Section8.7 The Binomial Theorem Formulas Factorial n ( n 1) . . - - PowerPoint PPT Presentation

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Section8.7 The Binomial Theorem Formulas Factorial n ( n 1) . . - - PowerPoint PPT Presentation

Feb 10, 2024 155 likes •656 views

Section8.7 The Binomial Theorem Formulas Factorial n ( n 1) . . . (2)(1) if n = 2 , 3 , 4 , . . . n ! = 1 if n = 0 or 1 For example: 5! = 5 4 3 2 1 = 120 Examples 1. Compute 6!. Examples 1. Compute 6!. 720 Examples 1.

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SLIDE 1

Section8.7

The Binomial Theorem

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SLIDE 2

Formulas

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SLIDE 3

Factorial

n! = n(n − 1) . . . (2)(1) if n = 2, 3, 4, . . . 1 if n = 0 or 1 For example: 5! = 5 · 4 · 3 · 2 · 1 = 120

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SLIDE 4

Examples

  • 1. Compute 6!.
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SLIDE 5

Examples

  • 1. Compute 6!.

720

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SLIDE 6

Examples

  • 1. Compute 6!.

720

  • 2. Compute 10!

8! .

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SLIDE 7

Examples

  • 1. Compute 6!.

720

  • 2. Compute 10!

8! . 90

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SLIDE 8

Binomial Coefficient

nCk =

n k

  • =

n! k!(n − k)! The binomial coefficient is read as “n choose k”. The number on the bottom is never bigger than the number on top. For example: 5 3

  • =

5! 3!(5 − 3)! = 5! 3!2! = 5 · 4 · 3 · 2 · 1 (3 · 2 · 1)(2 · 1) = 20 2 = 10

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SLIDE 9

Example

Compute 10 1

  • +

10 9

  • .
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SLIDE 10

Example

Compute 10 1

  • +

10 9

  • .

20

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SLIDE 11

Pascal’sTriangle

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SLIDE 12

Construction of Pascal’s Triangle

To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero.

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SLIDE 13

Construction of Pascal’s Triangle

To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero.

1

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SLIDE 14

Construction of Pascal’s Triangle

To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero.

1 1 1

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SLIDE 15

Construction of Pascal’s Triangle

To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero.

1 1 1 1 2 1

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SLIDE 16

Construction of Pascal’s Triangle

To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero.

1 1 1 1 2 1 1 3 3 1

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SLIDE 17

Construction of Pascal’s Triangle

To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero.

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

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SLIDE 18

Construction of Pascal’s Triangle

To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero.

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

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SLIDE 19

Construction of Pascal’s Triangle

To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero.

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

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SLIDE 20

Relationship to Binomial Coefficients

It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula.

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SLIDE 21

Relationship to Binomial Coefficients

It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula.

  • = 1
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SLIDE 22

Relationship to Binomial Coefficients

It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula.

  • = 1

1

  • = 1

1

1

  • = 1
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SLIDE 23

Relationship to Binomial Coefficients

It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula.

  • = 1

1

  • = 1

1

1

  • = 1

2

  • = 1

2

1

  • = 2

2

2

  • = 1
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SLIDE 24

Relationship to Binomial Coefficients

It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula.

  • = 1

1

  • = 1

1

1

  • = 1

2

  • = 1

2

1

  • = 2

2

2

  • = 1

3

  • = 1

3

1

  • = 3

3

2

  • = 3

3

3

  • = 1
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SLIDE 25

Relationship to Binomial Coefficients

It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula.

  • = 1

1

  • = 1

1

1

  • = 1

2

  • = 1

2

1

  • = 2

2

2

  • = 1

3

  • = 1

3

1

  • = 3

3

2

  • = 3

3

3

  • = 1

4

  • = 1

4

1

  • = 4

4

2

  • = 6

4

3

  • = 4

4

4

  • = 1
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SLIDE 26

Relationship to Binomial Coefficients

It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula.

  • = 1

1

  • = 1

1

1

  • = 1

2

  • = 1

2

1

  • = 2

2

2

  • = 1

3

  • = 1

3

1

  • = 3

3

2

  • = 3

3

3

  • = 1

4

  • = 1

4

1

  • = 4

4

2

  • = 6

4

3

  • = 4

4

4

  • = 1

5

  • = 1

5

1

  • = 5

5

2

  • = 10

5

3

  • = 10

5

4

  • = 5

5

5

  • = 1
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SLIDE 27

Relationship to Binomial Coefficients

It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula.

  • = 1

1

  • = 1

1

1

  • = 1

2

  • = 1

2

1

  • = 2

2

2

  • = 1

3

  • = 1

3

1

  • = 3

3

2

  • = 3

3

3

  • = 1

4

  • = 1

4

1

  • = 4

4

2

  • = 6

4

3

  • = 4

4

4

  • = 1

5

  • = 1

5

1

  • = 5

5

2

  • = 10

5

3

  • = 10

5

4

  • = 5

5

5

  • = 1

6

  • = 1

6

1

  • = 6

6

2

  • = 15

6

3

  • = 20

6

4

  • = 15

6

5

  • = 6

6

6

  • = 1
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SLIDE 28

Relationship to Binomial Coefficients (continued)

Each row of the triangle gives you all the coefficients for a particular n in n

k

  • :
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SLIDE 29

Relationship to Binomial Coefficients (continued)

Each row of the triangle gives you all the coefficients for a particular n in n

k

  • :

n = 0: 1

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SLIDE 30

Relationship to Binomial Coefficients (continued)

Each row of the triangle gives you all the coefficients for a particular n in n

k

  • :

n = 0: 1 n = 1: 1 1

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SLIDE 31

Relationship to Binomial Coefficients (continued)

Each row of the triangle gives you all the coefficients for a particular n in n

k

  • :

n = 0: 1 n = 1: 1 1 n = 2: 1 2 1

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SLIDE 32

Relationship to Binomial Coefficients (continued)

Each row of the triangle gives you all the coefficients for a particular n in n

k

  • :

n = 0: 1 n = 1: 1 1 n = 2: 1 2 1 n = 3: 1 3 3 1

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SLIDE 33

Relationship to Binomial Coefficients (continued)

Each row of the triangle gives you all the coefficients for a particular n in n

k

  • :

n = 0: 1 n = 1: 1 1 n = 2: 1 2 1 n = 3: 1 3 3 1 n = 4: 1 4 6 4 1

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SLIDE 34

Relationship to Binomial Coefficients (continued)

Each row of the triangle gives you all the coefficients for a particular n in n

k

  • :

n = 0: 1 n = 1: 1 1 n = 2: 1 2 1 n = 3: 1 3 3 1 n = 4: 1 4 6 4 1 n = 5: 1 5 10 10 5 1

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SLIDE 35

Relationship to Binomial Coefficients (continued)

Each row of the triangle gives you all the coefficients for a particular n in n

k

  • :

n = 0: 1 n = 1: 1 1 n = 2: 1 2 1 n = 3: 1 3 3 1 n = 4: 1 4 6 4 1 n = 5: 1 5 10 10 5 1 n = 6: 1 6 15 20 15 6 1

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SLIDE 36

MultiplyingBinomialEx- pressions

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SLIDE 37

The Binomial Theorem

The Binomial Theorem gives a formula for multiplying out binomial expressions: (A + B)n = n

  • An +

n 1

  • An−1B +

n 2

  • An−2B2 + . . .

+

  • n

n − 2

  • A2Bn−2 +
  • n

n − 1

  • ABn−1 +

n n

  • Bn
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SLIDE 38

Examples

  • 1. Expand (x + y)3, and compute the binomial coefficients using the

formula.

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SLIDE 39

Examples

  • 1. Expand (x + y)3, and compute the binomial coefficients using the

formula. x3 + 3x2y + 3xy2 + y3

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SLIDE 40

Examples

  • 1. Expand (x + y)3, and compute the binomial coefficients using the

formula. x3 + 3x2y + 3xy2 + y3

  • 2. Expand (2r + s2)4, using Pascal’s Triangle to find the coefficients.
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SLIDE 41

Examples

  • 1. Expand (x + y)3, and compute the binomial coefficients using the

formula. x3 + 3x2y + 3xy2 + y3

  • 2. Expand (2r + s2)4, using Pascal’s Triangle to find the coefficients.

16r4 + 32r3s2 + 24r2s4 + 8rs6 + s8

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SLIDE 42

Examples

  • 1. Expand (x + y)3, and compute the binomial coefficients using the

formula. x3 + 3x2y + 3xy2 + y3

  • 2. Expand (2r + s2)4, using Pascal’s Triangle to find the coefficients.

16r4 + 32r3s2 + 24r2s4 + 8rs6 + s8

  • 3. Expand
  • 2a − 1

2b

6

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SLIDE 43

Examples

  • 1. Expand (x + y)3, and compute the binomial coefficients using the

formula. x3 + 3x2y + 3xy2 + y3

  • 2. Expand (2r + s2)4, using Pascal’s Triangle to find the coefficients.

16r4 + 32r3s2 + 24r2s4 + 8rs6 + s8

  • 3. Expand
  • 2a − 1

2b

6 64a6 − 96a5b + 60a4b2 − 20a3b3 + 15

4 a2b4 − 3 8ab5 + 1 64b6

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SLIDE 44

Finding a Single Term

The (k + 1)st term in (A + B)n is n k

  • An−kBk
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SLIDE 45

Examples

  • 1. Find the fourth term in (5x − 2y)5.
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SLIDE 46

Examples

  • 1. Find the fourth term in (5x − 2y)5.

−2000x2y3

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SLIDE 47

Examples

  • 1. Find the fourth term in (5x − 2y)5.

−2000x2y3

  • 2. Find the tenth term in the expansion of
  • 2x + y

2

14

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SLIDE 48

Examples

  • 1. Find the fourth term in (5x − 2y)5.

−2000x2y3

  • 2. Find the tenth term in the expansion of
  • 2x + y

2

14

1001 8 x5y9