The Binomial Theorem Ms.M.M

Expanding Binomials Expanding a binomial such as (a + b)n means to write the factored form as a sum. (a + b)0 = 1 (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5 1 term 2 terms 3 terms 4 terms 5 terms 6 terms Ms.M.M

Expanding Binomials (a + b)0 = 1 (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5 The expansion of (a + b)n contains n + 1 terms. The first term is an and the last term is bn. The powers of a decrease by 1 for each term; the powers of b increase by 1 for each term. The sum of the exponents of a and b is n. Ms.M.M

Pascal’s Triangle There are also patterns in the coefficients of the terms. When written in a triangular array, the coefficients are called Pascal’s triangle. Ms.M.M

Pascal’s Triangle (a + b)0 (a + b)1 (a + b)2 (a + b)3 (a + b)4 1 1 1 2 1 1 3 3 1 4 6 4 1 1 5 10 10 5 1 n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 1 5 6 1 6 15 20 15 6 1 Add the consecutive numbers in the row for n = 5 and write each sum “between and below” the pair. Ms.M.M

Pascal’s Triangle Example: Expand (a + b)7. Use n = 7 row of Pascal’s triangle as the coefficients and the noted patterns. 1 6 15 20 15 6 1 n = 6 1 7 21 35 35 21 7 1 n = 7 (a + b)7 = 1a7 + 7a6b + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7ab6 + 1b7 Ms.M.M

Factorials Factorial of n: n! If n is a natural number, then An alternative method for determining the coefficients of (a + b)n is based on using factorials. The factorial of n, written n! (read “n factorial”), is the product of the first n consecutive natural numbers. Factorial of n: n! If n is a natural number, then n! = n(n – 1)(n – 2)(n – 3) . . . ∙ 3 ∙ 2 ∙ 1. The factorial of 0, written 0!, is defined to be 1. Ms.M.M

Evaluating Factorials Example: Evaluate each expression. Ms.M.M

Binomial Theorem Binomial Theorem If n is a positive integer, then It can be proved that the coefficients of terms in the expansion of (a + b)n can be expressed in terms of factorials. Following the earlier patterns and using the factorial expressions of the coefficients, we have the binomial theorem. Binomial Theorem If n is a positive integer, then Ms.M.M

Binomial Theorem Example: Use the binomial theorem to expand (x + 3)4. Ms.M.M

Binomial Theorem Example: Use the binomial theorem to expand (3a – 5b)6. Ms.M.M

Binomial Expansion (r + 1)st Term in a Binomial Expansion The (r + 1)st term of the binomial expansion of (a + b)n is Ms.M.M

Binomial Expansion Example: Find the ninth term in the expansion of (3x – 5y)10. n = 10, a = 3x, b = – 5y, r + 1 = 9, therefore r = 8 Ms.M.M