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.
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
Aim: Binomial Theorem Course: Alg. 2 & Trig.
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
Aim: Binomial Theorem Course: Alg. 2 & Trig.
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)
Aim: Binomial Theorem Course: Alg. 2 & Trig.
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
Aim: Binomial Theorem Course: Alg. 2 & Trig.
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
Aim: Binomial Theorem Course: Alg. 2 & Trig.
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
Aim: Binomial Theorem Course: Alg. 2 & Trig.
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
Aim: Binomial Theorem Course: Alg. 2 & Trig.
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
Aim: Binomial Theorem Course: Alg. 2 & Trig.
Regents Question
Write the binomial expansion of (2x − 1)5 as a polynomial in simplest form.
Aim: Binomial Theorem Course: Alg. 2 & Trig.
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
Aim: Binomial Theorem Course: Alg. 2 & Trig.
Regents Questions
What is the fourth term in the expansion of (3x – 2)5?
1) -720x2 2) -240x 3) 720x2 4) 1,080x3
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