Chapter 8: Further Topics in Algebra 8.1 Sequences and Series 8.2 Arithmetic Sequences and Series 8.3 Geometric Sequences and Series 8.4 The Binomial Theorem 8.5 Mathematical Induction 8.6 Counting Theory 8.7 Probability
8.4 The Binomial Theorem The binomial expansions reveal a pattern.
8.4 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-1y , x n-2y2 , x n-3y3and 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.
8.4 A Binomial Expansion Pattern Pascal’s Triangle Row
8.4 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-1y , x n-2y2 , x n-3y3, …y n in the expansion of (x + y)n.
8.4 n-Factorial n-Factorial For any positive integer n, and Example Evaluate (a) 5! (b) 7! Solution (a) (b)
8.4 Binomial Coefficients Binomial Coefficient For nonnegative integers n and r, with r<n,
8.4 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.
8.4 Evaluating Binomial Coefficients Example Evaluate (a) (b) Solution (a) (b)
8.4 The Binomial Theorem Binomial Theorem For any positive integers n,
8.4 Applying the Binomial Theorem Example Write the binomial expansion of . Solution Use the binomial theorem
8.4 Applying the Binomial Theorem Example Expand . Solution Use the binomial theorem with and n = 5,
8.4 Applying the Binomial Theorem Solution
8.4 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
8.4 Finding a Specific Term of a Binomial Expansion. Example Find the fourth term of . Solution Using n = 10, r = 4, x = a, y = 2b in the formula, we find the fourth term is