1632 the binomial theorem-02
TRANSCRIPT
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Copyright © 2007 Pearson Education, Inc. Slide 11-2
Chapter 11: Further Topics in Algebra
11.1 Sequences and Series
11.2 Arithmetic Sequences and Series
11.3 Geometric Sequences and Series
11.4 The Binomial Theorem
11.5 Mathematical Induction
11.6 Counting Theory
11.7 Probability
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Copyright © 2007 Pearson Education, Inc. Slide 11-3
11.4 The Binomial Theorem
The binomial expansions
reveal a pattern.
0
1
2 2 2
3 3 2 2 3
4 4 3 2 2 3 4
5 5 4 3 2 2 3 4 5
( ) 1
( )
( ) 2
( ) 3 3
( ) 4 6 4
( ) 5 10 10 5
x y
x y x y
x y x xy y
x y x x y xy y
x y x x y x y xy y
x y x x y x y x y xy y
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Copyright © 2007 Pearson Education, Inc. Slide 11-4
11.4 A Binomial Expansion Pattern
• The expansion of (x + y)n begins with x n and ends with y n .
• The variables in the terms after x n follow the pattern x n-1y , x n-2y2 , x n-3y3 and so on to y n . With each term the exponent on x decreases by 1 and the exponent on y increases by 1.
• In each term, the sum of the exponents on x and y is always n.
• The coefficients of the expansion follow Pascal’s triangle.
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Copyright © 2007 Pearson Education, Inc. Slide 11-5
11.4 A Binomial Expansion Pattern
Pascal’s Triangle
Row1 0
1 1 11 2 1 2
1 3 3 1 31 4 6 4 1 4
1 5 10 10 5 1 5
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Copyright © 2007 Pearson Education, Inc. Slide 11-6
11.4 Pascal’s Triangle
• Each row of the triangle begins with a 1 and ends with a 1.
• Each number in the triangle that is not a 1 is the sum of the two numbers directly above it (one to the right and one to the left.)
• Numbering the rows of the triangle 0, 1, 2, … starting at the top, the numbers in row n are the coefficients of x n, x n-1y , x n-2y2 , x n-3y3, … y n in the expansion of (x + y)n.
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Copyright © 2007 Pearson Education, Inc. Slide 11-7
11.4 n-Factorial
n-Factorial
For any positive integer n,
and! ( 1)( 2) (3)(2)(1),
0! 1 .n n n n
Example Evaluate (a) 5! (b) 7!
Solution (a)
(b)
5! 5 4 3 2 1 120
7! 7 6 5 4 3 2 1 5040
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Copyright © 2007 Pearson Education, Inc. Slide 11-8
11.4 Binomial Coefficients
Binomial Coefficient
For nonnegative integers n and r, with r < n,
!!( )!n r
n nCr r n r
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Copyright © 2007 Pearson Education, Inc. Slide 11-9
11.4 Binomial Coefficients
• The symbols and for the binomial
coefficients are read “n choose r”
• The values of are the values in the nth row
of Pascal’s triangle. So is the first number
in the third row and is the third.
n rC nr
nr
30
32
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Copyright © 2007 Pearson Education, Inc. Slide 11-10
11.4 Evaluating Binomial Coefficients
Example Evaluate (a) (b)
Solution
(a)
(b)
62
80
6 6! 6! 6 5 4 3 2 1 152 2!(6 2)! 2!4! 2 1 4 3 2 1
8 8! 8! 8! 10 0!(8 0)! 0!8! 1 8!
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Copyright © 2007 Pearson Education, Inc. Slide 11-11
11.4 The Binomial Theorem
Binomial Theorem
For any positive integers n,
1 2 2 3 3
1
( )1 2 3
... ...1
n n n n n
n r r n n
n n nx y x x y x y x y
n nx y xy y
r n
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Copyright © 2007 Pearson Education, Inc. Slide 11-12
11.4 Applying the Binomial Theorem
Example Write the binomial expansion of .
Solution Use the binomial theorem
9( )x y
9 9 8 7 2 6 3
5 4 4 5 3 6 2 7
8 9
9 9 9( )
1 2 3
9 9 9 94 5 6 7
98
x y x x y x y x y
x y x y x y x y
xy y
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Copyright © 2007 Pearson Education, Inc. Slide 11-13
11.4 Applying the Binomial Theorem
9 9 8 7 2 6 3
5 4 4 5 3 6 2 7
8 9
9 8 7 2 6 3 5 4 4 5
3 6 2 7 8 9
9! 9! 9!( )1!8! 2!7! 3!6!
9! 9! 9! 9!4!5! 5!4! 6!3! 7!2!9!
8!1!9 36 84 126 126
84 36 9
x y x x y x y x y
x y x y x y x y
xy y
x x y x y x y x y x y
x y x y xy y
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Copyright © 2007 Pearson Education, Inc. Slide 11-14
11.4 Applying the Binomial Theorem
Example Expand .
Solution Use the binomial theorem with
and n = 5,
5
2ba
2 35 5 4 3 2
4 5
5 5 5( )
1 2 32 2 2 2
54 2 2
b b b ba a a a a
b ba
,2bx a y
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Copyright © 2007 Pearson Education, Inc. Slide 11-15
11.4 Applying the Binomial Theorem
Solution
2 35 5 4 3 2
4 5
5 4 3 2 2 3 4 5
( ) 5 10 102 2 2 2
52 2
5 5 5 5 12 2 4 16 32
b b b ba a a a a
b ba
a a b a b a b ab b
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Copyright © 2007 Pearson Education, Inc. Slide 11-16
11.4 rth Term of a Binomial Expansion
rth Term of the Binomial Expansion
The rth term of the binomial expansion of (x + y)n,
where n > r – 1, is
( 1) 1
1n r rn
x yr
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Copyright © 2007 Pearson Education, Inc. Slide 11-17
11.4 Finding a Specific Term of a Binomial Expansion.
Example Find the fourth term of .
Solution Using n = 10, r = 4, x = a, y = 2b in the formula, we find the fourth term is
10( 2 )a b
7 3 7 3 7 310(2 ) 120 8 960 .
3a b a b a b