2-3 Factorials and the Binomial Theorem

Unit 2 Sequences and Series

Concepts and Objectives

Factorials and the Binomial Theorem (Obj. #8)

Be able to use the definition of factorials to simplify expressions containing factorials, or to express in factorial form expressions containing products of consecutive integers.

Given a binomial power, expand it as a binomial series in one step

Given a binomial power of the form , find term number k, or find the term which contains br, where kand r are integers from 0 through n.

n

a b

Factorials

The expression n! (read “n factorial”) means the product of the first n consecutive positive integers.

For example, 5! = 5 4 3 2 1 = 120

also, 5! = 5 4 3 2 1

= 5 4!

This behavior leads to a very important property:

! 1 !n n n

Factorials

Just as we can multiply n–1! by n to produce n!, we can reverse the process and divide n! by n to produce n–1! :

Thus, 0! = 1.

4! 24

3! 6

2! 2

1! 1

0! 1?

1

2

3

4

1

11

Factorials

Fractions which have factorials in the numerator and denominator can often be cancelled.

Example: Simplify 10!

7!

10! 10 9 8 7!

7! 7!

10 9 8

720

Factorials

We can also use these properties to write a product of consecutive integers as a ratio of factorials.

Example: Write the product 11 10 9 8 as a fraction.

7!

11 10 9 8 11 10 9 8 7!

11!

7!

Factorials

When dealing with variables, keep the definition of a factorial in mind.

Example: Simplify

1 !

1 !

n

n

1 ! 1 1 !

1 ! 1 !

n n n n

n n

1n n

Binomial Series

A binomial squared becomes

A binomial cubed becomes

2 2 22a b a ab b

3 2

a b a b a b

2 22a b a ab b

2 23 32 22 2a b a b aba b ba

3 2 2 33 3a a b ab b

Binomial Series

As you may recall from Algebra II, the coefficients correspond to rows from Pascal’s Triangle

0 1

1 1 1

2 1 2 1

3 1 3 3 1

4 1 4 6 4 1

5 1 5 10 10 5 1

6 1 6 15 20 15 6 1

Binomial Series

Example: Expand

a = 2x and b = 1; the exponents begin and end at 5 (a goes down while b goes up). Looking at row 5 on the triangle, our coefficients are 1, 5, 10, 10, 5, 1, so we write our expression as follows:

(Notice that the exponents apply to the entire term of the binomial, not just the variable.)

5

2 1x

5 4 3 2 2 3 4 5

5 10 11 1 12 2 2 12 15 20x x x x x

5 4 3 232 80 80 40 10 1x x x x x

5 4 3 2 2 3 4 55 10 10 5a a a a ab b b b b

Binomial Series

Consider the binomial series :

If we multiply the coefficient of a term by a fraction consisting of the exponent of a over the term number, we get the coefficient of the next number.

7

a b

7 6 5 2 4 3 3 4 2 5 6 77 21 35 35 21 7a a b a b a b a b a b ab b

87654321

exp. 7coeff. 1 7,

term # 1

67 21,

2

521 =35, ...

3

Binomial Series

Now let’s see what happens to if we don’t simplify the fractions as we calculate them:

8

a b

1

2

3

4

5

8a

78

1a b

6 28 7

1 2a b

5 38 7 6

1 2 3a b

4 48 7 6 5

1 2 3 4a b

Do you see the pattern?What is it?

Binomial Series

The coefficients of a binomial series can be written as factorials, much as we did earlier. For example, let’s look at the coefficient for the fourth term:

8 7 6 8 7 6

1 2 3 1 2 3

8 7 6 5!

1 2 3 5!

8!

3! 5!

Binomial Series

Looking back at the original expression:

Notice how the numbers in the coefficient expression are found elsewhere in the expression.

8 is the value of the exponent to which a + b is raised.

5 is the value of a’s exponent and 3 is the value of b’s.

The exponent of b is always one less than the term number.

8 5 38!

... ...3! 5!

a b a b

Binomial Theorem

The formula for the term containing br of a + bn, therefore, is

or nCr

Example: Find the term containing y6 of

!

! !n r rn

a br n r

n

r

10

8x y

610 6 4 610! 10!8 262144

6! 10 6 ! 6! 4!x y x y

4 610 9 8 7262144

4 3 2 1x y

3

4 655,050,240x y

Binomial Theorem

Example: Find the term in which contains f 23.

Since n = 58, n – r = 58 – 23 = 35. Therefore the term is

(When dealing with negative terms such as f, recall that even exponents will produce positive terms and odd exponents will produce negative terms.)

58

e f

35 2358!

35! 23!e f

Binomial Theorem

Similarly, the kth term of binomial expansion of is found by realizing that the exponent of b will be k – 1, which gives us the formula:

(replace r with k – 1)

n

a b

1 1!

1 ! 1 !

n k kna b

k n k

1 1

1

n k kna b

k

Binomial Theorem

Example: Find the 4th term of

n = 12, k = 4, which means that k – 1 = 3

12

2c d

9 3 9 312! 10 11 12

2 5123!9! 1 2 3

c d c d

5 4

9 3112,640c d

Binomial Theorem

Your calculator can also find the coefficient:

4th term of

n = 12, k – 1 = 3, n – k –1 = 9

12

2c d

Binomial Theorem Practice

Example: Expand 5

2 3x

5 4 3 25! 5!2 2 3 2 3

1!4! 2!3!x x x

2 3 4 55! 5!2 3 2 3 3

3!2! 4!1!x x

5 4 3 232 240 720 1080 810 243x x x x x

Binomial Theorem Practice

Example: Simplify ( ) 6

1 i 1i

6 5 4 2 3 36! 6! 6!1 1 1 1

1!5! 2!4! 3!3!i i i

2 4 5 66! 6!1 1

4!2! 5!1!i i i

2 3 4 5 61 6 15 20 15 6i i i i i i

1 6 15 20 15 6 1i i i

8i

Binomial Theorem Practice

Example: Expand 3

3

2 4

r r

2 33 2

3 3 3

2 2 4 2 4 43 3

r r rr r r

9 43

8 48 96 64

r rr r r r

13 39 4

2 2

8 48 96 64

r rr r

I will accept either