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1632 the binomial theorem-02
1.
2.
Copyright © 2007
Pearson Education, Inc. Slide 11-2 Chapter 11: Further Topics in Algebra 11.1 Sequences and Series 11.2 Arithmetic Sequences and Series 11.3 Geometric Sequences and Series 11.4 The Binomial Theorem 11.5 Mathematical Induction 11.6 Counting Theory 11.7 Probability
3.
Copyright © 2007
Pearson Education, Inc. Slide 11-3 11.4 The Binomial Theorem The binomial expansions reveal a pattern. 0 1 2 2 2 3 3 2 2 3 4 4 3 2 2 3 4 5 5 4 3 2 2 3 4 5 ( ) 1 ( ) ( ) 2 ( ) 3 3 ( ) 4 6 4 ( ) 5 10 10 5 x y x y x y x y x xy y x y x x y xy y x y x x y x y xy y x y x x y x y x y xy y + = + = + + = + + + = + + + + = + + + + + = + + + + +
4.
Copyright © 2007
Pearson Education, Inc. Slide 11-4 11.4 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-1 y , x n-2 y2 , x n-3 y3 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.
5.
Copyright © 2007
Pearson Education, Inc. Slide 11-5 11.4 A Binomial Expansion Pattern Pascal’s Triangle Row 1 0 1 1 1 1 2 1 2 1 3 3 1 3 1 4 6 4 1 4 1 5 10 10 5 1 5
6.
Copyright © 2007
Pearson Education, Inc. Slide 11-6 11.4 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-1 y , x n-2 y2 , x n-3 y3 , … y n in the expansion of (x + y)n .
7.
Copyright © 2007
Pearson Education, Inc. Slide 11-7 11.4 n-Factorial n-Factorial For any positive integer n, and ! ( 1)( 2) (3)(2)(1), 0! 1 . n n n n= − − ××× = Example Evaluate (a) 5! (b) 7! Solution (a) (b) 5! 5 4 3 2 1 120= × × × × = 7! 7 6 5 4 3 2 1 5040= × × × × × × =
8.
Copyright © 2007
Pearson Education, Inc. Slide 11-8 11.4 Binomial Coefficients Binomial Coefficient For nonnegative integers n and r, with r < n, ! !( )! n r n n C r r n r = = ÷ −
9.
Copyright © 2007
Pearson Education, Inc. Slide 11-9 11.4 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. n rC n r ÷ n r ÷ 3 0 ÷ 3 2 ÷
10.
Copyright © 2007
Pearson Education, Inc. Slide 11-10 11.4 Evaluating Binomial Coefficients Example Evaluate (a) (b) Solution (a) (b) 6 2 ÷ 8 0 ÷ 6 6! 6! 6 5 4 3 2 1 15 2 2!(6 2)! 2!4! 2 1 4 3 2 1 × × × × × = = = = ÷ − × × × × × 8 8! 8! 8! 1 0 0!(8 0)! 0!8! 1 8! = = = = ÷ − ×
11.
Copyright © 2007
Pearson Education, Inc. Slide 11-11 11.4 The Binomial Theorem Binomial Theorem For any positive integers n, 1 2 2 3 3 1 ( ) 1 2 3 ... ... 1 n n n n n n r r n n n n n x y x x y x y x y n n x y xy y r n − − − − − + = + + + ÷ ÷ ÷ + + + + + ÷ ÷ −
12.
Copyright © 2007
Pearson Education, Inc. Slide 11-12 11.4 Applying the Binomial Theorem Example Write the binomial expansion of . Solution Use the binomial theorem 9 ( )x y+ 9 9 8 7 2 6 3 5 4 4 5 3 6 2 7 8 9 9 9 9 ( ) 1 2 3 9 9 9 9 4 5 6 7 9 8 x y x x y x y x y x y x y x y x y xy y + = + + + ÷ ÷ ÷ + + + + ÷ ÷ ÷ ÷ + + ÷
13.
Copyright © 2007
Pearson Education, Inc. Slide 11-13 11.4 Applying the Binomial Theorem 9 9 8 7 2 6 3 5 4 4 5 3 6 2 7 8 9 9 8 7 2 6 3 5 4 4 5 3 6 2 7 8 9 9! 9! 9! ( ) 1!8! 2!7! 3!6! 9! 9! 9! 9! 4!5! 5!4! 6!3! 7!2! 9! 8!1! 9 36 84 126 126 84 36 9 x y x x y x y x y x y x y x y x y xy y x x y x y x y x y x y x y x y xy y + = + + + + + + + + + = + + + + + + + + +
14.
Copyright © 2007
Pearson Education, Inc. Slide 11-14 11.4 Applying the Binomial Theorem Example Expand . Solution Use the binomial theorem with and n = 5, 5 2 b a − ÷ 2 3 5 5 4 3 2 4 5 5 5 5 ( ) 1 2 32 2 2 2 5 4 2 2 b b b b a a a a a b b a − = + − + − + − ÷ ÷ ÷ ÷ ÷ ÷ + − + − ÷ ÷ ÷ , 2 b x a y= = −
15.
Copyright © 2007
Pearson Education, Inc. Slide 11-15 11.4 Applying the Binomial Theorem Solution 2 3 5 5 4 3 2 4 5 5 4 3 2 2 3 4 5 ( ) 5 10 10 2 2 2 2 5 2 2 5 5 5 5 1 2 2 4 16 32 b b b b a a a a a b b a a a b a b a b ab b − = + − + − + − ÷ ÷ ÷ + − + − ÷ ÷ = − + − + −
16.
Copyright © 2007
Pearson Education, Inc. Slide 11-16 11.4 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 ( 1) 1 1 n r rn x y r − − − ÷ −
17.
Copyright © 2007
Pearson Education, Inc. Slide 11-17 11.4 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 10 ( 2 )a b+ 7 3 7 3 7 310 (2 ) 120 8 960 . 3 a b a b a b = = ÷
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