Download Multiplying Two Binomials

Looking
5
Ahead
Multiplying Two Binomials
Why?
Then
You have already
multiplied a binomial by
a monomial.
The local youth club has a stage for karaoke competitions. The main stage is a
square. The technical crew needs to increase the length by 2 feet and the width
by 1 foot for their equipment. A plan for the stage is shown below.
(Looking Ahead 4)
xm
2m
main stage
sound
lights
storage
Now
Multiply two binomials
by using models.
xm
Multiply two binomials
by using the
Distributive Property.
New Vocabulary
FOIL Method
Math Online
1m
a. Write expressions for the area of each part of the stage.
b. Write an expression for the total area of the stage.
glencoe.com
Multiply Binomials As with multiplying a monomial by a binomial, you can
use algebra tiles to multiply two binomials. In the model, the length and width
of a rectangle represent the two binomials. The area represents the product.
EXAMPLE 1
Modeling Multiplication of Binomials
Find (x + 2)(x + 1).
Step 1 Make a rectangle with a width of
x + 1 and a length of x + 2.
x+2
x+1
Y
1
1
Y
1
Step 2 Fill in the rectangle with algebra tiles.
There is one x 2-tile, three x-tiles, and
two 1-tiles.
✓Guided Practice
Y
x+1
So, (x + 2)(x + 1) = x 2 + 3x + 2.
x+2
2
Y
Y
1
Y
1
1
Y Y
1
1
Multiply. Use a model.
1A. (x + 4)(x + 4)
1B. (2x + 1) (x + 3)
Personal Tutor glencoe.com
Lesson 5 Multiplying Two Binomials
LA17
Distributive Property The Distributive
Property can also be used to find the
product of two binomials. The figure at
the right shows the rectangle from
Example 1 separated into four parts.
Notice that each term from the first
parentheses (x + 2) is multiplied by each
term from the second parentheses (x + 1).
EXAMPLE 2
2
x
2
x
Y
1
Y
x·x
x Y Y
x·2
1·x
2
1 1 1
1·2
x
Use the Distributive Property
Find (2x + 4)(x + 5).
Method 1 Use a model.
2x + 4
x+5
Y
Watch Out!
Negative Signs
If one or both of the
binomials involve
negatives or
subtraction, remember
to distribute the
negatives.
Y
2
Y
Y
1
1
1
1
1
Y
Y
Y
Y
Y
Y
2
Y
Y
Y
Y
Y
1
1
1
1
Y Y Y Y
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Method 2 Use the Distributive Property.
(2x + 4)(x + 5) = 2x(x + 5) + 4(x + 5)
Distributive Property
= 2x 2 + 10x + 4x + 20
Distributive Property
2
= 2x + 14x + 20
Simplify.
So, (2x + 4) (x + 5) = 2x 2 + 14x + 20.
✓Guided Practice
Multiply. Use models if needed.
2A. (3x + 2)(2x - 3)
2B. (x + 6) (2x + 4)
Personal Tutor glencoe.com
A shortcut version of the Distributive Property is the FOIL method.
Key Concept
StudyTip
Special Products
Some pairs of binomials
have products that
follow a specific pattern.
(a + b)2 =
a2 + 2ab + b2
(a - b)2 =
a2 - 2ab + b2
(a + b)(a - b) =
a2 - b2
Words
FOIL Method
Symbols
(3x + 1)(x + 2)
F
Multiply the FIRST terms.
O Multiply the OUTER terms.
3x · x or 3x 2
3x · 2 or 6x
I
Multiply the INNER terms.
1 · x or x
L
Multiply the LAST terms.
1 · 2 or 2
So, (3x + 1)(x + 2) = 3x2 + 6x + x + 2 or 3x2 + 7x + 2.
LA18 Looking Ahead to Algebra 1
EXAMPLE 3
Use FOIL to Multiply Binomials
GEOMETRY A shipping box is shaped like a
rectangular prism. The volume V is equal to the
area of the base B times the height h. Express the
volume of the prism as a polynomial.
x- 1
First, find the area of the rectangular base.
B = w
= (x + 3)(x)
= x2 + 3x
x
x+ 3
Formula for area of a rectangle
Replace with x + 3 and w with x.
Distributive Property
To find the volume, multiply the area of the base by the height.
V = Bh
= (x 2 + 3x)(x - 1)
Formula for volume of a prism
Replace B with x 2 + 3x and h with x - 1.
F
O
I
L
2
= x · x - x · 1 + 3x · x - 3x · 1
= x 3 - x 2 + 3x 2 - 3x
= x 3 + 2x 2 - 3x
2
Use the FOIL method.
Multiply.
Simplify.
✓Guided Practice
3. GIFTS Express the volume of the gift
box at the right as a polynomial.
(x - 3) in.
2x in.
(x + 4) in.
Personal Tutor glencoe.com
✓ Check Your Understanding
Example 1
p. LA17
Multiply. Use a model.
1. (x + 2)(x + 3)
2. (x + 3)(x + 4)
x+2
1
1
Y
1
1
1
Y
1
Y
Y
x+4
x+3
Y
1
1
1
x+1
x+3
2x + 1
Y
3. (x + 1)(2x + 1)
1
1
1
1
Y
1
Multiply.
Example 3
p. LA19
4. ( y + 4)( y - 2)
5. (m - 3)(m + 1)
6. (x - 5)(x - 2)
7. (3n + 2)(n - 2)
8. (2x - 5)(x - 4)
9. (4b - 3)(2b + 4)
10. CEREAL A cereal box has a length of 2x inches, a width of x - 2 inches,
and a height of 2x + 5 inches. Express the volume as a polynomial.
Lesson 5 Multiplying Two Binomials
LA19
Practice and Problem Solving
Examples 1 and 2
pp. LA17–LA18
Example 3
p. LA19
Multiply.
11. (x + 4)(x + 8)
12. (r - 3)(r - 7)
13. (z + 6)(z - 4)
14. (2a + 5)(a - 7)
15. (5n + 2)(n - 3)
16. (2x + 5)(5x + 3)
17. (2x + 7)(x - 3)
18. (3a - b)(2a + b)
19. (n - 11)(n - 5)
20. (5n - 2p)(5n + 2p)
21. (3a + 1)(3a + 1)
22. (4h - 3)(3h + 2)
23. GEOMETRY A parcel box has a length of 3y centimeters, a width of y + 3
centimeters, and a height of 2y - 2 centimeters. Express the volume of the
package as a polynomial.
24. INTEREST Lauren deposited money into a savings account. The account earns
an interest rate of r%. For each dollar deposited, the amount in the account
after two years is given by the formula (1 + r)(1 + r). Find this product.
B Find each product.
25. (y + 3)(y 2 - 4)
26. (x 2 + 3)(3x 2 - 1)
27. (2y 2 + 1)(y + 1)
28. (x + 2y)(x + 3y)
29. (2a - 5b)(a + 2b)
30. (m 3 - 2m)(m + 3)
31. (x - 4)(3x + 2)
32. (2y - 4z)(3y - 6z)
33. (x 2 + 1)(x - 2)
34. ENVELOPES The mailing envelope shown has
a mailing label on the front.
6x + 5
a. Find the area of the label.
b. Find the area of the envelope not covered
by the label.
x+4
4x + 4
x+3
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35. GEOMETRY A square has sides of length s.
C A rectangle is 5 inches longer and 4 inches wider
than the square. Express the area of the rectangle as a polynomial.
36. GEOMETRY The model at the right represents
the square of a binomial.
a
a. What product does this model represent?
a
b. What is the area of each tile?
b
b
c. Write the area of the square as a polynomial.
H.O.T. Problems
Use Higher-Order Thinking Skills
37. OPEN ENDED Find two binomials that have 6x as one of the terms in their
product.
38. CHALLENGE Find (x + 1) 2, (x + 2) 2, and (x + 3) 2. Is there a pattern in the
products of binomials? If so, use the pattern to find (x + 6) 2 and (x + n) 2.
39. REASONING Does the product of two binomials always have three terms? If
so, explain why. If not, give a counterexample.
40. WRITING IN MATH Compare and contrast the procedure for multiplying two
binomials and the procedure for multiplying a binomial by a monomial.
LA20 Looking Ahead to Algebra 1