THINK AND DISCUSS
1. Use the FOIL method to verify that (a + b)(a - b) = a 2 - b 2.
2. When a binomial is squared, the middle term of the resulting trinomial
is twice the
?
of the first and last terms.
3. GET ORGANIZED Copy
and complete the graphic
organizer. Complete the
special product rules and
give an example of each.
6-6
-«iVˆ>Ê*Àœ`ÕVÌÃʜvʈ˜œ“ˆ>Ã
*iÀviV̇-µÕ>Ài
/Àˆ˜œ“ˆ>Ã
­>ÊÊL®ÓÊʶ
ˆvviÀi˜Viʜv
/ܜÊ-µÕ>ÀiÃ
­>ÊÊL®ÓÊʶ
­>ÊÊL®­>ÊÊL®Êʶ
Exercises
Homework Help Online
Parent Resources Online
GUIDED PRACTICE
1. Vocabulary In your own words, describe a perfect-square trinomial.
Multiply.
SEE EXAMPLE
SEE EXAMPLE
SEE EXAMPLE
1
2
3
2. (x + 7)2
3. (2 + x)2
4. (x + 1)2
5. (2x + 6)2
6. (5x + 9)2
7.
8. (x - 6)2
9. (x - 2)2
11. (8 - x)2
12.
14. (x + 5)(x - 5)
15. (x + 6)(x - 6)
17.
SEE EXAMPLE 4
(2x 2 + 3)(2x 2 - 3)
18.
(2a + 7b)2
10. (2x - 1) 2
2
13. (7a - 2b)
(6p - q)2
16. (5x + 1)(5x - 1)
(9 - x 3)(9 + x 3)
19.
(2x - 5y)(2x + 5y)
ÝÊÊÎ
20. Geometry Write a polynomial that represents
the area of the figure.
ÝÊÊ£
ÝÊÊ£
ÝÊÊÎ
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
21–26
27–32
33–38
39
1
2
3
4
Extra Practice
See Extra Practice for
more Skills Practice and
Applications Practice
exercises.
Multiply.
21. (x + 3)2
22. (4 + z)2
23.
(x 2 + y 2)2
26.
(r 2 + 5t)2
24.
(p + 2q 3)2
25. (2 + 3x)2
27.
(s 2 - 7)2
28.
33. (a - 10)(a + 10)
36.
(x
29. (a - 8)2
31. (3x - 4)2
30. (5 - w)2
2
(2c - d 3)2
- 2)(x + 2)
2
34.
37.
32.
(y + 4)(y - 4)
(5a 2 + 9)(5a 2 - 9)
(1 - x 2)2
35. (7x + 3)(7x - 3)
38.
(x 3 + y 2)(x 3 - y 2)
6-6 Special Products of Binomials
CC13_A1_MESE647036_C06L06.indd 437
437
4/30/11 2:03:46 AM
39. Entertainment Write a polynomial that
represents the area of the circular puzzle.
Remember that the formula for area of a circle
is A = πr 2, where r is the radius of the circle.
Leave the symbol π in your answer.
40. Multi-Step A square has sides that are
(x - 1) units long and a rectangle has a length
of x units and a width of (x - 2) units.
a. What are the possible values of x ? Explain.
b. Which has the greater area, the square or
the rectangle?
c. What is the difference in the areas?
r=x+4
Multiply.
41.
(x + y)2
42. (x - y)2
44.
(x 2 + 4)2
45.
43.
(x 2 - 4) 2
(x 2 + 4)(x 2 - 4)
46. (1 - x) 2
47. (1 + x)2
48. (1 - x)(1 + x)
49.
50. (5 + n)(5 + n)
51. (6a - 5b)(6a + 5b)
52.
(x 3 - a 3)(x 3 - a 3)
(r - 4t 4)(r - 4t 4)
Math History
Beginning about 3000
B.C.E., the Babylonians
lived in what is now
Iraq and Turkey. Around
575 B.C.E., they built
the Ishtar Gate to
serve as one of eight
main entrances into
the city of Babylon.
The image above is a
relief sculpture from a
restoration of the
Ishtar Gate.
438
a
b
(a - b)2
a 2 - 2ab + b 2
1
4
(1 - 4)2 = 9
1 2 - 2(1)(4) + 4 2 = 9
53.
2
4
54.
3
2
a
b
(a + b)2
a 2 + 2ab + b 2
55.
1
4
56.
2
5
57.
3
0
a
b
(a + b)(a - b)
a2 - b2
58.
1
4
59.
2
3
60.
3
2
61. Math History The Babylonians used tables of squares and the formula
(a + b)2 - (a - b)2
ab = _____________
to multiply two numbers. Use this formula to find the product
4
35 · 24.
62. Critical Thinking Find a value of c that makes 16 x 2 - 24x + c a perfect-square
trinomial.
63.
ANALYSIS/////
/////ERROR
2
(a - b) = a 2 - b 2
Chapter 6 Exponents and Polynomials
Explain the error below. What is the correct product?
(tr),Molly Eckler - puzzle, photo by Victoria Smith/HMH; (bl),© Ruggero Vanni/CORBIS
Copy and complete the tables to verify the special products of binomials.
64. a. Michael is fencing part of his yard. He started with a square of length x on each
side. He then added 3 feet to the length and subtracted 3 feet from the width.
Make a sketch to show the fenced area with the length and width labeled.
b. Write a polynomial that represents the area of the fenced region.
c. Michael bought a total of 48 feet of fencing. What is the area of his fenced region?
65. Critical Thinking The polynomial ax 2 - 49 is a difference of two squares. Find all
possible values of a between 1 and 100 inclusive.
66. Write About It When is the product of two binomials also a binomial? Explain and
give an example.
67. What is (5x - 6y)(5x - 6y)?
25x 2 - 22xy + 36y 2
25x 2 + 22xy + 36y 2
25x 2 - 60xy + 36y 2
25x 2 + 60xy + 36y 2
ÓxÝÓ
£äÝ
£äÝ
{
68. Which product is represented by the model?
(2x + 5)(2x + 5)
(5x + 2)(5x - 2)
(5x + 2)(5x + 2)
(5x - 2)(5x - 2)
69. If a + b = 12 and a 2 - b 2 = 96 what is the value of a?
2
4
8
10
70. If rs = 15 and (r + s)2 = 64, what is the value of r 2 + s 2?
25
30
34
49
CHALLENGE AND EXTEND
71. Multiply (x + 4)(x + 4)(x - 4).
72. Multiply (x + 4)(x - 4)(x - 4).
73. If x 2 + bx + c is a perfect-square trinomial, what is the relationship between b and c ?
74. You can multiply two numbers by rewriting the numbers as the difference of two
squares. For example:
36 · 24 = (30 + 6)(30 - 6) = 30 2 - 6 2 = 900 - 36 = 864
HMH
Use this method to multiply 27 · 19. Explain how you rewrote the numbers.
6-6 Special Products of Binomials
439