1. No category

7-5
Factoring Special Products
Connection: Area
Essential question: How can you represent factoring special products
geometrically?
TE ACH
Standards for
Mathematical Content
1
A-SSE.1.1b Interpret complicated expressions
by viewing one or more of their parts as a single
entity.*
A-SSE.1.2 Use the structure of an expression to
identify ways to rewrite it.
EXPLORE
Questioning Strategies
• How does the perfect-square trinomial help you
label the dimensions of the shapes in Step A?
The terms of the trinomial represent the areas of
the shapes and, thus, their factors represent the
various lengths and widths.
Prerequisites
Factoring trinomials
• What is the length of each side of the square in
Step B? a + b
Math Background
A trinomial is a perfect-square trinomial if it is of the
form a2 + 2ab + b2 or a2 - 2ab + b2. Patterns that
can be used to factor perfect-square trinomials are:
2
EXPLORE
Questioning Strategies
• In Step A, what does the shaded part of the model
represent? the part of a square with sides of
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2
length a that is left after a square with sides of
length b is removed
A polynomial is the difference of two squares if it
has two terms, one subtracted from the other, that
are both perfect squares. A pattern that can be used
to factor the difference of two squares is:
• What are the dimensions of the shaded parts of
the model in Step A? square: (a - b) and (a - b);
rectangles: (a - b) and b
a2 - b2 = (a + b)(a - b)
CLOS E
Essential Question
How can you represent factoring special products
geometrically?
Because the area of a square or rectangle is the
number of square units in the product of its
dimensions, squares and rectangles can be used
to represent parts of algebraic expressions that
are perfect-square trinomials or the difference of
two squares. You can partition and combine these
squares and rectangles into other squares and
rectangles so that you can model factorizations of
these special products.
IN T RO DUC E
Sketch a square with sides labeled a and a rectangle
with sides labeled a and b. Ask students to use
algebraic expressions to represent the areas of
both shapes. Tell them that they can use similar
models to explore the patterns for factoring special
polynomials.
Summarize
Have students write a journal entry in which they
state the factorizations of a2 + 2ab + b2 and of
a2 - b2. Ask them to sketch a geometric model for
each factorization.
Chapter 7
393
Lesson 5
© Houghton Mifflin Harcourt Publishing Company
These patterns can be derived algebraically using
the factoring methods from previous lessons.
However, because the areas of rectangles and
squares are the product of their lengths and widths,
these patterns can also be derived geometrically.
Name
Class
Notes
7-5
Date
Factoring Special Products
Connection: Area
Essential question: How can you represent factoring special products geometrically?
Recall that perfect square trinomials and the difference of two
squares are special polynomials.
A-SSE.1.2
1
EXPLORE
Representing the Factoring of a Perfect Square
Trinomial
Use area models to factor a2 + 2ab + b2.
A
Finish labeling this model of a2 + 2ab + b2. Use a and b.
a
b
b
a
a
a
b
b
+
a2
B
2ab
+
a
© Houghton Mifflin Harcourt Publishing Company
b2
Draw dashed lines inside the square below to show how the squares and rectangles
from Step A could be placed together to form a larger square. Label the dimension of
each part of the length and width of the larger square.
b
a
b
C
Use the dimensions of the square in Step B to write the area of the square in Step B.
D
Because the square has the same area as the model of the polynomial, the factorization
A = ( a + b )( a + b )
of a2 + 2ab + b2 is (a + b) (a + b) .
393
Chapter 7
Lesson 5
b
a–b
REFLECT
1a. How does the model at the right
show that the factorization of
a2 - 2ab + b2 is (a - b)(a - b)?
a–b
The area of the large square is a . The area of the smaller square is (a - b)(a - b).
The smaller area is the result of removing two rectangles, each of which has one
dimension of a, because a - b and b are added together, and one dimension of b.
So the area of both rectangles is 2ab. Because these areas overlap, the area of the
smallest square, b2, must be added back.
2
A-SSE.1.2
EXPLORE
Representing the Factoring of the Difference of Two
Squares
a
Use area models to factor a2 - b2.
A
Finish labeling this model of a2 - b2.
Use a and b.
B
Make a drawing that shows the shaded parts of the model
arranged to show a rectangle. Label each segment of the length
and width of the rectangle.
a–b
b
a
b
b
b
a–b
C
What is the length of the longer side of the rectangle? Explain.
a + b; (a - b) + b + b = a + b
What is the length of the shorter side of the rectangle?
a-b
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
b
2
What is the area of the rectangle? ( a + b )( a - b )
D
Because the rectangle has the same area as the model of the polynomial, the
factorization of a2 - b2 is (a + b) (a - b) .
Chapter 7
Chapter 7
394
Lesson 5
394
Lesson 5
ADD I T I O N A L P R AC T I C E
AND PRO BL E M S O LV I N G
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
Answers
Additional Practice
1. yes; (x + 3)2
2. yes; (2x + 5)2
3. no; 24x ≠ 2(6x · 4)
4. yes; (3x − 2)2
5. 4(2x + 3) ft; 28 ft
6. yes; (x + 4)(x − 4)
7. no; 200 is not a perfect square.
8. yes; (1 + m3)(1 − m3)
9. yes; (6s + 2t)(6s − 2t)
10. no; the operation between the two squares is
addition.
Problem Solving
1. 16x + 4; 36 feet
3. (x + 2y)(x − 2y)
4. π (x + 6)(x − 6)
5. C
© Houghton Mifflin Harcourt Publishing Company
2. 12x − 60; 240 cm
6. F
Chapter 7
395
Lesson 5
Name
Class
Date
Notes
7-5
© Houghton Mifflin Harcourt Publishing Company
Additional Practice
395
Chapter 7
Lesson 5
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Problem Solving
Chapter 7
Chapter 7
396
Lesson 5
396
Lesson 5