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binomial_theorem_notes.ppt
1.
The Binomial Theorem Digital
Lesson
2.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 2 The binomial theorem provides a useful method for raising any binomial to a nonnegative integral power. Consider the patterns formed by expanding (x + y)n. (x + y)0 = 1 (x + y)1 = x + y (x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2y + 3xy2 + y3 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 Notice that each expansion has n + 1 terms. 1 term 2 terms 3 terms 4 terms 5 terms 6 terms Example: (x + y)10 will have 10 + 1, or 11 terms.
3.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 3 Consider the patterns formed by expanding (x + y)n. (x + y)0 = 1 (x + y)1 = x + y (x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2y + 3xy2 + y3 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 1. The exponents on x decrease from n to 0. The exponents on y increase from 0 to n. 2. Each term is of degree n. Example: The 5th term of (x + y)10 is a term with x6y4.”
4.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 4 The coefficients of the binomial expansion are called binomial coefficients. The coefficients have symmetry. The coefficient of xn–ryr in the expansion of (x + y)n is written or nCr . n r (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 The first and last coefficients are 1. The coefficients of the second and second to last terms are equal to n. 1 1 Example: What are the last 2 terms of (x + y)10 ? Since n = 10, the last two terms are 10xy9 + 1y10. So, the last two terms of (x + y)10 can be expressed as 10C9 xy9 + 10C10 y10 or as xy9 + y10. 9 10 10 10
5.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 5 The triangular arrangement of numbers below is called Pascal’s Triangle. Each number in the interior of the triangle is the sum of the two numbers immediately above it. The numbers in the nth row of Pascal’s Triangle are the binomial coefficients for (x + y)n . 1 1 1st row 1 2 1 2nd row 1 3 3 1 3rd row 1 4 6 4 1 4th row 1 5 10 10 5 1 5th row 0th row 1 6 + 4 = 10 1 + 2 = 3
6.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 6 Example: Use the fifth row of Pascal’s Triangle to generate the sixth row and find the binomial coefficients , , 6C4 and 6C2 . 6 1 6 5 5th row 1 5 10 10 5 1 6th row 6 0 6 1 6 2 6 3 6 4 6 5 6 6 6C0 6C1 6C2 6C3 6C4 6C5 6C6 = 6 = and 6C4 = 15 = 6C2. 6 1 6 5 There is symmetry between binomial coefficients. nCr = nCn–r 1 6 15 20 15 6 1
7.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Use Pascal’s Triangle to expand (2a + b)4. (2a + b)4 = 1(2a)4 + 4(2a)3b + 6(2a)2b2 + 4(2a)b3 + 1b4 = 1(16a4) + 4(8a3)b + 6(4a2b2) + 4(2a)b3 + b4 = 16a4 + 32a3b + 24a2b2 + 8ab3 + b4 1 1 1st row 1 2 1 2nd row 1 3 3 1 3rd row 1 4 6 4 1 4th row 0th row 1
8.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 8 The symbol n! (n factorial) denotes the product of the first n positive integers. 0! is defined to be 1. n! = n(n – 1)(n – 2) 3 • 2 • 1 1! = 1 4! = 4 • 3 • 2 • 1 = 24 6! = 6 • 5 • 4 • 3 • 2 • 1 = 720 Formula for Binomial Coefficients For all nonnegative integers n and r, ! ( )! ! n r n C n r r Example: ) 1 2 3 ( ) 1 2 3 4 ( ) 1 2 3 ( ) 4 5 6 7 ( • • • • • • • • • • • • ! 3 ! 4 7 ! 3 ! 4 ! 7 ! 3 )! 3 7 ( ! 7 • • • 3 7 C 35 1 2 3 4 4 5 6 7 • • • • • •
9.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 9 Example: Use the formula to calculate the binomial coefficients 10C5, 15C0, and . 50 48 12 1 ! 5 )! 5 10 ( ! 10 • 5 10 C ! 0 )! 0 10 ( ! 10 • 0 10 C ! 48 )! 48 50 ( ! 50 48 50 • ! 1 )! 1 12 ( ! 12 1 12 • ! 5 ! 5 ! 10 • ! 5 ! 5 ! 5 ) 6 7 8 9 10 ( • • • • • • 252 1 2 3 4 5 6 7 8 9 10 • • • • • • • • ! 0 ! 10 ! 10 • 1 1 1 ! 0 ! 1 ! 48 ! 2 ! 48 ) 49 50 ( ! 48 ! 2 ! 50 • • • • 1225 1 2 49 50 • • ! 1 ! 11 ! 11 12 ! 1 ! 1 ! 12 • • • 12 1 12
10.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 10 Binomial Theorem 1 1 ( )n n n n r r n n n r x y x nx y C x y nxy y ! with ( )! ! n r n C n r r Example: Use the Binomial Theorem to expand (x4 + 2)3. 0 3C 1 3C 2 3C 3 3C 3 4 ) 2 (x 3 4 ) (x ) 2 ( ) ( 2 4 x 2 4 ) 2 )( (x 3 ) 2 ( 1 3 4 ) (x 3 ) 2 ( ) ( 2 4 x 3 2 4 ) 2 )( (x 1 3 ) 2 ( 8 12 6 4 8 12 x x x
11.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 11 Although the Binomial Theorem is stated for a binomial which is a sum of terms, it can also be used to expand a difference of terms. Simply rewrite (x + y)n as (x + (– y))n and apply the theorem to this sum. Example: Use the Binomial Theorem to expand (3x – 4)4. 4 ) 4 3 ( x 4 )) 4 ( 3 ( x 4 3 2 2 3 4 ) 4 ( 1 ) 4 )( 3 ( 4 ) 4 ( ) 3 ( 6 ) 4 ( ) 3 ( 4 ) 3 ( 1 x x x x 256 ) 64 )( 3 ( 4 ) 16 )( 9 ( 6 ) 4 )( 27 ( 4 81 2 3 4 x x x x 256 768 864 432 81 2 3 4 x x x x
12.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 12 Example: Use the Binomial Theorem to write the first three terms in the expansion of (2a + b)12 . ... ) 2 ( 2 12 ) 2 ( 1 12 ) 2 ( 0 12 ) 2 ( 2 10 11 12 12 b a b a a b a ... ) 2 ( ! 2 )! 2 12 ( ! 12 ) 2 ( ! 1 )! 1 12 ( ! 12 ) 2 ( ! 0 )! 0 12 ( ! 12 2 10 10 11 11 12 12 • • • b a b a a ... ) 2 )( 11 12 ( ) 2 ( 12 ) 2 ( 2 10 10 11 11 12 12 • b a b a a ... 135168 24576 4096 2 10 11 12 b a b a a
13.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 13 Example: Find the eighth term in the expansion of (x + y)13 . Think of the first term of the expansion as x13y0 . The power of y is 1 less than the number of the term in the expansion. The eighth term is 13C7 x6y7. Therefore, the eighth term of (x + y)13 is 1716 x6y7. ! 7 ! 6 ! 7 ) 8 9 10 11 12 13 ( ! 7 ! 6 ! 13 • • • • • • • • 7 13 C 1716 1 2 3 4 5 6 8 9 10 11 12 13 • • • • • • • • • •
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