LESSON
12.4 Multiplying Binomials
Investigate
Find the product (3x + 1)(x + 4) using algebra tiles.
1 Represent each binomial using algebra tiles. Arrange the first
binomial vertically and the second binomial horizontally, as shown.
1
LESSON
12.4 Multiplying Binomials
Investigate
Find the product (3x + 1)(x + 4) using algebra tiles.
2 The binomials define a rectangular region with length (3x + 1) units
and width (x + 4) units. Fill in the region with the appropriate tiles.
2
LESSON
12.4 Multiplying Binomials
Investigate
Find the product (3x + 1)(x + 4) using algebra tiles.
3 The tiles covering the rectangular region represent 3x 2 + 13x + 4.
This expression is the product of the binomials.
3
LESSON
12.4 Multiplying Binomials
Investigate
Find the product (3x + 1)(x + 4) using algebra tiles.
You
use
a table
to multiply
binomials.
model
2 + 12x
Fromcan
thealso
table,
you
can see
that (3xtwo
+ 1)(x
+ 4) is 3xThe
+ x + 4,
and
below
2 + 13x
or 3xtable
+ 4. show the product (3x + 1)(x + 4).
Terms of the first binomial
x
3x
3x 2
1
x
4
12x
Terms of
the second
binomial
4
Multiply terms to fill in the
table. For example, 1 • x = x.
4
LESSON
12.4 Multiplying Binomials
EXAMPLE
1
Multiplying Binomials Using a Table
Find the product (–2x + 5)(3x – 1).
Write any subtractions in the binomial as additions.
(–2x + 5)(3x – 1) = (–2x + 5)[3x + (–1)]
3x
–1
–2x
–6x 2
2x
5
15x
–5
First binomial
Second binomial
The product is –6x 2 + 2x + 15x – 5, or –6x 2 + 17x – 5.
5
LESSON
12.4 Multiplying Binomials
Distributive Property Another way to multiply binomials is to
use the distributive property.
6
LESSON
12.4 Multiplying Binomials
EXAMPLE
2
Using the Distributive Property
Photograph You are enlarging a photo that is 7
inches long and 5 inches wide. The length and width
of the enlargement are x times the length and width
of the original photo. The enlargement will have a 2
inch mat. Write a polynomial expression for the
combined area of the enlargement and mat.
SOLUTION
The total length of the enlargement and mat is (7x + 4) inches.
The total width is (5x + 4) inches. To find the area, multiply.
(7x + 4)(5x + 4) = 7x(5x + 4) + 4(5x + 4)
ANSWER
Distribute 5x + 4.
= 35x 2 + 28x + 20x + 16
Distribute 7x and 4.
= 35x 2 + 48x + 16
Combine like terms.
The area is (35x 2 + 48x + 16) square inches.
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LESSON
12.4 Multiplying Binomials
FOIL Method Notice that using the distributive property to multiply
binomials produces four products that are then added. These four
products are the products of the first terms, the outer terms, the inner
terms, and the last terms of the binomials. You can use the shorthand
FOIL to remind you of the words First, Outer, Inner, and Last.
First
Outer
Inner
Last
(2x + 5)(6x + 1) = 12x 2 + 2x + 30x + 5
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LESSON
12.4 Multiplying Binomials
EXAMPLE
3
Using the FOIL Method
Find the product (4x + 3)(5x – 2).
First
+ Outer
+ Inner
+ Last
(4x)(5x) + (4x)(–2) + (3)(5x) + (3)(–2)
Write products of terms.
20x 2
Multiply.
+ (–8x)
+ 15x
20x 2 + 7x – 6
+ (–6)
Combine like terms.
9