1. Year 12 C1 Binomial Theorem

2. Task Expand the following: 1. (x + y) 1 2. (x + y) 2 3. (x + y) 3 4. (x + y) 4 What do you notice? Powers of x start from left with the power from the question, then successively reduce by 1. Powers of y increase by 1 until they reach the power in the question. Coefficients of the terms follow ‘Pascal’s Triangle’.

3. Pascal’s Triangle 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Start and end the row with a 1, add pairs of numbers above to get the entry positioned below and between them.

4. Examples Write down the expansion of (2x + 3) 4 1(2x) 4 + 4(2x) 3 (3) + 6(2x) 2 (3 2 ) + 4(2x)(3 3 ) +1(3 4 ) = 16x 4 + 96x 3 + 216x 2 + 216x + 81 Find the coefficient of x 3 in the expansion of (3x – 4) 5 Coefficients will be 1, 5, 10, 5, 1. The x 3 will come in the 3 rd term which is: 10 x (3x) 3 (-4) 2 = 4320x 3

5. PPQ

6. Further Example Expand (1 + 2x + 3 x 2 ) 3 Write 1 + 2x + 3 x 2 in a form with 2 terms rather than 3 (1 + (2x + 3x 2 )) 3 = 1 3 + 3(1 2 )(2x + 3x 2 ) + 3(1)(2x + 3x 2 ) 2 + (2x + 3x 2 ) 3 Use binomial theorem to expand brackets. = 1 + 6x + 9x 2 + 3(4x 2 + 12x 3 + 9x 4 ) + (8x 3 + 36x 4 + 54x 5 + 27x 6 ) = 1 + 6x + 21x 2 + 44x 3 + 63x 4 + 54x 5 + 27x 6

7. Whiteboards Ready! 1.Expand (1 + y) 6 1 + 6y + 15y 2 + 20y 3 + 15y 4 + 6y 5 + y 6 2. Expand (x 2 + 2) 3 x 6 + 6x 4 + 12x 2 + 8 3. Find the coefficient of x 2 in the expansion of (1 – 3x) 4 54 4.Expand (p + 2q) 5 p 5 + 10p 4 q + 40p 3 q 2 + 80p 2 q 3 + 80pq 4 + 32q 5

8. Task Neill and Quadling: Page 270 Questions 1 a, f, h, 4 c, 5 b, 9.