2. Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4

3. Aim: Binomial Theorem Course: Alg. 2 & Trig. Permutations & Combinations A permutation is an arrangement of objects in a specific order. The number of permutation of n things taken n at a time is n P n = n! = n(n – 1)(n – 2)(n – 3)... 3, 2, 1 The number of permutation of n things taken r at a time is r factors

4. Aim: Binomial Theorem Course: Alg. 2 & Trig. Permutations & Combinations A combination is an arrangement of objects in which there is no specific order. The number of combinations of n things taken n at a time is The number of combinations of n things taken r at a time is 1

5. Aim: Binomial Theorem Course: Alg. 2 & Trig. Pascal’s Triangle 1 11 121 1331 14641 1510 51 1615201561 172135 2171 The first & last numbers in each row are 1 Every other number in each row is formed by adding the two numbers above the number. In each expansion there is n + 1 terms (n is the row number)

6. Aim: Binomial Theorem Course: Alg. 2 & Trig. Pascal’s Triangle & Expansion of (x + y) n (x + y) 0 = 1 (x + y) 1 = 1x + 1y (x + y) 2 = 1x 2 + 2xy + 1y 2 (x + y) 3 = 1x 3 + 3x 2 y + 3xy 2 + 1y 3 (x + y) 4 = 1x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + 1y 4 (x + y) 5 = 1x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5xy 4 + 1y 5 In each expansion there is n + 1 terms. In each expansion the x and y have symmetric roles. The sum of the powers of each term is n. The coefficients increase & decrease symmetrically. 5 5 4 4 expansion of (x + y) n zero row 1st row

7. Aim: Binomial Theorem Course: Alg. 2 & Trig. The Binomial Theorem In the expansion of (x + y) n, (x + y) n = x n + nx n-1 y +... + n C r x n-r y r +.... + nxy n-1 + y n, the coefficient of x n-r y r is given by Example: Bernoulli Experiment - probability of success & failure

8. Aim: Binomial Theorem Course: Alg. 2 & Trig. Coefficients of the ninth row 99 Pascal’s Triangle & the Binomial Theorem (x + y) 0 = 1 (x + y) 1 = 1x + 1y (x + y) 2 = 1x 2 + 2xy + 1y 2 (x + y) 3 = 1x 3 + 3x 2 y + 3xy 2 + 1y 3 (x + y) 4 = 1x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + 1y 4 (x + y) 5 = 1x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5xy 4 + 1y 5 5 5 4 4 5C05C05C15C15C25C25C35C35C45C45C55C59C09C09C19C19C29C29C39C39C49C49C59C59C69C69C79C79C89C89C99C9 1136 84 126 expansion of (x + y) n (x + y) n = x n + nx n-1 y +...+ n C r x n-r y r +.... + nxy n-1 + y n

9. Aim: Binomial Theorem Course: Alg. 2 & Trig. 3C03C03C13C13C23C23C33C3 3131 4 terms (n + 1) x n-r y r Binomial Expansion Coefficients of the third row n = 3 Write the expansion of (x + 1) 3 3C03C03C13C13C23C23C33C3 3131 4 terms (n + 1) 1x 3 + 3x 2 + 3x + 1 x n-r y r marry coefficients with terms and exponents of binomial Write the expansion of (x - 1) 3 1x 3 – 3x 2 + 3x – 1 expanded binomials with differences – alternate signs n = highest exponent value in row

10. Aim: Binomial Theorem Course: Alg. 2 & Trig. Model Problem Coefficients of the fourth row n = 4 Write the expansion of (x - 2y) 4 1x 4 (2y) 0 - 4x 3 (2y) 1 + 6x 2 (2y) 2 - 4x(2y) 3 + 1x 0 (2y) 4 1x 4 - 4x 3 + 6x 2 - 4x + 1x 0 1 - 4 + 6 - 4 + 1 x n-r y r 5 terms (n + 1) 4C04C04C14C14C24C24C34C3 4164 4C44C4 1 (x + y) n x 4 – 8x 3 y + 24x 2 y 2 – 32xy 3 + 16y 4

11. Aim: Binomial Theorem Course: Alg. 2 & Trig. Regents Question Write the binomial expansion of (2x − 1) 5 as a polynomial in simplest form.

12. Aim: Binomial Theorem Course: Alg. 2 & Trig. n C r x n-r y r General Formula n = 12 12th row Model Problem Find the sixth term of the expansion of (3a + 2b) 12 13 terms (n + 1) xn-ryrxn-ryr 6 th term key: r = ? 12 C 0 = 11 st term 5 792(3a) 7 (2b) 5 = 55427328a 7 b 5 x = 3ay = 2b (3a) 12-5 (2b) 5 x n-r y r 12 C 5 coeff. 12 C 0 1 st term 12 C 1 2 nd term 12 C 2 3 rd term 12 C 3 4 th term 12 C 4 5 th term 12 C 5 6 th term 1 st term 2 nd term 3 rd term 4 th term 5 th term 6 th term x 12 y 0 x 11 y 1 x 10 y 2 x9y3x9y3 x8y4x8y4 x7y5x7y5

13. Aim: Binomial Theorem Course: Alg. 2 & Trig. Regents Questions What is the fourth term in the expansion of (3x – 2) 5? 1) -720x 2 2) -240x 3) 720x 2 4) 1,080x 3