Pc Binomial Theorem

The Binomial Theorem

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

1 term

(x + y)1 = x + y

2 terms

(x + y)2 = x2 + 2xy + y2

3 terms

(x + y)3 = x3 + 3x2y + 3xy2 + y3

4 terms

5 terms 6 terms (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4

Notice that each expansion has n + 1 terms. Example: (x + y)10 will have 10 + 1, or 11 terms.

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.”

The coefficients of the binomial expansion are called binomial coefficients. The coefficients have symmetry. (x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5 The first and last coefficients are 1. The coefficients of the second and second to last terms are equal to n. Example: What are the last 2 terms of (x + y)10 ? Since n = 10, the last two terms are 10xy9 + 1y10. n

The coefficient of xn–ryr in the expansion of (x + y)n is written   r or nCr . So, the last two terms of (x + y)10 can be expressed  

10  9  + as 10C9 xy + 10C10 y or as  xy 9  9

10

10  10  y . 10 

The triangular arrangement of numbers below is called Pascal’s Triangle. 1 0th row

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

1+2=3 6 + 4 = 10

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 .

Example: Use the fifth row of Pascal’s Triangle to generate the

6  , 6C4 and 6C2 . 5

6 sixth row and find the binomial coefficients  1 ,  

5th row 6th row

1

5

10

10

5

1

1

6

15

20

15

6   0

6   1

6   2

6   3

6 6     4 5

6

C0

C1

6

6

C2

6  =6= 1

6

C3

6

C4

6

6

C5

1 6   6

C6

6

6   and 6C4 = 15 = 6C2. 5

There is symmetry between binomial coefficients. C = nCn–r n r

Example: Use Pascal’s Triangle to expand (2a + b)4. 1

0th row

1 1

1st row

1 2 1

2nd row

1 3 3 1

3rd row

1 4 6 4 1

4th row

(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

The symbol n! (n factorial) denotes the product of the first n positive integers. 0! is defined to be 1. 1! = 1 4! = 4 3 2 1 = 24 6! = 6 5 4 3 2 1 = 720 n! = n(n – 1)(n – 2) … 3 2 1 •

•

•

•

•

•

•

•

•

•

Formula for Binomial Coefficients For all nonnegative n! integers n and r, n

Cr =

(n − r )!r !

7! 7! 7 = = Example: 7 C3 = (7 − 3)! • 3! 4! • 3! 4! • 3! (7 • 6 • 5 • 4) • (3 • 2 • 1) 7 • 6 • 5 • 4 = = = 35 (4 • 3 • 2 • 1) • (3 • 2 • 1) 4 • 3 • 2 •1

Example: Use the formula to calculate the binomial coefficients  50  12  C, C, and  48 . 10 5 15 0    1

10! 10! (10 • 9 • 8 • 7 • 6) • 5! 10 • 9 • 8 • 7 • 6 = = = = 252 10 C5 = (10 − 5)! • 5! 5! • 5! 5! • 5! 5 • 4 • 3• 2 •1 10 C0 =

10! 10! 1! 1 = = = =1 (10 − 0)! • 0! 10! • 0! 0! 1

50! (50 • 49) • 48! 50 • 49  50  50! = = = = 1225   = 2! • 48! 2 •1  48  (50 − 48)! • 48! 2! • 48! 12  12! 12 • 11! 12 12! = = = = 12   = • • • 1 1  (12 − 1)! 1! 1! 1! 11! 1!

Binomial Theorem

( x + y ) n = x n + nx n −1 y + L + nCr x n − r y r + L + nxy n −1 + y n n! with nCr = (n − r )!r !

Example: Use the Binomial Theorem to expand (x4 + 2)3. (x 4 + 2) 3 = 3 C0 (x 4 ) 3 + 3 C1( x 4 ) 2 (2) + 3 C2(x 4 )(2) 2 + 3 C3(2) 3 = 1 (x 4 ) 3 + 3 ( x 4 ) 2 (2) + 3 (x 4 )(2) 2 + 1 (2) 3 = x12 + 6 x 8 + 12 x 4 + 8

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. (3 x − 4) 4 = (3 x + (−4)) 4 = 1(3 x) 4 + 4(3 x) 3 (−4) + 6(3 x) 2 (−4) 2 + 4(3 x)(−4) 3 + 1(−4) 4 = 81x 4 + 4(27 x 3 )(−4) + 6(9 x 2 )(16) + 4(3 x)(−64) + 256 = 81x 4 − 432 x 3 + 864 x 2 − 768 x + 256

Example: Use the Binomial Theorem to write the first three terms in the expansion of (2a + b)12 . 12  12  12  12 11 (2a + b) =  (2a) +  (2a ) b +  (2a)10 b 2 + ... 0  1  2  12

12! 12! 12! 12 12 11 11 = (2 a ) + (2 a )b + (210 a10 )b 2 + ... (12 − 0)! • 0! (12 − 1)! • 1! (12 − 2)! • 2!

= (212 a12 ) + 12(211 a11 )b + (12 • 11)(210 a10 )b 2 + ...

= 4096 a12 + 24576 a11b + 135168 a10b 2 + ...

Example: Find the eighth term in the expansion of (x + y)13 . Think of the first term of the expansion as x13y 0 . The power of y is 1 less than the number of the term in the expansion. The eighth term is 13C7 x 6 y7. 13! (13 • 12 • 11 • 10 • 9 • 8) • 7! = 13 C7 = 6! • 7! 6! • 7! 13 • 12 • 11 • 10 • 9 • 8 = = 1716 • • • • • 6 5 4 3 21

Therefore, the eighth term of (x + y)13 is 1716 x 6 y7.