Section8.7 The Binomial Theorem Formulas Factorial n ( n 1) . . - PowerPoint PPT Presentation

Section8.7 The Binomial Theorem

Formulas

Factorial � n ( n − 1) . . . (2)(1) if n = 2 , 3 , 4 , . . . n ! = 1 if n = 0 or 1 For example: 5! = 5 · 4 · 3 · 2 · 1 = 120

Examples 1. Compute 6!.

Examples 1. Compute 6!. 720

Examples 1. Compute 6!. 720 2. Compute 10! 8! .

Examples 1. Compute 6!. 720 2. Compute 10! 8! . 90

Binomial Coefficient � n � n ! n C k = = k !( n − k )! k The binomial coefficient is read as “ n choose k ”. The number on the bottom is never bigger than the number on top. For example: � 5 � 3!(5 − 3)! = 5! 5! = 3 3!2! 5 · 4 · 3 · 2 · 1 = (3 · 2 · 1)(2 · 1) = 20 2 = 10

Example � 10 � � 10 � Compute + . 1 9

Example � 10 � � 10 � Compute + . 1 9 20

Pascal’sTriangle

Construction of Pascal’s Triangle To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero.

Construction of Pascal’s Triangle To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero. 1

Construction of Pascal’s Triangle To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero. 1 1 1

Construction of Pascal’s Triangle To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero. 1 1 1 1 2 1

Construction of Pascal’s Triangle To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero. 1 1 1 1 2 1 1 3 3 1

Construction of Pascal’s Triangle To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

Construction of Pascal’s Triangle To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

Construction of Pascal’s Triangle To generate Pascal’s Triangle, you start with a 1 at the top. Then, the numbers for each subsequent row is found by adding the two numbers above it. You can imagine that everything “outside” of the triangle is a zero. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

Relationship to Binomial Coefficients It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula.

Relationship to Binomial Coefficients It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula. � 0 � = 1 0

Relationship to Binomial Coefficients It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula. � 0 � = 1 0 � 1 � 1 � = 1 � = 1 0 1

Relationship to Binomial Coefficients It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula. � 0 � = 1 0 � 1 � 1 � = 1 � = 1 0 1 � 2 � � 2 � � 2 � = 1 = 2 = 1 0 1 2

Relationship to Binomial Coefficients It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula. � 0 � = 1 0 � 1 � 1 � = 1 � = 1 0 1 � 2 � � 2 � � 2 � = 1 = 2 = 1 0 1 2 � 3 � 3 � 3 � 3 � � � � = 1 = 3 = 3 = 1 0 1 2 3

Relationship to Binomial Coefficients It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula. � 0 � = 1 0 � 1 � 1 � = 1 � = 1 0 1 � 2 � � 2 � � 2 � = 1 = 2 = 1 0 1 2 � 3 � 3 � 3 � 3 � � � � = 1 = 3 = 3 = 1 0 1 2 3 � 4 � 4 � 4 � 4 � 4 � � � � � = 1 = 4 = 6 = 4 = 1 0 1 2 3 4

Relationship to Binomial Coefficients It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula. � 0 � = 1 0 � 1 � 1 � = 1 � = 1 0 1 � 2 � � 2 � � 2 � = 1 = 2 = 1 0 1 2 � 3 � 3 � 3 � 3 � � � � = 1 = 3 = 3 = 1 0 1 2 3 � 4 � 4 � 4 � 4 � 4 � � � � � = 1 = 4 = 6 = 4 = 1 0 1 2 3 4 � 5 � 5 � 5 � 5 � 5 � 5 � � � � � � = 1 = 5 = 10 = 10 = 5 = 1 0 1 2 3 4 5

Relationship to Binomial Coefficients It turns out, that you can use Pascal’s Triangle to compute the binomial coefficients rather than using the formula. � 0 � = 1 0 � 1 � 1 � = 1 � = 1 0 1 � 2 � � 2 � � 2 � = 1 = 2 = 1 0 1 2 � 3 � 3 � 3 � 3 � � � � = 1 = 3 = 3 = 1 0 1 2 3 � 4 � 4 � 4 � 4 � 4 � � � � � = 1 = 4 = 6 = 4 = 1 0 1 2 3 4 � 5 � 5 � 5 � 5 � 5 � 5 � � � � � � = 1 = 5 = 10 = 10 = 5 = 1 0 1 2 3 4 5 � 6 � 6 � 6 � 6 � 6 � 6 � 6 � = 1 � = 6 � = 15 � = 20 � = 15 � = 6 � = 1 0 1 2 3 4 5 6

Relationship to Binomial Coefficients (continued) Each row of the triangle gives you all the coefficients for a particular n in � n � : k

Relationship to Binomial Coefficients (continued) Each row of the triangle gives you all the coefficients for a particular n in � n � : k n = 0: 1

Relationship to Binomial Coefficients (continued) Each row of the triangle gives you all the coefficients for a particular n in � n � : k n = 0: 1 n = 1: 1 1

Relationship to Binomial Coefficients (continued) Each row of the triangle gives you all the coefficients for a particular n in � n � : k n = 0: 1 n = 1: 1 1 n = 2: 1 2 1

Relationship to Binomial Coefficients (continued) Each row of the triangle gives you all the coefficients for a particular n in � n � : k n = 0: 1 n = 1: 1 1 n = 2: 1 2 1 n = 3: 1 3 3 1

Relationship to Binomial Coefficients (continued) Each row of the triangle gives you all the coefficients for a particular n in � n � : k n = 0: 1 n = 1: 1 1 n = 2: 1 2 1 n = 3: 1 3 3 1 n = 4: 1 4 6 4 1

Relationship to Binomial Coefficients (continued) Each row of the triangle gives you all the coefficients for a particular n in � n � : k n = 0: 1 n = 1: 1 1 n = 2: 1 2 1 n = 3: 1 3 3 1 n = 4: 1 4 6 4 1 n = 5: 1 5 10 10 5 1

Relationship to Binomial Coefficients (continued) Each row of the triangle gives you all the coefficients for a particular n in � n � : k n = 0: 1 n = 1: 1 1 n = 2: 1 2 1 n = 3: 1 3 3 1 n = 4: 1 4 6 4 1 n = 5: 1 5 10 10 5 1 n = 6: 1 6 15 20 15 6 1

MultiplyingBinomialEx- pressions

The Binomial Theorem The Binomial Theorem gives a formula for multiplying out binomial expressions: � n � � n � � n � ( A + B ) n = A n + A n − 2 B 2 + . . . A n − 1 B + 0 1 2 � � � � � n � n n A 2 B n − 2 + AB n − 1 + B n + n − 2 n − 1 n

Examples 1. Expand ( x + y ) 3 , and compute the binomial coefficients using the formula.

Examples 1. Expand ( x + y ) 3 , and compute the binomial coefficients using the formula. x 3 + 3 x 2 y + 3 xy 2 + y 3

Examples 1. Expand ( x + y ) 3 , and compute the binomial coefficients using the formula. x 3 + 3 x 2 y + 3 xy 2 + y 3 2. Expand (2 r + s 2 ) 4 , using Pascal’s Triangle to find the coefficients.

Examples 1. Expand ( x + y ) 3 , and compute the binomial coefficients using the formula. x 3 + 3 x 2 y + 3 xy 2 + y 3 2. Expand (2 r + s 2 ) 4 , using Pascal’s Triangle to find the coefficients. 16 r 4 + 32 r 3 s 2 + 24 r 2 s 4 + 8 rs 6 + s 8

Examples 1. Expand ( x + y ) 3 , and compute the binomial coefficients using the formula. x 3 + 3 x 2 y + 3 xy 2 + y 3 2. Expand (2 r + s 2 ) 4 , using Pascal’s Triangle to find the coefficients. 16 r 4 + 32 r 3 s 2 + 24 r 2 s 4 + 8 rs 6 + s 8 � 6 2 a − 1 � 3. Expand 2 b

Examples 1. Expand ( x + y ) 3 , and compute the binomial coefficients using the formula. x 3 + 3 x 2 y + 3 xy 2 + y 3 2. Expand (2 r + s 2 ) 4 , using Pascal’s Triangle to find the coefficients. 16 r 4 + 32 r 3 s 2 + 24 r 2 s 4 + 8 rs 6 + s 8 � 6 2 a − 1 � 3. Expand 2 b 64 a 6 − 96 a 5 b + 60 a 4 b 2 − 20 a 3 b 3 + 15 8 ab 5 + 1 4 a 2 b 4 − 3 64 b 6