Chapter 8: Further Topics in Algebra

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

Copyright © 2007 Pearson Education, Inc. Slide 8-2

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

Copyright © 2007 Pearson Education, Inc. Slide 8-3

8.4 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

Copyright © 2007 Pearson Education, Inc. Slide 8-4

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.

Copyright © 2007 Pearson Education, Inc. Slide 8-5

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

Copyright © 2007 Pearson Education, Inc. Slide 8-6

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.

Copyright © 2007 Pearson Education, Inc. Slide 8-7

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

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

8.4 Binomial Coefficients

Binomial Coefficient

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

!

!( )!n r

n nC

r r n r

Copyright © 2007 Pearson Education, Inc. Slide 8-9

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

Copyright © 2007 Pearson Education, Inc. Slide 8-10

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!

Copyright © 2007 Pearson Education, Inc. Slide 8-11

8.4 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

Copyright © 2007 Pearson Education, Inc. Slide 8-12

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

Copyright © 2007 Pearson Education, Inc. Slide 8-13

8.4 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

Copyright © 2007 Pearson Education, Inc. Slide 8-14

8.4 Applying the Binomial Theorem

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

Copyright © 2007 Pearson Education, Inc. Slide 8-15

8.4 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 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

Copyright © 2007 Pearson Education, Inc. Slide 8-16

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

Copyright © 2007 Pearson Education, Inc. Slide 8-17

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