11 1 11 2 Binomial Theorem Binomial Expansion

11. 1 -11. 2 Binomial Theorem & Binomial Expansion

Pascal’s Triangle

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

Expanding Binomials… What pattern(s) do you see? . . .

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 Example 2 1 1 1 1 8 9 6 28 36 15 56 1 10 20 35 84 4 10 21 1 6 5 7 3 4 1 1 3 1 5 15 35 70 126 1 6 21 56 126 1 7 28 84 1 8 36 1 9 1

Expand (a + 6 b) • 1 6 15 20 15 6 1 (coefficients from Pascal’s triangle) • 1 a 6 6 a 5 15 a 4 20 a 3 15 a 2 6 a 1 1 a 0 (exponents of a begin with 6 and decrease) • 1 a 6 b 0 6 a 5 b 1 15 a 4 b 2 20 a 3 b 3 15 a 2 b 4 6 a 1 b 5 1 a 0 b 6 (exponents of b begin with 0 and increase by 1) • a 6 + 6 a 5 b + 15 a 4 b 2 + 20 a 3 b 3 + 15 a 2 b 4 + 6 ab 5 + b 6 (expansion in standard, simplified form) Return to Triangle

Expand (x – 3 2) (let a = x and b = – 2) • 1 3 3 1 (coefficients from Pascal’s triangle) • 1 a 3 3 a 2 3 a 1 1 a 0 • 1 a 3 b 0 3 a 2 b 1 3 a 1 b 2 1 a 0 b 3 • Now substitute x for a and – 2 for b: • 1(x)3(– 2)0 3(x)2(– 2)1 3(x)1(– 2)2 1(x)0(– 2)3 • x 3 – 6 x 2 + 12 x – 8 (expansion in standard, simplified form) Return to Triangle

Factorial Notation Read as “n factorial”

Finding a particular term in a Binomial Expansion: Formula for finding a particular term in expansion of (a + b)n is: Ex. : Find the 4 th term in expansion of (a + b)9: This is the 4 th 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 4 th term: a 6 b 3 Coefficient is:

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