
where there is no term for a given power of the
variable. Advise students to use 0x 2, 0x, or 0 when
necessary to make sure that all places contain a term.
Reflect
7.
Discussion Is the product of two polynomials always another polynomial?
Yes. The product, after using the Distributive Property and multiplying monomials,
consists of a monomial or a sum or difference of monomials.
9.
Can the Distributive Property be used to multiply two trinomials?
Yes. Multiply each term in the first trinomial by each term in the second trinomial.
(3x + 1)(x 3 + 4x 2 - 7)
= 3x(x 3 + 4 x 2 -7) + 1(x 3 + 4 x 2 - 7)
= 3x(x 3) + 3x(4 x 2) + 3x(-7) + x 3 + 4x 2 - 7
= 3x 4 + 12 x 3 -21x + x 3 + 4x 2 -7
= 3x 4 + 13 x 3 + 4x 2 - 21x -7
Module 18
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Explain that polynomials can also be
© Houghton Mifflin Harcourt Publishing Company
8.
multiplied using a vertical format. Multiplying
vertically is similar to multiplying multi-digit whole
numbers. Multiply the top polynomial by each term
of the bottom polynomial in turn, then add the
results.
Lesson 2
858
DIFFERENTIATE INSTRUCTION
A1_MNLESE368187_U7M18L2.indd 858
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Graphic Organizers
Students may find it easier to do polynomial multiplication in a table, as shown here
for (4x – 7)(x 2 + 2x + 3). Each cell in the table is the product of one term from the
factor written at the top and one term from the factor written at the left.
x2
+2x
+3
4x
4x
8x 2
12x
−7
−7x 2
−14x
−21
3
(4x – 7)(x + 2x + 3) = 4x + 8x − 7x + 12x − 14x − 21 = 4x 3 + x 2 − 2x − 21
2
3
2
2
Multiplying Polynomial Expressions
858
Explain 4
EXPLAIN 4
Polynomial multiplication is sometimes necessary in problem solving.
A
Modeling with Polynomial
Multiplication
Gardening Trina is building a garden. She designs a rectangular garden with length
(x + 4) feet and width (x + 1) feet. When x = 4, what is the area of the garden?
Let y represent the area of Trina’s garden. Then the equation for this situation is
y = (x + 4) (x + 1).
y = (x + 4) (x + 1)
QUESTIONING STRATEGIES
Use FOIL.
How could drawing a diagram help you to
solve a real-world problem? You can label the
diagram with the information that you know, and
then write an expression to model the situation.
y = x 2 + x + 4x + 4
y = x 2 + 5x + 4
Now substitute 4 for x to finish the problem.
y = x 2 + 5x + 4
2
y = (4) + 5(4) + 4
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Technology
MP.5 After substituting a value for the
y = 16 + 20 + 4
y = 40
The area of Trina’s garden is 40 ft 2.
B
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Tim
Pannell/Corbis
variable in a polynomial that models a
real-world situation, students can use a graphing
calculator to evaluate the result.
Modeling with Polynomial Multiplication
Design Orik has designed a rectangular mural that measures
20 feet in width and 30 feet in length. Laura has also designed
a rectangular mural, but it measures x feet shorter on each side.
When x = 6, what is the area of Laura’s mural?
Let y represent the area of Laura’s mural. Then the equation for this
situation is
y = (20 - x)(30 - x).
y = (20 - x)(30 - x)
Use FOIL.
y=
600
y = 1 x2 -
-
20 x - 30 x + 1 x 2
50 x +
600
Now substitute 6 for x to finish the problem.
2
y= 6
y=
y=
36
-
50
⋅ 6
300
+
+
600
600
336
The area of Laura’s mural is
336
Module 18
ft 2.
859
Lesson 2
LANGUAGE SUPPORT
A1_MNLESE368187_U7M18L2 859
Cognitive Strategies
The English language often uses acronyms. Explain to students that an acronym
is a word formed from the initial letters of other words. The acronym FOIL
represents the order of the steps used in multiplying binomials: First terms, Outer
terms, Inner terms, Last terms.
FOIL = First, Outer, Inner, Last
Have students discuss other acronyms they have encountered at school or in their
community.
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Your Turn
ELABORATE
10. Landscaping A landscape architect is designing a rectangular garden in a local park. The garden will
be 20 feet long and 15 feet wide. The architect wants to place a walkway with a uniform width all the way
around the garden. What will be the area of the garden, including the walkway?
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 Tell students that the product of a polynomial
Let x be the width of the walkway. The length of the garden, including the walkway,
is (20 + 2x) feet. The width of the garden, including the walkway, is (15 + 2x) feet.
(20 + 2x) (15 + 2x) = 300 + 40x + 30x + 4x 2
= 300 + 70x + 4x 2
with m terms and a polynomial with n terms has mn
terms before you simplify it. Have students offer
explanations for why this is true.
= 4x 2+ 70x + 300
So the area, including the walkway, is (4x 2+ 70x + 300)ft 2.
v
Elaborate
11. How is the FOIL method different from the Distributive Property? Explain.
There is no difference. FOIL simply gives an order in which to use the Distributive Property.
SUMMARIZE THE LESSON
How is the FOIL method similar to using
the Distributive Property to multiply larger
polynomials? The FOIL method is a double use of
the Distributive Property. The first term of the first
binomial is distributed across the second binomial
(F and O), and then the second term of the first
binomial is distributed across the second binomial (I
and L).
12. Why can FOIL not be used for polynomials with three or more terms?
FOIL refers to four partial products within a product: first, outer, inner, and last.
When polynomials with three or more terms are multiplied, there are more than
four partial products.
13. Essential Question Check–In How do you multiply two binomials?
Use the FOIL method to find the partial products of the first terms, the outer terms, the
inner terms, and the last terms in the binomials. Then, add the partial products to find the
product of the binomials.
© Houghton Mifflin Harcourt Publishing Company
Module 18
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860
Lesson 2
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Multiplying Polynomial Expressions
860
Evaluate: Homework and Practice
EVALUATE
Multiply by using the Distributive Property.
(x + 6) (x - 4)
1.
(2x + 5)(x - 3) = 2x (x - 3) + 5(x - 3)
(x + 6)(x - 4) = x (x - 4) + 6 (x - 4)
= 2x(x) + 2x(-3) + 5(x) + 5(-3)
= x (x) + x (-4) + 6 (x) + 6(-4)
= x 2 - 4x + 6x - 24
= 2x 2 - 6x + 5x - 15
= x 2 + 2x - 24
ASSIGNMENT GUIDE
Concepts and Skills
Exercise 24
Example 1
Multiplying Binomials Using the
Distributive Property
Exercises 1–6, 27
Example 2
Multiplying Binomials Using FOIL
Exercises 7–12,
25
= 2x 2 - x - 15
(x - 6)(x + 1)
3.
Practice
Explore
Modeling Binomial Multiplication
(x
(x - 6) (x + 1) = x (x + 1) -6 (x + 1)
= x (x) + x (1) - 6(x) - 6 (1)
Example 4
Modeling with Polynomial
Multiplication
Exercises 19–22,
26
+ 3)(x - 4) = x 2(x - 4) + 3 (x - 4)
= x 2 (x) + x 2(-4) + 3 (x) + 3 (-4)
= x 3 - 4x 2 + 3x - 12
= x - 5x - 6
2
(x 2 + 11) (x + 6)
(x 2 + 11)(x + 6)
6.
= x 2(x + 6) + 11 (x + 6)
= x (x) + x (6) + 11 (x) + 11 (6)
2
2
= x 2 (x) + x 2(-5) + 8 (x) + 8 (-5)
= x 3 - 5x 2 + 8x - 40
Multiply by using the FOIL method.
(x + 3) (x + 7)
7.
8.
(x + 3)(x + 7) = x + 7x + 3x + 21
= 4x 2 - x - 14
© Houghton Mifflin Harcourt Publishing Company
10. (x 2 - 6) (x - 4)
(3x + 2) (2x + 5)
9.
(x 2 - 6)(x - 4) = x 3 - 4x 2 - 6x + 24
(3x + 2)(2x + 5) = 6x 2 + 15x + 4x + 10
= 6x 2 + 19x + 10
(x 2 + 9 )(x - 3)
11.
12. (4x 2 - 4) (2x + 1)
(x 2 + 9)(x - 3) = x 3 - 3x 2 + 9x - 27
Module 18
Exercise
A1_MNLESE368187_U7M18L2 861
(4x + 2)(x - 2)
(4x + 7)(x - 2) = 4x 2 - 8x + 7x - 14
= x + 10x + 21
Show students the “FOIL face” to help them keep
track of which terms to multiply when finding a
product of binomials.
(x 2 + 8)(x - 5) = x 2(x - 5) + 8 (x - 5)
2
2
VISUAL CUES
(x 2 + 8)(x - 5)
= x + 6x + 11x + 66
3
2
Lesson 18.2
2
= x + x - 6x - 6
5.
Exercises 13–18,
23
861
(x 2 + 3) (x - 4)
4.
2
Example 3
Multiplying Polynomials
(x + 3) (x + 2)
(2x + 5)(x - 3)
2.
• Online Homework
• Hints and Help
• Extra Practice
(4x 2 - 4)(2x + 1) = 8x 3 + 4x 2 - 8x - 4
Lesson 2
861
Depth of Knowledge (D.O.K.)
COMMON
CORE
Mathematical Practices
1–16
1 Recall of Information
MP.5 Using Tools
17–18
2 Skills/Concepts
MP.5 Using Tools
19–22
1 Recall of Information
MP.4 Modeling
23
1 Recall of Information
MP.5 Using Tools
24
1 Recall of Information
MP.4 Modeling
25–26
2 Skills/Concepts
MP.2 Reasoning
27
2 Skills/Concepts
MP.3 Logic
3/24/14 7:12 PM
Multiply the polynomials.
CURRICULUM INTEGRATION
13. (x - 3) (x 2 + 2x + 1)
In biology, a Punnett square is used to show possible
ways that genes can combine at fertilization. Discuss
how filling out a Punnett square is similar to
multiplying binomials using a table or algebra tiles.
(x - 3)(x 2 + 2x + 1) = x (x 2 + 2x + 1) - 3 (x 2 + 2x + 1)
= x (x 2) + x(2x) + x(1) - 3(x 2) -3 (2x) - 3(1)
= x 3 + 2x 2 + x - 3x 2 - 6x - 3
= x 3 - x 2 - 5x - 3
14. (x + 5) (x 3 + 6x 2 + 18x)
(x + 5)(x 3 + 6x 2 + 18x) = x (x 3 + 6x 2 + 18x) + 5 (x 3 + 6x 2 + 18x)
= x (x 3) + x (6x 2) + x(18x) + 5(x 3) + 5 (6x 2)+ 5(18x)
= x 4 + 6x 3 + 18x 2 + 5x 3 + 30x 2 + 90x
= x 4 + 11x 3 + 48x 2 + 90x
15. (x + 4) (x 4 + x 2 + 1)
(x + 4)(x 4 + x 2 + 1) = x (x 4+ x 2 + 1) + 4(x 4 + x 2 + 1)
= x (x 4) + x (x 2) + x(1) + 4(x 4) + 4(x 2) + 4(1)
= x 5 + x 3 + x + 4x 4 + 4x 2 + 4
= x 5 + 4x 4 + x 3 + 4x 2 + x + 4
16. (x - 6)(x 5 + 4x 3 + 6x 2 + 2x)
(x - 6)(x 5 + 4x 3 + 6x 2 + 2x) = x(x 5 + 4x 3 + 6x 2 + 2x) - 6(x 5 + 4x 3 + 6x 2 + 2x)
= x(x 5)+ x(4x 3)+ x(6x 2) + x(2x) - 6(x 5) - 6(4x 3) - 6(6x 2) -6(2x)
= x 6 + 4x 4 + 6x 3+ 2x 2 - 6x 5 - 24x 3 - 36x 2 - 12x
= x 6 - 6x 5 + 4x 4 - 18x 3 - 34x 2 - 12x
17.
(x 2 + x + 3)(x 3 - x 2 + 4) = x 2(x 3 - x 2 + 4) + x(x 3 - x 2 + 4) + 3(x 3 - x 2 + 4)
= x 2(x 3) + x 2(-x 2) + x 2(4) + x(x 3) + x(-x 2) + x(4) + 3(x 3)
+ 3(-x 2) + 3(4)
= x 5 - x 4 + 4x 2 + x 4 - x 3 + 4x + 3x 3 - 3x 2 + 12
= x 5 + 2x 3 + x 2 + 4x + 12
18.
(x 3 + x 2 + 2x)(x 4 - x 3 + x 2)
(x 3 + x 2 + 2x)(x 4 - x 3 + x 2) = x 3(x 4 - x 3 + x 2) + x 2(x 4 - x 3 + x 2) + 2x(x 4 - x 3 + x 2)
= x 3(x 4) + x 3(-x 3) + x 3(x 2) + x 2(x 4) + x 2(-x 3) + x 2(x 2) + 2x(x 4)
+ 2x(-x 3) + 2x(x 2)
© Houghton Mifflin Harcourt Publishing Company
(x 2 + x + 3)(x 3 - x 2 + 4)
= x 7 - x 6 + x 5 + x 6 - x 5 + x 4 + 2x 5 - 2x 4 + 2x 3
= x 7 + 2x 5 - x 4 + 2x 3
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Multiplying Polynomial Expressions
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Write a polynomial equation for each situation.
AVOID COMMON ERRORS
19. Gardening Cameron is creating a garden. He
designs a rectangular garden with a length of
(x + 6) feet and a width of (x + 2) feet. When
x = 5, what is the area of the garden?
When using the FOIL method, students may forget to
combine like terms. Remind them that the terms that
result from multiplying the inner and outer terms of
two binomials need to be combined.
Let y represent the area of Cameron’s garden.
Then the equation for this situation is
y = (x + 6)(x + 2). Use FOIL.
y = x 2 + 2x + 6x + 12
y = x 2 + 8x + 12
Now substitute 5 for x to finish the problem.
y = x 2 + 8x + 12
y = 5 2 + 8 ⋅ 5 + 12
y = 25 + 40 + 12
y = 77
The area of Cameron’s garden is 77 ft 2.
20. Design Sabrina has designed a rectangular painting that measures 50 feet in length
and 40 feet in width. Alfred has also designed a rectangular painting, but it measures
x feet shorter on each side. When x = 3, what is the area of Alfred’s painting?
Let y represent the area of Alfred’s painting. Then the equation for this
situation is y = (50 - x)(40 - x).
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Paul
Burns/Corbis
y = (50 - x)(40 - x)
Use FOIL.
y = 2000 - 50x - 40x + x 2
y = x 2 - 90x + 2000
Now substitute 3 for x to finish the problem.
y = x 2 - 90 + 2000
y = 3 2 - 90 ⋅ 3 + 2000
y = 9 - 270 + 2000
y = 1739
The area of Alfred’s painting is 1739 ft 2.
21. Photography Karl is putting a frame around a rectangular photograph. The
photograph is 12 inches long and 10 inches wide, and the frame is the same width
all the way around. What will be the area of the framed photograph?
Let x be the width of the frame. The length of the framed photograph is
(12 + 2x) inches and the width is (10 + 2x) inches.
(12 + 2x)(10 + 2x) = 120 + 24x + 20x + 4x 2
= 120 + 44x + 4x 2
= 4x 2 + 44x + 120
The area of the framed photograph is (4x 2 + 44x + 120) in 2.
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22. Sports A tennis court is surrounded
by a fence so that the distance from each
boundary of the tennis court to the fence
is the same. If the tennis court is 78 feet
long and 36 feet wide, what is the area of
the entire surface inside the fence?
GRAPHIC ORGANIZERS
Have students refer to the following patterned
graphic organizer, in which each column represents
one term of a binomial, as a model for the FOIL
method of multiplying binomials.
Let x be the distance between each
side of the court and the fence.
First
Outer
Inner
Last
The length of the fenced area is
(78 + 2x) feet, and the width is
(36 + 2x) feet.
(78 + 2x)(36 + 2x) = 2808 + 156x + 72x + 4x 2
= 2808 + 228x + 4x 2
AVOID COMMON ERRORS
= 4x 2 + 228x + 2808
The area of fenced surface is (4x 2 + 228x + 2808)ft 2.
When finding the product of polynomials with many
terms, students may miss some terms. Remind
students that the product of a polynomial with m
terms and a polynomial with n terms has mn terms
before you simplify it. Encourage them to count the
number of terms in the product to make sure they
have enough.
23. State the first term of each product.
a. (2x + 1)(3x + 4)
b.
c.
d.
x(x + 9)
(x 2 + 9)(3x + 4)(2x + 6)
(x 3 + 4)(x 2 + 6)(x + 5)
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Berna
Namoglu/Shutterstock
e.
(x 4 + x 2)(3x 8 + x 11)
a. 6x 2
b. 3x 12
c. x 2
d. 6x 4
e. x 6
24. Draw algebra tiles to model the factors in the polynomial multiplication modeled on
the mat. Then write the factors and the product in simplest form.
×
(x - 2)(x + 3) = x 2 + x - 6
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JOURNAL
H.O.T. Focus on Higher Order Thinking
25. Critical Thinking The product of 3 consecutive odd numbers is 2145. Write an
expression for finding the numbers.
Have students make a table summarizing methods
for multiplying polynomials. They should include
examples for multiplying monomials, binomials, and
trinomials.
n(n + 2)(n + 4) = n 3 + 6n 2 + 8n
26. Represent Real-World Problems The town swimming pool is d feet deep. The
width of the pool is 10 feet greater than 5 times its depth. The length of the pool is
35 feet greater than 5 times its depth. Write and simplify an expression to represent
the volume of the pool.
(d)(5d + 10)(5d + 35)
= (5d 2 + 10d)(5d + 35)
= 25d 3 + 175d 2 + 50d 2 + 350d
= 25d 3 + 225d + 350d
2
27. Explain the Error Bill argues that (x + 1)(x + 19) simplifies to x 2 + 20x + 20.
Explain his error.
Bill added the constants in the binomials. He should have multiplied the
© Houghton Mifflin Harcourt Publishing Company
constant of each binomial together instead.
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Lesson Performance Task
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Encourage students to draw and label a
Roan is planning a large vegetable garden in her yard. She plans to have at least six x by x regions for rotating crops
and some 2 or 3 feet by x strips for fruit bushes like blueberries and raspberries.
Design a rectangular garden for Roan and write a polynomial that will give its area.
diagram of the garden, and then use the labels to find
the polynomials that represent the length and width
of the garden.
The answers will vary widely but the method for finding the
polynomial will be multiplying the length of the garden by its width.
The dimensions will just be the sum of the defined regions along the
horizontal edge and the vertical edge.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 Have students create a drawing or diagram to
explain the method they used to multiply the
polynomials representing the length and width of
their gardens. Have students share their drawings
with the class.
If each shaded region is one of the x by x plots and the others are 2 by x
regions, then the dimensions are represented as follows:
w = 3x + 4
© Houghton Mifflin Harcourt Publishing Company
ℓ = 2x + 2
The area will be represented by the following:
A=ℓ⋅w
= (2x + 2)(3x + 4)
= 2x ⋅ 3x + 2x ⋅ 4 + 2 ⋅ 3x + 2 ⋅ 4
= 6x 2 + 8x + 6x + 8
= 6x 2 + 14x + 8
If the regions for fruit bushes are 3 feet by x feet, the width is 3x + 6,
the length is 2x + 3, and the area is 6x 2 + 21x + 18.
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EXTENSION ACTIVITY
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Have students select a value of x, find the perimeter and area of their garden
designs, and then revise their designs to maximize the area without increasing the
perimeter of their gardens.
10/16/14 12:38 PM
Students may find that designing a square garden allows them to increase the area
while keeping the same perimeter.
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Multiplying Polynomial Expressions
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