Maths C4

1
Maths C4

• Binomial Theorem

2
Pascals

• Click here or
• Answer the questions on the sheet

3
Pascals Triangle

• These numbers are known as Pascals triangle and
  it will help us with the next section on binomial
  expansion.
• Even though it was named after Blaise Pascal the
  Chinese discovered this triangle long before he
  was born.
• 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1
  etc.

4
Pascals triangle

• We can use Pascals triangle to find the
  coefficients when expanding a binomial
  expression.
• Have a look at these examples on the next slide

5
Algebraic expressions

• (1 x)2 1 2x x2
• (1 x)3 1 3x 3x2 x3
• If you look at the coefficients (the numbers on
  their own and in front of the x's) of the results
  you will see that for the first one they are
  1,2,1 and for the second one they are 1, 3, 3, 1.
  These, of course, are the lines from Pascal's
  Triangle. And yes, it does work for all positive
  whole number values of the index. Prove to
  yourself by algebra that,
• (1 x)4 1 4x 6x2 4x3 x4
• (a b)4 a4 4a3b 6a2b2 4ab3 b4

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Example 1

• Expand (1x)7. Hence find the first 4 terms of
  (14x)7 in ascending powers of x.
• Looking at line 7 of Pascals triangle the
  coefficients are 1,7,21,35,35,21,7and 1
• So (1x)7 1 7x 21x235x3 35x421x5 7x6x7
• (14x)7 1 7(4x) 21(4x)235(4x)3
  35(4x)421(4)x5
• 7(4x)6(4x)7 Can you see
  why?
• Please simplify only the first 4 terms.
• 128x336x22240x3

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The general form for Binomial expansion

• When n is a positive integer, this expansion is
  finite and exact.

8
Expand the following

• (1x)4
• 14x6x24x3x4
• (1-2x)4
• 14(-2x) 6(-2x)24(-2x)3(-2x)4.

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Three steps

• 1) write coefficients
• 2) Make first term 1
• 3) second term ascending

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What happens is n is a fraction or negative
number?

• Click here to investigate

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What happens when n is negative or fractional?

• Use the binomial expansion to find the first four
  terms of
• This expansion is infinite and convergent when
  x lt 1 or
• -1ltxlt1. This is very important condition!

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Another example

• Find the first four terms of The condition is
  3x lt 1 or xlt1/3

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Problem A

• Find the binomial expansions of up to and
  including the term in x3 stating the range of
  values for which the expansions are valid.

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Problem B

• Find the binomial expansions of up to and
  including the term in x3 stating the range of
  values for which the expansions are valid.

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Problem C

• Find the binomial expansions of up to and
  including the term in x3 stating the range of
  values for which the expansions are valid.

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Problem D

• Find the binomial expansions of up to and
  including the term in x3 stating the range of
  values for which the expansions are valid.
• Hint make sure the first term is 1!!!

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Problem E

• Find the binomial expansions of up to and
  including the term in x3 stating the range of
  values for which the expansions are valid. Hint
  make sure the first term is 1!!!

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Lets use our Partial Fractions knowledge!

• Example 1 Express the following a partial
  fractions first and then expand up to the x3
  term.

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A summary of the key points

• Look at this summary and take notes accordingly

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What are the real life applications of BE?

• Click here
• Mainly problems in probability and physics