Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4

Aim: Binomial Theorem Course: Alg. 2 & Trig.

Do Now:

Aim: What is the Binomial Theorem and how is it useful?

Expand (x + 3)4


Permutations & Combinations

A permutation is an arrangement of objects in a specific order.

The number of permutation of n things taken n at a time is

nPn = n! = n(n – 1)(n – 2)(n – 3) . . . 3, 2, 1

The number of permutation of n things taken r at a time is

( 1)( 2)n rP n n n L

r factors

!

( )!

n

n r


Permutations & Combinations

A combination is an arrangement of objects in which there is no specific order.

The number of combinations of n things taken n at a time is

The number of combinations of n things taken r at a time is

!n r

n r

PC

r

!( )!

!

nn r

r

1

1n nC

!

( )! !

n

n r r


Pascal’s Triangle

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

1 7 21 35 35 21 7 1

The first & last numbers in each row are 1

Every other number in each row is formed by adding the two numbers above the number.

In each expansion there is n + 1 terms (n is the row number)


Pascal’s Triangle & Expansion of (x + y)n

(x + y)0 = 1

(x + y)1 = 1x + 1y

(x + y)2 = 1x2 + 2xy + 1y2

(x + y)3 = 1x3 + 3x2y + 3xy2 + 1y3

(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4

(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5

In each expansion there is n + 1 terms.

In each expansion the x and y have symmetric roles.

The sum of the powers of each term is n.

The coefficients increase & decrease symmetrically.

5 5

44

expansion of (x + y)n

zero row

1st row


The Binomial Theorem

In the expansion of (x + y)n,

(x + y)n = xn + nxn-1y + . . .

+ nCrxn-ryr + . . . .

+ nxyn-1 + yn,

the coefficient of xn-ryr is given by

!

( )! !n r

nC

n r r

Example: 37 C!3)!37(

!7

35

123

567

351

1

77

07

C

C

Bernoulli Experiment

- probability of success &

failure


Coefficients of the ninth row

9 9

Pascal’s Triangle & the Binomial Theorem

(x + y)0 = 1

(x + y)1 = 1x + 1y

(x + y)2 = 1x2 + 2xy + 1y2

(x + y)3 = 1x3 + 3x2y + 3xy2 + 1y3

(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4

(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5

5 5

44

!

( )! !n r

nC

n r r

5C0 5C1 5C2 5C3 5C4 5C5

9C0 9C1 9C2 9C3 9C4 9C5 9C6 9C7 9C8 9C9

1 136 368484 126 126

expansion of (x + y)n

(x + y)n = xn + nxn-1y + . . .+ nCrxn-ryr + . . . . + nxyn-1 + yn


3C0 3C1 3C2 3C3

31 3 1

4 terms (n + 1)

xn-ryr

Binomial Expansion

Coefficients of the third row n = 3

Write the expansion of (x + 1)3

3C0 3C1 3C2 3C3

31 3 1

4 terms (n + 1)

1x3 + 3x2 + 3x + 1

xn-ryr

marry coefficients with terms and exponents of binomial

Write the expansion of (x - 1)3

1x3 – 3x2 + 3x – 1

expanded binomials with differences – alternate signs

n = highest exponent value in row


Model Problem

Coefficients of the fourth row n = 4

Write the expansion of (x - 2y)4

1x4(2y)0 - 4x3(2y)1 + 6x2(2y)2 - 4x(2y)3 + 1x0(2y)4

1x4 - 4x3 + 6x2 - 4x + 1x0

1 - 4 + 6 - 4 + 1

xn-ryr

5 terms (n + 1)

4C0 4C1 4C2 4C3

41 6 44C4

1

(x + y)n

x4 – 8x3y + 24x2y2 – 32xy3 + 16y4


Regents Question

Write the binomial expansion of (2x − 1)5 as a polynomial in simplest form.


nCrxn-ryr

General Formula

n = 12 12th row

Model Problem

Find the sixth term of the expansion of (3a + 2b)12

13 terms (n + 1)

xn-ryr

6th term

key: r = ?

12C0 = 11st term

5

792(3a)7(2b)5 = 55427328a7b5

x = 3a y = 2b

(3a)12-5(2b)5

xn-ryr

12C5

coeff.

12C01st term

12C12nd term

12C23rd term

12C34th term

12C45th term

12C56th term

1st term2nd term

3rd term4th term

5th term6th term

x12y0

x11y1

x10y2

x9y3

x8y4

x7y5


Regents Questions

What is the fourth term in the expansion of (3x – 2)5?

1) -720x2 2) -240x 3) 720x2 4) 1,080x3

Documents

Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4