5 multiplying two binomials -...
TRANSCRIPT
ThenThenYou have already multiplied a binomial by a monomial. (Looking Ahead 4)
NowNow Multiply two binomials
by using models.
Multiply two binomials by using the Distributive Property.
New VocabularyNew VocabularyFOIL Method
Math Online
glencoe.com
5Looking
Ahead
Why?Why?
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.
main stage
lights storage
sound
x m
x m
1 m
2 m
a. Write expressions for the area of each part of the stage.
b. Write an expression for the total area of the stage.
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.
Step 2 Fill in the rectangle with algebra tiles. There is one x 2 -tile, three x-tiles, and two 1-tiles.
So, (x + 2)(x + 1) = x 2 + 3x + 2.
Guided Practice✓Multiply. Use a model.
1A. (x + 4)(x + 4) 1B. (2x + 1) (x + 3)
Personal Tutor glencoe.com
1 1
1
x + 2x
+ 1
1 1
1
x + 2x
+ 1
1 1
1 11
x + 2
x+
1
2
1 1
1 11
x + 2
x+
1
2
Lesson 5 Multiplying Two Binomials LA17
Multiplying Two Binomials
LA18 Looking Ahead to Algebra 1
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 Use the Distributive Property
Find (2x + 4)(x + 5).
Method 1 Use a model.
Method 2 Use the Distributive Property.
(2x + 4)(x + 5) = 2x(x + 5) + 4(x + 5) Distributive Property
= 2 x 2 + 10x + 4x + 20 Distributive Property
= 2x 2 + 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.
FOIL Method
Words Symbols
(3x + 1)(x + 2)
F Multiply the FIRST terms. 3x · x or 3 x 2
O Multiply the OUTER terms. 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.
Key Concept
1 1
x
x
x xx · x x · 2
1 1
2
21 · x 1 · 2
2
1 1
x
x
x xx · x x · 2
1 1
2
21 · x 1 · 2
2
Watch Out!
Negative SignsIf one or both of the binomials involve negatives or subtraction, remember to distribute the negatives.
1 1
1 11
2x + 4
x+
5
1 1
1 1
2
1 11 1 11 11 1 11 11 1 11 11 1 1
2
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
Lesson 5 Multiplying Two Binomials LA19
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.
First, find the area of the rectangular base.
B = �w Formula for area of a rectangle
= (x + 3)(x) Replace � with x + 3 and w with x.
= x2 + 3x Distributive Property
To find the volume, multiply the area of the base by the height.
V = Bh Formula for volume of a prism
= ( x 2 + 3x)(x - 1) Replace B with x 2 + 3x and h with x - 1.
F O I L= x 2 · x - x 2 · 1 + 3x · x - 3x · 1 Use the FOIL method.
= x 3 - x 2 + 3 x 2 - 3x Multiply.
= x 3 + 2 x 2 - 3x Simplify.
Guided Practice✓ 3. GIFTS Express the volume of the gift
box at the right as a polynomial.
Personal Tutor glencoe.com
x - 1
xx + 3
x - 1
xx + 3
(x - 3) in.
(x + 4) in.
2x in.
(x - 3) in.
(x + 4) in.
2x in.
✓ Check Your Understanding
Multiply. Use a model.
1. (x + 2)(x + 3) 2. (x + 3)(x + 4) 3. (x + 1)(2x + 1)
1 1
1
x + 2
x+
3
11
1 1 1
1
x + 3
x+
4
111
1
x + 1
2x+
1
1
Multiply.
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.
Example 1p. LA17
Example 1p. LA17
Example 3p. LA19
Example 3p. LA19
LA20 Looking Ahead to Algebra 1
Practice and Problem Solving
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.
Find each product.
25. (y + 3)( y 2 - 4) 26. ( x 2 + 3)(3 x 2 - 1) 27. (2 y 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.
a. Find the area of the label.
b. Find the area of the envelope not covered by the label.
35. GEOMETRY A square has sides of length s. 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. What product does this model represent?
b. What is the area of each tile?
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.
Examples 1 and 2pp. LA17–LA18
Examples 1 and 2pp. LA17–LA18
Example 3p. LA19
Example 3p. LA19
6x + 5
x + 3
x + 44x + 4
6x + 5
x + 3
x + 44x + 4
a
b
ba
a
b
ba
B
C