Download Guided Notes 5_7

The Binomial Theorem
5-7
Vocabulary
Review
1. Identify the next number in each pattern.
1, 2, 1, 2, 1,
1, 3, 5, 7, 9,
100, 90, 80, 70, 60,
Vocabulary Builder
(2x 5)2 4x2 20x 25
expand (verb) ek SPAND
binomial to
a power
expansion of
the binomial
Math Usage: You expand a power of a polynomial by doing the multiplying
to find all the terms.
Use Your Vocabulary
2. Place a ✓ if the sentence shows the correct use of expand. Place an ✗ if it does not.
The writer expanded the short story into a novel.
Heat expands most metals.
The balloon expanded as all of the air was let out of it.
The bird expanded its wings.
The expanded form of (3x2 2 4)2 is 9x4 2 24x2 1 16.
3. Circle a rule for squaring binomials that you could use to expand (a 2 b)2 .
(a 2 b)2 5 (a 1 b)(a 2 b)
(a 2 b)2 5 a2 2 2ab 1 b2
(a 2 b)2 5 a2 1 2ab 1 b2
4. The expression 25 2 10x 1 x2 is the expansion of which power of a binomial?
Circle your answer.
52 2 x2
Chapter 5
(25 2 x)2
(5 2 x)2
142
(5 1 x)2
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Definition: You expand something by increasing it,
stretching it out, or giving it more detail.
Pascal’s Triangle is a triangular array of numbers. 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.
5. Use the description above to complete Pascal’s Triangle.
1
1
1
1
1
2
1
1
1
1
Using Pascal’s Triangle
Problem 1
Got It? What is the expansion of (a 1 b)8 ? Use Pascal’s Triangle.
6. Circle the row of Pascal’s Triangle that you would use to expand (a 1 b)8.
Pascal’s Triangle
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
1
1
1
1
2
1
1
1
1
1
1
1
9
10
28
4
20
35
15
70
126
1
5
35
56
84
1
10
15
21
36
3
6
5
7
8
3
4
6
1
1
6
1
21
56
126
7
28
84
1
8
36
1
9
1
7. Use the row you circled to complete the expansion.
(a 1 b)8 5
a8 1
1
a7b 1
a2b6 1
a6b2 1
ab7 1
a5b3 1
a4b4 1
a3b5
b8
143
Lesson 5-7
Theorem
Binomial Theorem
For every positive integer n, (a 1 b)n 5 P0an 1 P1an21b 1 P2an22b2 1 c 1 Pn21abn21 1 Pnbn ,
where P0 , P1 , c, Pn are the numbers in the nth row of Pascal’s Triangle.
8. Complete the row of Pascal’s Triangle that you would use to expand (a 1 b)5 .
1
5
9. Write the row of Pascal’s Triangle that you would use to expand (a 1 b)6 .
Expanding a Binomial
Problem 2
Got It? What is the expansion of (2x 2 3)4 ? Use the Binomial Theorem.
10. Identify the values of a, b, and n.
a5
b5
n5
11. Write the expression in the form (a 1 b)n .
(2x 2 3)4 5 Q
R
1
12. Circle the row of Pascal’s Triangle that you would use to expand (2x 2 3)4 .
Pascal’s Triangle
1
1
2
1
1
1
3
4
1
1
3
6
5
1
4
10
10
1
5
1
13. Use the row you circled to complete the expansion of (2x 2 3)4 . Simplify the result.
(2x 2 3)4 5
5
4
Q
R 1
x4 2
(2x)3 Q
x3 1
R1
x2 2
(2x)2 Q
2
R 1
(2x) Q
3
R 1
x1
Lesson Check • Do you UNDERSTAND?
Vocabulary Tell whether each expression can be expanded using the Binomial Theorem.
(2a 2 6)4
(5x2 1 1)5
(x2 2 3x 2 4)3
14. Circle the expressions that are written in the form (a 1 b)n .
(2a 2 6)4
Chapter 5
(5x2 1 1)5
(x2 2 3x 2 4)3
144
Q
R
4
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
1
15. Place a ✓ if you can use the Binomial Theorem to expand the expression. Place an ✗ if
you cannot.
(2a 2 6)4
(5x2 1 1)5
(x2 2 3x 2 4)3
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 ?
16. Circle the row of Pascal’s Triangle you would use to expand (x 1 a)3 .
1
1
1
1
2
1
3
1
1
3
4
1
6
4
1
17. Write the expansion of (x 1 a)3 .
1
1
1
18. Compare. Write ,, ., or 5. Then complete the last expression.
the number of terms in
531
the expansion of (x 1 a)3
the number of terms in
the row of Pascal’s Triangle
19. Circle the number of terms in the row of Pascal’s Triangle you would use to expand
(x 1 a)12 .
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
1
2
10
11
12
13
20. Circle the number of terms in the expansion of (x 1 a)12 .
1
2
10
11
12
13
n
n11
21. Circle the number of terms in the expansion of (x 1 a)n .
1
2
n21
10
Math Success
Check off the vocabulary words that you understand.
binomial
Pascal’s Triangle
Binomial Theorem
Rate how well you can expand a binomial.
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get it!
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145
Lesson 5-7