the binomial theorem unit 10.5. binomial theorem (a + b) 0 = 1a 0 b 0 (a + b) 1 = 1a 1 b 0 + 1a 0 b...

The Binomial Theorem Unit 10.5

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The Binomial Theorem

Unit 10.5

Binomial Theorem

(a + b)0 = 1a0b0

(a + b)1 = 1a1b0 + 1a0b1

(a + b)2 = 1a2b0 + 2a1b1 + 1a0b2

(a + b)3= 1a3b0 + 3a2b1 + 3a1b2 + 1a0b3

(a + b)4 = 1a4b0 + 4a3b1 +6a2b2 + 4a1b3 + 1a0b4

(a + b)5 =1a5b0 + 5a4b1 +10a3b2 +10a2b3 + 5a1b4 + 1a0b5

Complete the next 5 binomial expansions

Page 3: The Binomial Theorem Unit 10.5. Binomial Theorem (a + b) 0 = 1a 0 b 0 (a + b) 1 = 1a 1 b 0 + 1a 0 b 1 (a + b) 2 = 1a 2 b 0 + 2a 1 b 1 + 1a 0 b 2 (a +

Practical Implication

1. Computer programming 1 0

2. Physics: Comparing earths’ radius to its height or weight

3. Easier way to add numbers

Page 4: The Binomial Theorem Unit 10.5. Binomial Theorem (a + b) 0 = 1a 0 b 0 (a + b) 1 = 1a 1 b 0 + 1a 0 b 1 (a + b) 2 = 1a 2 b 0 + 2a 1 b 1 + 1a 0 b 2 (a +

Practice Problems

Page 629 1b

(2x + 3y)5

Identify the polynomial that will be computed

1a5b0 + 5a4b1 +10a3b2 +10a2b3 + 5a1b4 + 1a0b5

32x5 + 240x4y +720x3y2+1080x2y3 + 810xy4 + 243y5

Page 5: The Binomial Theorem Unit 10.5. Binomial Theorem (a + b) 0 = 1a 0 b 0 (a + b) 1 = 1a 1 b 0 + 1a 0 b 1 (a + b) 2 = 1a 2 b 0 + 2a 1 b 1 + 1a 0 b 2 (a +

Practice Problems

Page 629 Guided 2a page 629

(2x – 7)3

Pascal 1a3b0 + 3a2b1 + 3a1b2 + 1a0b3

8x3 - 84x2 + 294x3 – 343

Page 633 Problems 1 - 10

Page 6: The Binomial Theorem Unit 10.5. Binomial Theorem (a + b) 0 = 1a 0 b 0 (a + b) 1 = 1a 1 b 0 + 1a 0 b 1 (a + b) 2 = 1a 2 b 0 + 2a 1 b 1 + 1a 0 b 2 (a +

Find binomial coefficients

Formula nCr = n!

(n – r)!r!

r = term in the coefficient r = k – 1

(x + y)9, 6th term

n = 9 r = 5

nCr = 9! = 9! = 9*8*7*6*5*4*3*2*1 = 126 (9

– 5)!5! 4!5! 4*3*2*1*5*4*3*2*1

Page 7: The Binomial Theorem Unit 10.5. Binomial Theorem (a + b) 0 = 1a 0 b 0 (a + b) 1 = 1a 1 b 0 + 1a 0 b 1 (a + b) 2 = 1a 2 b 0 + 2a 1 b 1 + 1a 0 b 2 (a +

Binomials with coefficients other than 1

(2x – 3y)8, x3y5

n = 8 r = 5 a = 2x b = (-3y)

nCr = n! 8! → 8*7*6 = 56

(n – r)!r! 3!5! 3*2*1

56(2x)3(-3y)5 = 56(8)(-243) = -108,864

Page 8: The Binomial Theorem Unit 10.5. Binomial Theorem (a + b) 0 = 1a 0 b 0 (a + b) 1 = 1a 1 b 0 + 1a 0 b 1 (a + b) 2 = 1a 2 b 0 + 2a 1 b 1 + 1a 0 b 2 (a +

Exercises

Page 633 Problems 11 - 18

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