The Binomial Theorem

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The Binomial Theorem

• Lecture 34
• Section 6.7
• Wed Mar 28 2007

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The Binomial Theorem

• Theorem Given any numbers a and b and any
  nonnegative integer n

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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.

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Proof continued

• Inductive step
• Suppose the statement is true when n k for some
  k 0.
• Then

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Proof continued
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Proof continued

• Therefore the statement is true when n k 1.
• Thus the statement is true for all n 0.

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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.

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Example Binomial Theorem

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

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Example Calculating 1.018

• Compute 1.018 on a calculator.
• What do you see

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Example Calculating 1.018

• Compute 1.018 on a calculator.
• What do you see
• 1.018 1.0828567056280801.

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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.

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Example Approximating (1x)n

• Theorem For small values of x
• and so on.

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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.

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Expanding Trinomials

• Expand (a b c)3.

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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.

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Expanding Trinomials

• a3 3a2b 3ab2 b3 3a2c 6abc 3b2c
  3ac2 3bc2 c3.
• What is the pattern

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Expanding Trinomials

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

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The Multinomial Theorem

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

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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.

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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.

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Example The Multinomial Theorem

• Find (a 2b 1)4.

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Actuary Exam Problem

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