Unit 1 The Binomial Theorem The binomial expansions

Unit 1. The Binomial Theorem The binomial expansions reveal a pattern.

1. 1 A Binomial Expansion Pattern • The expansion of (x + y)n begins with x n and ends with y n. • The variables in the terms after x n follow the pattern x n-1 y , x n-2 y 2 , x n-3 y 3 and so on to y n. With each term the exponent on x decreases by 1 and the exponent on y increases by 1. • In each term, the sum of the exponents on x and y is always n. • The coefficients of the expansion follow Pascal’s triangle.

1. 2 A Binomial Expansion Pattern Pascal’s Triangle Row

1. 3 Pascal’s Triangle • Each row of the triangle begins with a 1 and ends with a 1. • Each number in the triangle that is not a 1 is the sum of the two numbers directly above it (one to the right and one to the left. ) • Numbering the rows of the triangle 0, 1, 2, … starting at the top, the numbers in row n are the coefficients of x n, x n-1 y , x n-2 y 2 , x n-3 y 3, … y n in the expansion of (x + y)n.

1. 4 n-Factorial (Optional) n-Factorial For any positive integer n, and Example Evaluate (a) 5! (b) 7! Solution (a) (b)

1. 5 Binomial Coefficients Binomial Coefficient For nonnegative integers n and r, with r < n,

1. 5 Binomial Coefficients • The symbols and for the binomial coefficients are read “n choose r” • The values of are the values in the nth row of Pascal’s triangle. So is the first number in the third row and is the third.

1. 6 Evaluating Binomial Coefficients Example Evaluate (a) Solution (a) (b)

1. 7 The Binomial Theorem For any positive integers n,

1. 8 Applying the Binomial Theorem Example Write the binomial expansion of Solution Use the binomial theorem .

1. 8 Applying the Binomial Theorem

1. 8 Applying the Binomial Theorem Example Expand . Solution Use the binomial theorem with and n = 5,

1. 8 Applying the Binomial Theorem Solution

1. 9 rth Term of a Binomial Expansion rth Term of the Binomial Expansion The rth term of the binomial expansion of (x + y)n, where n > r – 1, is

1. 9 Finding a Specific Term of a Binomial Expansion. Example Find the fourth term of . Solution Using n = 10, r = 4, x = a, y = 2 b in the formula, we find the fourth term is