Download The Binomial Theorem

5-7
The Binomial Theorem
Content Standard
A.APR.5 Know and apply the Binomial Theorem for
the expansion of (x 1 y)n in powers of x and y for a
positive integer n, where x and y are any numbers, with
coefficients determined for example by Pascal’s Triangle.
Objectives To expand a binomial using Pascal’s Triangle
To use the Binomial Theorem
When counting
seems complicated,
it helps to be
systematic.
MATHEMATICAL
PRACTICES
Lesson
Vocabulary
V
•e
expand
• Pascal’s Triangle
• Binomial
Theorem
How many unique letter combinations are
possible using each of the following?
a. 2 of 5 letters
b. 3 of 5 letters
c. 2 of 6 letters
d. 4 of 6 letters
e. 3 of 6 letters
Justify your reasoning.
Hint: Use the diagram, a previous
response, or both. The same letters in
different orders are one combination.
There is a connection between the triangular pattern of numbers in the Solve It and the
expansion of (a 1 b)n .
Essential Understanding You can use a pattern of coefficients and the pattern
an, an21b, an22b 2, c, a 2b n22, ab n21, b n to write the expansion of (a 1 b)n.
You can expand (a 1 b)3 using the Distributive Property.
(a 1 b)3 5 (a 1 b)(a 1 b)(a 1 b) 5 a 3 1 3a 2b 1 3ab 2 1 b 3
To expand the power of a binomial in general, first multiply as needed. Then
write the polynomial in standard form.
Consider the expansions of (a 1 b)n for the first few values of n:
Row
326
Chapter 5
0326_hsm11a2se_cc_0507.indd 326
Power
Expanded Form
Coefficients Only
0
(a 1 b)0
1
1
1
(a 1 b)1
1a1 1 1b1
1 1
2
(a 1 b)2
1a 2 1 2a1b1 1 1b 2
1 2 1
3
(a 1 b)3
1a 3 1 3a2b1 1 3a1b 2 1 1b 3
1 3 3 1
4
(a 1 b)4
1a 4 1 4a3b1 1 6a 2b 2 1 4a1b 3 1 1b4
1 4 6 4 1
Polynomials and Polynomial Functions
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The “coefficients only” column matches the numbers in Pascal’s Triangle. Pascal’s
Triangle, named for the French mathematician Blaise Pascal (1623–1662), is a
triangular array of numbers in which the first and last number of each row is 1. Each
of the other numbers in the row is the sum of the two numbers above it.
For example, to generate row 5, use the sums of the adjacent elements in the row
above it.
Row
0
Pascal’s Triangle
1
1
1
2
1
3
1
5
1
7
1
1
7
8
5
6
3
6
10
15
21
28
1
3
4
1
6
8
2
1
4
1
4
10
20
35
56
1
1
15
6
21
56
28
6
4
5
1
5
35
70
1
10
4
10
1
5
1
7
1
8
1
Problem 1 Using Pascal’s Triangle
What row of Pascal’s
Triangle should
you use for this
expansion?
The expression is raised
to the 6th power so use
the 6th row.
What is the expansion of (a 1 b) 6 ? Use Pascal’s Triangle.
The exponents for a begin with 6 and decrease to 0.
1a6b0 6a5b1 15a4b2 20a3b3 15a2b4 6a1b5 1a0b6
The exponents for b begin with 0 and increase to 6.
(a 1 b) 6 5 a 6 1 6a 5b 1 15a 4b 2 1 20a 3b 3 1 15a 2b 4 1 6ab 5 1 b 6.
Got It? 1. What is the expansion of (a 1 b)8 ? Use Pascal’s Triangle.
The Binomial Theorem gives a general formula for expanding a binomial.
Theorem Binomial Theorem
For every positive integer n,
(a 1 b)n 5 P0a n 1 P1a n21b 1 P2a n22b 2 1 c 1 Pn21ab n21 1 Pnb n
where P0, P1, c, Pn are the numbers in the nth row of Pascal’s Triangle.
Lesson 5-7 The Binomial Theorem
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When you use the Binomial Theorem to expand (x 2 2) 4, a 5 x and b 5 22. To
expand a binomial such as (3x 2 2) 5, a 5 3x so remember that a 4 5 (3x) 4 not 3x 4 .
Problem 2 Expanding a Binomial
What is the expansion of (3x 2 2) 5 ? Use the Binomial Theorem.
For (3x 2 2)5, use
the 5th row of Pascal’s
Triangle.
Pascal’s Triangle
1
1
1
1
1
1
The Binomial Theorem
uses a binomial sum.
3
4
5
1
2
1
3
6
10
1
4
10
1
5
1
(3x 2 2)5 5 (3x 1 (22))5
Apply the Binomial
Theorem.
5 (3x) 5 1 5(3x) 4(22) 1 1 10(3x) 3(22) 2
Simplify.
5 243x5 2 81 0x4 1 1 080x3 2 720x2 1 240x 2 32
1 10(3x) 2(22) 3 1 5(3x) 1(22) 4 1 1(22) 5
Got It? 2. a. What is the expansion of (2x 2 3) 4 ? Use the Binomial Theorem.
b. Reasoning Consider the following:
11 0 5 1
11 1 5 11
112 5 121
113 5 1331
114 5 14641
Why do these powers of 11 have digits that mirror Pascal’s Triangle?
Lesson Check
Do you know HOW?
Use Pascal’s Triangle to expand each binomial.
1. (x 1 a)3
2. (x 2 2)5
4. (3a 2 2)3
Chapter 5
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MATHEMATICAL
PRACTICES
5. Vocabulary Tell whether each expression can be
expanded using the Binomial Theorem.
a. (2a 2 6)4
b. (5x 2 1 1)5
c. (x 2 2 3x 2 4)3
6. Writing Describe the relationship between Pascal’s
Triangle and the Binomial Theorem.
3. (2x 1 4)2
328
Do you UNDERSTAND?
7. Reasoning Using Pascal’s Triangle, determine the
number of terms in the expansion of (x 1 a)12 . How
many terms are there in the expansion of (x 1 a)n ?
Polynomials and Polynomial Functions
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Practice and Problem-Solving Exercises
A
B
Practice
Apply
MATHEMATICAL
PRACTICES
See Problems 1 and 2.
Expand each binomial.
8. (x 2 y)3
9. (a 1 2)4
10. (6 1 a)6
11. (x 2 5)3
12. (y 1 1)8
13. (x 1 2)10
14. (b 2 4)7
15. (b 1 3)9
16. (2x 2 y)7
17. (a 1 3b)4
18. (4x 1 2)6
19. (4 2 x)8
20. (4x 1 5)2
21. (3a 2 7)3
22. (2a 1 16)6
23. (3y 2 11)4
24. Think About a Plan The side length of a cube is Q x 2 2 12 R . Determine the volume
of the cube.
• Rewrite the binomial as a sum.
• Consider (a 1 b)n. Identify a and b in the given binomial.
• Which row of Pascal’s Triangle can be used to expand the binomial?
25. In the expansion of (2m 2 3n)9, one of the terms contains m3.
a. What is the exponent of n in this term?
b. What is the coefficient of this term?
Find the specified term of each binomial expansion.
26. Fourth term of (x 1 2)5
27. Third term of (x 2 3)6
28. Third term of (3x 2 1)5
29. Fifth term of (a 1 5b2)4
30. Reasoning Explain why the coefficients in the expansion of (x 1 2y)3 do not
match the numbers in the 3rd row of Pascal’s Triangle.
31. Compare and Contrast What are the benefits and challenges of using the Binomial
Theorem when expanding (2x 1 3)2? Using FOIL? Which method would you
choose when expanding (2x 1 3)6? Why?
Expand each binomial.
32. (2x 2 2y)6
33. (x 2 1 4)10
34. (x 2 2 y 2) 3
35. (a 2 b 2)5
36. (3x 1 8y) 3
37. (4x 2 7y)4
38. (7a 1 2y)10
39. (4x 3 1 2y 2) 6
40. (3b 2 36)7
41. (5a 1 2b)3
42. (b 2 2 2)8
43. (22y 2 1 x)5
44. Geometry The side length of a cube is given by the expression (2x 1 8). Write a
binomial power for the area of a face of the cube and for the volume of the cube. Then
use the Binomial Theorem to expand and rewrite the powers in standard form.
45. Writing Explain why the terms of (x 2 y)n have alternating positive and negative signs.
46. Error Analysis A student expands (3x 2 8)4 as shown below. Describe and correct
the student’s error.
(3x – 8)4 = (3x)4 + 4(3x)3(–8) + 6(3x)2(–8)2 + 4(3x)(–8)3 + (–8)4
= 3x4 – 96x3 + 1152x2 – 6144x + 4096
Lesson 5-7 The Binomial Theorem
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C
Challenge
Use the Binomial Theorem to expand each complex expression.
47. (7 1 !216)5
48. ( !281 2 3)3
49. (x2 2 i)7
50. The first term in the expansion of a binomial (ax 1 by)n is 1024x 10. Find a and n.
8
51. Determine the coefficient of x7y in the expansion of Q 12 x 1 14 y R .
52. a. Expand (1 1 i)4.
b. Verify that 1 2 i is a fourth root of 24 by repeating the process in part (a) for
(1 2 i)4.
53. Verify that 21 1 !3i is a cube root of 8 by expanding (21 1 !3i)3.
Standardized Test Prep
54. What is the fourth term in the expansion of (2a 1 4b)5 ?
SAT/ACT
256a4b
768a3b 2
2560a 2b 3
2048ab4
55. Suppose y varies directly with x. If x is 30 when y is 10, what is x when y is 9?
3
27
300
9
29
56. Which of following is a root of 9x 2 2 30x 1 25 5 0?
3
x55
5
5
x53
3
x 5 23
x 5 25
57. One company charges a monthly fee of $7.95 and $2.25 per hour for Internet
access. Another company does not charge a monthly fee, but charges $2.75 per
hour for Internet access. Write a system of equations to represent the cost c for t
hours of access in one month for each company. Then find how many hours of use
it will take for the costs to be equal.
Extended
Response
Mixed Review
See Lesson 5-6.
Find all the roots of each equation.
58. x 4 1 7x 3 1 20x 2 1 29x 1 15 5 0
59. x 5 2 x 4 1 10x 3 2 10x 2 1 9x 2 9 5 0
60. 2x 3 1 11x 2 1 14x 1 8 5 0
61. x 4 2 x 3 1 6x 2 2 13x 1 7 5 0
See Lesson 4-8.
Simplify each expression.
62. (5i 2 4)(22i 1 7)
63. (23i)(20i)(10i)
26 2 2i
11i 1 9
64. 3 1 i
65. 2 2 i
Get Ready! To prepare for Lesson 5-8, do Exercises 66–68.
Write each polynomial in standard form. Then classify it by degree and by
number of terms.
66. 5x 2 2 x 1 2x 3 1 9
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Chapter 5
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67. 1 1 4x 2 7x 2
See Lesson 5-1.
68. 29x 2 1 x 2 3x 3 2 8 1 12x 4
Polynomials and Polynomial Functions
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