9 5 The Binomial Theorem Lets look at

9. 5 The Binomial Theorem Let’s look at the expansion of (x + y)n (x + y)0 = 1 (x + y)1 = x + y (x + y)2 = x 2 +2 xy + y 2 (x + y)3 = x 3 + 3 x 2 y + 3 xy 2 + y 3 (x + y)4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 xy 3 + y 4

Expanding a binomial using Pascal’s Triangle 1 1 1 1 2 3 4 5 6 1 3 6 10 15 1 1 4 10 20 1 5 15 Write the next row. 1 6 1

Expand (x + 3)4 From Pascal’s triangle write down the 4 th row. 1 4 6 4 1 These numbers are the same numbers that are the coefficients of the binomial expansion. The expansion of (a + b)4 is: 1 a 4 b 0 + 4 a 3 b 1 + 6 a 2 b 2 + 4 a 1 b 3 + 1 a 0 b 4 Notice that the exponents always add up to 4 with the a’s going in descending order and the b’s in ascending order. Now substitute x in for a and 3 in for b.

1 a 4 b 0 + 4 a 3 b 1 + 6 a 2 b 2 + 4 a 1 b 3 + 1 a 0 b 4 x 4 + 4 x 3(3)1 + 6 x 2(3)2 + 4 x(3)3 + 34 This simplifies to Expand (x – 2 y)4 x 4 + 12 x 3 + 54 x 2 + 108 x + 81 This time substitute x in for a and -2 y for b. Use ( ). x 4 + 4 x 3(-2 y)1 + 6 x 2(-2 y)2 + 4 x(-2 y)3 + (-2 y)4 The final answer is: x 4 – 8 x 3 y + 24 x 2 y 2 – 32 xy 3 + 16 y 4

The Binomial Theorem In the expansion of (x + y)n = xn + nxn-1 y + … + n. Cmxn-mym + … +nxyn-1 + yn the coefficient of xn-mym is given by

Find the following binomial coefficients. 8 C 2 = 10 C 3 = 7 C 4 = 28 120 35 35

Find the 6 th term in the expansion of (3 a + 2 b)12 Using the Binomial Theorem, let x = 3 a and y = 2 b and note that in the 6 th term, the exponent of y is m = 5 and the exponent of x is n – m = 12 – 5 = 7. Consequently, the 6 th term of the expansion is: = 55, 427, 328 a 7 b 5