College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson

College AlgebraSixth EditionJames Stewart Lothar Redlin Saleem Watson

Sequences

and Series8

The Binomial Theorem8.6

Binomial

An expression of the form a + b

is called a binomial.

• Although in principle it’s easy to raise a + b to any power, raising it to a very high power would be tedious.

• Here, we find a formula that gives the expansion of (a + b)n for any natural number n and then prove it using mathematical induction.

Expanding (a + b)n

Expanding (a + b)n

To find a pattern in the expansion of (a + b)n,

we first look at some special cases:

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

( )

( ) 2

( ) 3 3

( ) 4 6 4

( ) 5 10 10 5

a b a b

a b a ab b

a b a a b ab b

a b a a b a b ab b

a b a a b a b a b ab b

Expanding (a + b)n

The following simple patterns emerge

for the expansion of (a + b)n:

1. There are n + 1 terms, the first being an and the last bn.

2. The exponents of a decrease by 1 from term to term while the exponents of b increase by 1.

3. The sum of the exponents of a and b in each term is n.

Expanding (a + b)n

For instance, notice how the exponents

of a and b behave in the expansion of

(a + b)5.

• The exponents of a decrease.• The exponents of b increase.

5

5 4 1 3 2 2 3 1 4 5

( )

5 10 10 5

a b

a a b a b a b a b b

Expanding (a + b)n

With these observations, we can write

the form of the expansion of (a + b)n for

any natural number n.

• For example, writing a question mark for the missing coefficients, we have:

• To complete the expansion, we need to determine these coefficients.

8 8 7 6 2 5 3 4 4

3 5 2 6 7 8

( ) ? ? ? ?? ? ?

a b a a b a b a b a ba b a b ab b

Expanding (a + b)n

To find a pattern, let’s write the coefficients

in the expansion of (a + b)n for the first few

values of n in a triangular array, which is

called Pascal’s triangle.

Pascal’s Triangle

The row corresponding to (a + b)0

is called the zeroth row.• It is included to show the symmetry of the array.

0

1

2

3

4

5

( ) 1

( ) 1 1

( ) 1 2 1

( ) 1 3 3 1

( ) 1 4 6 4 1

( ) 1 5 10 10 5 1

a b

a b

a b

a b

a b

a b

Key Property of Pascal’s Triangle

The key observation about Pascal’s

triangle is the following property.

Every entry (other than a 1) is the sum

of the two entries diagonally above it.

• From this property, it’s easy to find any row of Pascal’s triangle from the row above it.

Key Property of Pascal’s Triangle

For instance, we find the sixth and

seventh rows, starting with the fifth row:

5

6

7

( ) 1 5 10 10 5 1

( ) 1 6 15 20 15 6 1

( ) 1 7 21 35 35 21 7 1

a b

a b

a b

Key Property of Pascal’s Triangle

To see why this property holds, let’s

consider the following expansions:

5

5 4 3 2 2 3 4 5

6

6 5 4 2 3 3 2 4 5 6

( )

5 10 10 5

( )

6 15 20 15 6

a b

a a b a b a b ab b

a b

a a b a b a b a b ab b

Key Property of Pascal’s Triangle

We arrive at the expansion of (a + b)6

by multiplying (a + b)5 by (a + b).• Notice, for instance, that the circled term

in the expansion of (a + b)6 is obtained via this multiplication from the two circled terms above it.

5

5 4 3 2 2 3 4 5

6

6 5 4 2 3 3 2 4 5 6

( )

5 10 10 5

( )

6 15 20 15 6

a b

a a b a b a b ab b

a b

a a b a b a b a b ab b

Key Property of Pascal’s Triangle

We get this when the two terms above it

are multiplied by b and a, respectively.

• Thus, its coefficient is the sum of the coefficients of these two terms.

5

5 4 3 2 2 3 4 5

6

6 5 4 2 3 3 2 4 5 6

( )

5 10 10 5

( )

6 15 20 15 6

a b

a a b a b a b ab b

a b

a a b a b a b a b ab b

Key Property of Pascal’s Triangle

We will use this observation at

the end of the section when we prove

the Binomial Theorem.

• Having found these patterns, we can now easily obtain the expansion of any binomial, at least to relatively small powers.

E.g. 1—Expanding a Binomial Using Pascal’s Triangle

Find the expansion of (a + b)7 using

Pascal’s triangle.

• The first term in the expansion is a7, and the last term is b7.

• Using the fact that the exponent of a decreases by 1 from term to term and that of b increases by 1 from term to term, we have:

7 7 6 5 2 4 3

3 4 2 5 6 7

( ) ? ? ?

? ? ?

a b a a b a b a b

a b a b ab b

The appropriate coefficients appear

in the seventh row of Pascal’s triangle.

Thus,

E.g. 1—Expanding a Binomial Using Pascal’s Triangle

7 7 6 5 2 4 3

3 4 2 5 6 7

( ) 7 21 35

35 21 7

a b a a b a b a b

a b a b ab b

E.g. 2—Expanding a Binomial Using Pascal’s Triangle

Use Pascal’s triangle to expand

(2 – 3x)5

• We find the expansion of (a + b)5 and then substitute 2 for a and –3x for b.

• Using Pascal’s triangle for the coefficients, we get:

5 5 4 3 2 2 3

4 5

( ) 5 10 10

5

a b a a b a b a b

ab b

Substituting a = 2 and b = –3x gives:

E.g. 2—Expanding a Binomial Using Pascal’s Triangle

5

5 4 3 2

2 3 4 5

2 3

4 5

(2 3 )

(2) 5(2) ( 3 ) 10(2) ( 3 )

10(2) ( 3 ) 5(2)( 3 ) ( 3 )

32 240 720 1080

810 243

x

x x

x x x

x x x

x x

The Binomial Coefficients

The Binomial Coefficients

Although Pascal’s triangle is useful in finding

the binomial expansion for reasonably small

values of n, it isn’t practical for finding (a + b)n

for large values of n.

• The reason is that the method we use for finding the successive rows of Pascal’s triangle is recursive.

• Thus, to find the 100th row of this triangle, we must first find the preceding 99 rows.

The Binomial Coefficients

We need to examine the pattern in

the coefficients more carefully to develop

a formula that allows us to calculate directly

any coefficient in the binomial expansion.

• Such a formula exists, and the rest of the section is devoted to finding and proving it.

• However, to state this formula, we need some notation.

n factorial

The product of the first n natural

numbers is denoted by n! and is called

n factorial:

n! = 1 · 2 · 3 · … · (n – 1) · n

0 factorial

We also define 0! as follows:

0! = 1

• This definition of 0! makes many formulas involving factorials shorter and easier to write.

The Binomial Coefficient

Let n and r be nonnegative integers

with r ≤ n.

The binomial coefficient is denoted by

and is defined by:

n

r

!

!( )!

n n

r r n r

E.g. 3—Calculating Binomial Coefficients

9 9! 9!

4 4!(9 4)! 4!5!

1 2 3 4 5 6 7 8 9

(1 2 3 4)(1 2 3 4 5)

6 7 8 9

1 2 3 4126

Example (a)

100 100!

3 3!(100 3)!

1 2 3 97 98 99 100

(1 2 3)(1 2 3 97)

98 99 100

1 2 3161,700

E.g. 3—Calculating Binomial Coefficients Example (b)

100 100!

97 97!(100 97)!

1 2 3 97 98 99 100

(1 2 3 97)(1 2 3)

98 99 100

1 2 3161,700

E.g. 3—Calculating Binomial Coefficients Example (c)

Binomial Coefficients

Although the binomial coefficient is

defined in terms of a fraction, all the results

of Example 3 are natural numbers.

• In fact, is always a natural number.

• See Exercise 54.

n

r

n

r

Binomial Coefficients

Notice that the binomial coefficients in

parts (b) and (c) of Example 3 are equal.

• This is a special case of the following relation.

• You are asked to prove this in Exercise 52.

n nr n r

Binomial Coefficients

To see the connection between the binomial

coefficients and the binomial expansion

of (a + b)n, let’s calculate these binomial

coefficients:

5 5 51 5 10

0 1 2

5 5 510 5 1

3 4 5

Binomial Coefficients

• These are precisely the entries in the fifth row of Pascal’s triangle.

• In fact, we can write Pascal’s triangle as follows.

5 5 51 5 10

0 1 2

5 5 510 5 1

3 4 5

Binomial Coefficients

0

0

1 1

0 1

2 2 2

0 1 2

3 3 3 3

0 1 2 3

4 4 4 4 4

0 1 2 3 4

5 5 5 5 5 5

0 1 2 3 4 5

. . . . . . .

.0 1 2

n n n

. .1

n n

n n

0

0

1 1

0 1

2 2 2

0 1 2

3 3 3 3

0 1 2 3

4 4 4 4 4

0 1 2 3 4

5 5 5 5 5 5

0 1 2 3 4 5

. . . . . . .

.0 1 2

n n n

. .1

n n

n n

0

0

1 1

0 1

2 2 2

0 1 2

3 3 3 3

0 1 2 3

4 4 4 4 4

0 1 2 3 4

5 5 5 5 5 5

0 1 2 3 4 5

. . . . . . .

.0 1 2

n n n

. .1

n n

n n

0

0

1 1

0 1

2 2 2

0 1 2

3 3 3 3

0 1 2 3

4 4 4 4 4

0 1 2 3 4

5 5 5 5 5 5

0 1 2 3 4 5

. . . . . . .

.0 1 2

n n n

. .1

n n

n n

0

0

1 1

0 1

2 2 2

0 1 2

3 3 3 3

0 1 2 3

4 4 4 4 4

0 1 2 3 4

5 5 5 5 5 5

0 1 2 3 4 5

. . . . . . .

.0 1 2

n n n

. .1

n n

n n

0

0

1 1

0 1

2 2 2

0 1 2

3 3 3 3

0 1 2 3

4 4 4 4 4

0 1 2 3 4

5 5 5 5 5 5

0 1 2 3 4 5

. . . . . . .

.0 1 2

n n n

. .1

n n

n n

Binomial Coefficients

To demonstrate that this pattern holds,

we need to show that any entry in this version

of Pascal’s triangle is the sum of the two

entries diagonally above it.

• That is, we must show that each entry satisfies the key property of Pascal’s triangle.

• We now state this property in terms of the binomial coefficients.

Key Property of the Binomial Coefficients

For any nonnegative integers r and k

with r ≤ k,

• The two terms on the left side are adjacent entries in the kth row of Pascal’s triangle.

• The term on the right side is the entry diagonally below them, in the (k + 1)st row.

1

1

k k k

r r r

Key Property of the Binomial Coefficients

Thus, this equation is a restatement

of the key property of Pascal’s triangle

in terms of the binomial coefficients.

• A proof of this formula is outlined in Exercise 53.

The Binomial Theorem

The Binomial Theorem

• We prove this at the end of the section.

• First, let’s look at some of its applications.

1 2 2

1

( )0 1 2

1

n n n n

n n

n n na b a a b a b

n nab b

n n

E.g. 4—Expanding a Binomial Using Binomial Theorem

Use the Binomial Theorem

to expand (x + y)4

• By the Binomial Theorem,

4 4 3 2 2

3 4

4 4 4( )

0 1 2

4 4

3 4

x y x x y x y

xy y

Verify that:

• It follows that:

4 4 4 4 41 4 6 4 1

0 1 2 3 4

E.g. 4—Expanding a Binomial Using Binomial Theorem

4 4 3 2 2 3 4( ) 4 6 4x y x x y x y xy y

E.g. 5—Expanding a Binomial Using the Binomial Theorem

Use the Binomial Theorem

to expand

• We first find the expansion of (a + b)8.

• Then, we substitute for a and –1 for b.

8( 1)x

x

Using the Binomial Theorem, we have:

E.g. 5—Expanding a Binomial Using the Binomial Theorem

8

8 7 6 2 5 3 4 4

3 5 2 6 7 8

( )

8 8 8 8 8

0 1 2 3 4

8 8 8 8

5 6 7 8

a b

a a b a b a b a b

a b a b ab b

Verify that:

• So,

E.g. 5—Expanding a Binomial Using the Binomial Theorem

8 8 8 8 81 8 28 56 70

0 1 2 3 4

8 8 8 856 28 8 1

5 6 7 8

8 8 7 6 2 5 3 4 4

3 5 2 6 7 8

( ) 8 28 56 70

56 28 8

a b a a b a b a b a b

a b a b ab b

Performing the substitutions

a = x1/2 and b = –1 gives:

E.g. 5—Expanding a Binomial Using the Binomial Theorem

8 1/ 2 8 1/ 2 7 1/ 2 6 2

1/ 2 5 3 1/ 2 4 4

1/ 2 3 5 1/ 2 2 6

1/ 2 7 8

( 1) ( ) 8( ) ( 1) 28( ) ( 1)

56( ) ( 1) 70( ) ( 1)

56( ) ( 1) 28( ) ( 1)

8( )( 1) ( 1)

x x x x

x x

x x

x

This simplifies to:

E.g. 5—Expanding a Binomial Using the Binomial Theorem

8 4 7 / 2 3 5 / 2

2 3 / 2

1/ 2

( 1) 8 28 56

70 56

28 8 1

x x x x x

x x

x x

General Term of the Binomial Expansion

The Binomial Theorem can be used to find

a particular term of a binomial expansion

without having to find the entire expansion.

The term that contains ar in the expansion

of (a + b)n is:r n rn

a bn r

E.g. 6—Finding a Particular Term in a Binomial Expansion

Find the term that contains x5

in the expansion of (2x + y)20.

• The term that contains x5 is given by the formula for the general term with:

a = 2x, b = y, n = 20, r = 5

So, this term is:

E.g. 6—Finding a Particular Term in a Binomial Expansion

5 15 5 15

5 15

5 15

20 20!(2 )

15 15!(20 15)!

20!32

15!5!

496,128

a b x y

x y

x y

E.g. 7—Finding a Particular Term in a Binomial Expansion

Find the coefficient of x8

in the expansion of

• Both x2 and 1/x are powers of x.

• So, the power of x in each term of the expansion is determined by both terms of the binomial.

102 1

xx

To find the required coefficient, we first

find the general term in the expansion.

• By the formula, we have: a = x2, b = 1/x, n = 10

• So, the general term is:10

2 2 1 10

3 10

10 101( ) ( )

10 10

10

10

rr r r

r

x x xr x r

xr

E.g. 7—Finding a Particular Term in a Binomial Expansion

Thus, the term that contains x8

is the term in which

3r – 10 = 8

r = 6• So, the required coefficient is:

10 10210

10 6 4

E.g. 7—Finding a Particular Term in a Binomial Expansion

Proof of the Binomial Theorem

Binomial Theorem

We now give a proof of

the Binomial Theorem using

mathematical induction.

Binomial Theorem—Proof

Let P(n) denote the statement

1 2 2

1

( )0 1 2

1

n n n n

n n

n n na b a a b a b

n nab b

n n

Binomial Theorem—Proof

Step 1: We show that P(1) is true.

• However, P(1) is just the statement

which is certainly true.

1 1 11 1( ) 1 1

0 1a b a b a b a b

Binomial Theorem—Proof

Step 2: We assume that P(k) is true.

• Thus, our induction hypothesis is:

• We use this to show that P(k + 1) is true.

1 2 2

1

( )0 1 2

1

k k k k

k k

k k ka b a a b a b

k kab b

k k

Binomial Theorem—Proof

1

1 2 2

1

( )

( )[( ) ]

( )0 1 2

1

k

k

k k k

k k

a b

a b a b

k k ka b a a b a b

k kab b

k k

Binomial Theorem—Proof

1 2 2

1

1 2 2

1

0 1 2

1

0 1 2

1

k k k

k k

k k k

k k

k k ka a a b a b

k kab b

k k

k k kb a a b a b

k kab b

k k

Binomial Theorem—Proof

1 1 2

2 1

1 2 2 3

1

0 1 2

1

0 1 2

1

k k k

k k

k k k

k k

k k ka a b a b

k ka b ab

k k

k k ka b a b a b

k kab b

k k

Binomial Theorem—Proof

1

1 2

1

0 0 1

1 2

1

k k

k

k k

k k ka a b

k ka b

k k kab b

k k k

Binomial Theorem—Proof

Using the key property of the binomial

coefficients, we can write each of

the expressions in square brackets as

a single binomial coefficient.

Binomial Theorem—Proof

Also, writing the first and last coefficients

as

(these are equal to 1 by Exercise 50)

gives the following result.

1 10 1andk k

k

Binomial Theorem—Proof

• However, this last equation is precisely P(k + 1).

• This completes the induction step.

1 1

1 2

1

1 1( )

0 1

1

2

1 1

1

k k k

k

k k

k ka b a a b

ka b

k kab b

k k

Binomial Theorem—Proof

Having proved Steps 1 and 2, we

conclude, by the Principle of Mathematical

Induction, that the theorem is true for

all natural numbers n.