11.6 Binomial Theorem & Binomial Expansion

1. 11.6 Binomial Theorem & Binomial Expansion

2. Pascal’s Triangle

3. Pascal’s Triangle with even and odd numbers colored differently:

4. Binomial Expansion

5. Pascal's Triangle Use this triangle to expand binomials of the form (a+b)n. Each row corresponds to a whole number n. The first row consists of the coefficients of (a+b)n when n = 0. Example 1 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 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 Example 2

6. Expand (a + b)6 • 1 6 15 20 15 6 1 (coefficients from Pascal’s triangle) • 1a6 6a5 15a4 20a3 15a2 6a1 1a0 (exponents of a begin with 6 and decrease) • 1a6b0 6a5b1 15a4b2 20a3b3 15a2b4 6a1b5 1a0b6 (exponents of b begin with 0 and increase by 1) • a6 + 6a5b+ 15a4b2 + 20a3b3 + 15a2b4 + 6ab5 + b6(expansion in standard, simplified form) Return to Triangle

7. Expand (x – 2)3 (let a = x and b = –2) • 1 3 3 1 (coefficients from Pascal’s triangle) • 1x3 3x2 3x1 1x0 • 1x3(–2)0 3x2(–2)1 3x1(–2)2 1x0(–2)3 • x3 – 6x2 + 12x – 8 (expansion in standard, simplified form) (do example 2 in text, p. 850) Return to Triangle

8. Factorial Notation Read as “n factorial”

9. Finding a particular term in a Binomial Expansion: Formula for finding a particular term in expansion of (a + b)n is: Ex.: Find the 4th term in expansion of (a + b)9: This is the 4th term, so value of r (b’s exponent) is 4 – 1 = 3. This means the exponent for a is 9 – 3, or 6. So, we have the variables of the 4th term: a6b3 Coefficient is:

10. Finding a particular term in a Binomial Expansion: Formula for finding a particular term in expansion of (a + b)n is: Ex.: Find the 8th term in expansion of (2x – y)12: This is 8th term, exponent for b is 8 – 1 = 7. This means the exponent for a is 12 – 7, or 5. So, the variables of the 8th term: a5b7, or (2x)5(–y)7. Coefficient:

11. Binomial Theorem: (do example 4 in text, p. 853)