Slide 11.5- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

The Binomial Theorem

Learn to use Pascal’s Triangle to compute binomial coefficients.Learn to use Pascal’s Triangle to expand a binomial power.Learn to use the Binomial Theorem to expand a binomial power.Learn to find the coefficient of a term in a binomial expansion.

SECTION 11.5

1

2

3

4

Slide 11.5- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

PASCAL’S TRIANGLE

When expanding (x + y)n the coefficients of each term can be determined using Pascal’s Triangle. The top of the triangle, that is, the first row, which contains only the number 1, represents the coefficients of (x + y)0 and is referred to as Row 0. Row 2 represents the coefficients of (x + y)1. Each row begins and ends with 1. Each entry of Pascal’s Triangle if found by adding the two neighboring entries in the previous row.

Slide 11.5- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

PASCAL’S TRIANGLE

An infinite sequence is a function whose

Slide 11.5- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Expand (4y – 2x)5 .

x y 5 x5 5x4y 10x3y2 10x2y3 5xy4 y5

Solution

Row 5 of Pascal’s Triangle yields the binomial coefficients 1, 5, 10, 10, 5, 1.

Replace x with 4y and y with –2x.

EXAMPLE 1Using Pascal’s Triangle to Expand a Binomial Power

Slide 11.5- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 1Using Pascal’s Triangle to Expand a Binomial Power

x y 5 4y 2x 5

Solution continued

Expanding a difference results in alternating signs.

1024y5 2560y4x 2560y3x2

1280y2x3 320yx4 32x5

4y 5 5 4y 4 2x 10 4y 3 2x 2

10 4y 2 2x 3 5 4y 2x 4 2x 5

Slide 11.5- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

DEFINITION OF

If r and n are integers with 0 ≤ r ≤ n, the we define

n

r

n!

r! n r !

n

r

n

0

1 and

n

n

1

Slide 11.5- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

THE BINOMIAL THEOREM

If n is a natural number, then the binomial expansion of (x + y)n is given by

x y n

n

0

xn

n

1

xn 1y

n

2

xn 2y2 L

n

n

yn

n

r

xn ryr

i0

n

.

The coefficient of xn–ryr isn

r

n!

r! n r ! .

Slide 11.5- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

PARTICULAR TERM IN A BINOMIAL EXPRESSION

The term containing the factor xr in the expansion of (x + y)n is

n

n r

xryn r .

Slide 11.5- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 5Finding a Particular Term in a Binomial Expansion

Find the term containing x10 in the expansion of (x + 2a)15 .

Solution

Use the formula for the term containing xr.

n

n r

xryn r

15

15 10

x10 2a 15 10

15

5

x10 2a 5

Slide 11.5- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 5Finding a Particular Term in a Binomial Expansion

Solution continued

15!

5! 15 5 ! x10 25a5

15!

5! 10 !32x10a5

1514 13121110!

5!10!32x10a5

96,096x10a5