Chapter 8: Further Topics in Algebra

Copyright © 2007 Pearson Education, Inc. Slide 8-1


Chapter 8: Further Topics in Algebra

8.1 Sequences and Series

8.2 Arithmetic Sequences and Series

8.3 Geometric Sequences and Series

8.4 The Binomial Theorem

8.5 Mathematical Induction

8.6 Counting Theory

8.7 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


8.4 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.


8.4 A Binomial Expansion Pattern

Pascal’s Triangle

Row

1 0

1 1 1

1 2 1 2

1 3 3 1 3

1 4 6 4 1 4

1 5 10 10 5 1 5


8.4 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.


8.4 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


8.4 Binomial Coefficients

Binomial Coefficient

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

!

!( )!n r

n nC

r r n r


8.4 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 rCn

r

n

r

3

0

3

2


8.4 Evaluating Binomial Coefficients

Example Evaluate (a) (b)

Solution

(a)

(b)

6

2

8

0

6 6! 6! 6 5 4 3 2 115

2 2!(6 2)! 2!4! 2 1 4 3 2 1

8 8! 8! 8!1

0 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


8.4 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 9

4 5 6 7

9

8

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

2

ba

2 35 5 4 3 2

4 5

5 5 5( )

1 2 32 2 2 2

5

4 2 2

b b b ba a a a a

b ba

,2

bx 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 1

2 2 4 16 32

b b b ba a a a a

b ba

a a b a b a b ab b


8.4 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


8.4 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

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Chapter 8: Further Topics in Algebra