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The Binomial Theorem

- Lecture 34
- Section 6.7
- Wed, Mar 28, 2007

The Binomial Theorem

- Theorem Given any numbers a and b and any

nonnegative integer n,

The Binomial Theorem

- Proof Use induction on n.
- Base case Let n 0. Then
- (a b)0 1 and
- Therefore, the statement is true when n 0.

Proof, continued

- Inductive step
- Suppose the statement is true when n k for some

k ? 0. - Then

Proof, continued

Proof, continued

- Therefore, the statement is true when n k 1.
- Thus, the statement is true for all n ? 0.

Example Binomial Theorem

- Expand (a b)8.
- C(8, 0) C(8, 8) 1.
- C(8, 1) C(8, 7) 8.
- C(8, 2) C(8, 6) 28.
- C(8, 3) C(8, 5) 56.
- C(8, 4) 70.

Example Binomial Theorem

- Therefore,
- (a b)8 a8 8a7b 28a6b2 56a5b3
- 70a4b4 56a3b5 28a2b6 8ab7 b8.

Example Calculating 1.018

- Compute 1.018 on a calculator.
- What do you see?

Example Calculating 1.018

- Compute 1.018 on a calculator.
- What do you see?
- 1.018 1.0828567056280801.

Example Calculating 1.016

- 1.018 (1 0.01)8
- 1 8(0.01) 28(0.01)2 56(0.01)3
- 70(0.01)4 56(0.01)5 28(0.01)6

8(0.01)7 (0.01)8 - 1 .08 .0028 .000056 .00000070
- .0000000056 .000000000028

.00000000000008 - .0000000000000001
- 1.0828567056280801.

Example Approximating (1x)n

- Theorem For small values of x,
- and so on.

Example

- For example,
- (1 x)8 ? 1 8x 28x2
- when x is small.
- Compute the value of (1 x)8 and the

approximation when x .03. - Do it again for x -.03.

Expanding Trinomials

- Expand (a b c)3.

Expanding Trinomials

- Expand (a b c)3.
- (a b c)3 ((a b) c)3
- (a b)3 3(a b)2c
- 3(a b)c2 c3,
- (a3 3a2b 3ab2 b3)
- 3(a2 2ab b2)c
- 3(a b)c2
- c3.

Expanding Trinomials

- a3 3a2b 3ab2 b3 3a2c 6abc 3b2c

3ac2 3bc2 c3. - What is the pattern?

Expanding Trinomials

- (a b c)3 (a3 b3 c3)
- 3(a2b a2c ab2 b2c ac2 bc2)
- 6abc.

The Multinomial Theorem

- Theorem In the expansion of
- (a1 ak)n,
- the coefficient of a1n1a2n2aknk is

Example The Multinomial Theorem

- Expand (a b c d)3.
- The terms are
- a3, b3, c3, d3, with coefficient 3!/3! 1.
- a2b, a2c, a2d, ab2, b2c, b2d, ac2, bc2, c2d, ad2,

bd2, cd2, with coefficient 3!/(1!2!) 3. - abc, abd, acd, bcd, with coefficient 3!/(1!1!1!)

6.

Example The Multinomial Theorem

- Therefore,
- (a b c d)3 a3 b3 c3 d3 3a2b
- 3a2c 3a2d 3ab2 3b2c 3b2d
- 3ac2 3bc2 3c2d 3ad2 3bd2
- 3cd2 6abc 6abd 6acd 6bcd.

Example The Multinomial Theorem

- Find (a 2b 1)4.

Actuary Exam Problem

- If we expand the expression
- (a 2b 3c)4,
- what will be the sum of the coefficients?

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