Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it...
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Transcript of Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it...
![Page 1: Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/1.jpg)
Aim: Binomial Theorem Course: Alg. 2 & Trig.
Do Now:
Aim: What is the Binomial Theorem and how is it useful?
Expand (x + 3)4
![Page 2: Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/2.jpg)
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
![Page 3: Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/3.jpg)
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
![Page 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.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/4.jpg)
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)
![Page 5: Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/5.jpg)
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
![Page 6: Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/6.jpg)
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
![Page 7: Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/7.jpg)
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
![Page 8: Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/8.jpg)
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
![Page 9: Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/9.jpg)
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
![Page 10: Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/10.jpg)
Aim: Binomial Theorem Course: Alg. 2 & Trig.
Regents Question
Write the binomial expansion of (2x − 1)5 as a polynomial in simplest form.
![Page 11: Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/11.jpg)
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
![Page 12: Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.](https://reader035.fdocuments.in/reader035/viewer/2022062518/56649ef35503460f94c05083/html5/thumbnails/12.jpg)
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