Transcript Document
Copyright © 2011 Pearson Education, Inc.
Slide 11.5-1
Chapter 11: Further Topics in Algebra
11.1Sequences and Series
11.2Arithmetic Sequences and Series
11.3Geometric Sequences and Series
11.4Counting Theory
11.5The Binomial Theorem
11.6Mathematical Induction
11.7Probability
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Slide 11.5-2
11.5 The Binomial Theorem
The binomial expansions
( x y )0 1
( x y )1 x y
( x y ) 2 x 2 2 xy y 2
( x y )3 x 3 3 x 2 y 3xy 2 y 3
( x y ) 4 x 4 4 x 3 y 6 x 2 y 2 4 xy 3 y 4
( x y )5 x 5 5 x 4 y 10 x 3 y 2 10 x 2 y 3 5 xy 4 y 5
reveal a pattern.
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Slide 11.5-3
11.5 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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Slide 11.5-4
11.5 Pascal’s Triangle
Pascal’s Triangle
Row
0
1
1
1
1
1
1
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2
3
4
5
1
1
3
6
10
1
2
1
4
10
3
1
5
4
1
5
Slide 11.5-5
11.5 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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Slide 11.5-6
11.5 Binomial Coefficients
Binomial Coefficient
For nonnegative integers n and r, with r < n,
n
n!
r r !(n r )!
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Slide 11.5-7
11.5 Binomial Coefficients
n
• The symbols n Cr and for the binomial
r
coefficients are read “n choose r”
n
• The values of are the values in the nth row
r
3
of Pascal’s triangle. So is the first number
0
3
in the third row and is the third.
2
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Slide 11.5-8
11.5 Evaluating Binomial Coefficients
6
Example Evaluate (a)
2
8
(b)
0
Solution
6
6!
6! 6 5 4 3 2 1
15
(a)
2 2!(6 2)! 2!4! 2 1 4 3 2 1
8
8!
8!
8!
1
(b)
0 0!(8 0)! 0!8! 1 8!
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Slide 11.5-9
11.5 The Binomial Theorem
Binomial Theorem
For any positive integer n,
n n 1 n n 2 2 n n 3 3
( x y) x x y x y x y
1
2
3
n nr r
n n 1
n
... x y ...
xy
y
r
n 1
n
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n
Slide 11.5-10
11.5 Applying the Binomial Theorem
Example Write the binomial expansion of ( x y )9 .
Solution Use the binomial theorem
9 8 9 7 2 9 6 3
( x y) x x y x y x y
1
2
3
9 5 4 9 4 5 9 3 6 9 2 7
x y x y x y x y
4
5
6
7
9 8
xy y 9
8
9
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9
Slide 11.5-11
11.5 Applying the Binomial Theorem
9! 8
9! 7 2 9! 6 3
( x y) x
x y
x y
x y
1!8!
2!7!
3!6!
9! 5 4 9! 4 5 9! 3 6
9! 2 7
x y
x y
x y
x y
4!5!
5!4!
6!3!
7!2!
9! 8
xy y 9
8!1!
x9 9 x8 y 36 x 7 y 2 84 x 6 y 3 126 x 5 y 4 126 x 4 y 5
9
9
84 x3 y 6 36 x 2 y 7 9 xy 8 y 9
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Slide 11.5-12
11.5 Applying the Binomial Theorem
5
b
Example Expand a .
2
Solution Use the binomial theorem with
b
xa , y
and n = 5,
2
2
5
5
4 b 3 b
b
5
a a 1a 2 a
2
2 2
5
3
4
5
5
5
2 b b b
a a
3 2 4 2 2
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Slide 11.5-13
11.5 Applying the Binomial Theorem
Solution
5
b
5
4 b
3 b
a a 5a 10a
2
2
2
3
4
2
5
b
b b
10a 5a
2
2 2
5 4 5 3 2 5 2 3 5 4 1 5
5
a a b a b a b ab b
2
2
4
16
32
2
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Slide 11.5-14
11.5 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
n n( r 1) r 1
y .
x
r 1
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Slide 11.5-15
11.5 Finding a Specific Term of a
Binomial Expansion
10
(
a
2
b
)
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 7
3
7
3
7 3
a (2b) 120a 8b 960a b .
3
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Slide 11.5-16