Chapter 8: Further Topics in Algebra

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## Chapter 8: Further Topics in Algebra

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**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