1632 the binomial theorem-02

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



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


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.


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


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.


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


11.4 Binomial Coefficients

Binomial Coefficient

For nonnegative integers n and r, with r < n,

!!( )!n r

n nCr r n r


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


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!



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


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



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



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



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


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


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

1632 the binomial theorem-02

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