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15-5 The Binomial Theorem Pascal’s Triangle

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Page 1: 15-5 The Binomial Theorem Pascal’s Triangle. At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1)

15-5 The Binomial Theorem

Pascal’s Triangle

Page 2: 15-5 The Binomial Theorem Pascal’s Triangle. At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1)
Page 3: 15-5 The Binomial Theorem Pascal’s Triangle. At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1)

At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1) contains two 1's, both

formed by adding the two numbers above them to the left and the right, in this case 1 and 0 (all numbers outside the

Triangle are 0's). Do the same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1. And the third: 0+1=1; 1+2=3; 2+1=3;

1+0=1. In this way, the rows of the triangle go on infinitely. A number in the triangle can also be found by nCr (n Choose r) where n is the number of the row and r is the element in

that row. For example, in row 3, 1 is the zeroth element, 3 is element number 1, the next three is the 2nd element, and

the last 1 is the 3rd element. The formula for nCr is:

n!--------r!(n-r)!

Page 4: 15-5 The Binomial Theorem Pascal’s Triangle. At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1)

The Binomial Theorem is a quick way (okay, it's a less slow way) of expanding (or multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power. For instance, the expression (3x – 2)10 would be very painful to multiply out by hand. Thankfully, somebody figured out a formula for this expansion, and we can plug the binomial 3x – 2 and the power 10 into that formula to get that expanded (multiplied-out) form.

The Binomial Theorem

Page 5: 15-5 The Binomial Theorem Pascal’s Triangle. At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1)

0

nn n k k

k

na b a b

k

0 1 1 2 20 1 2

3 3 03

n n nn n n

n nn n n

C a b C a b C a b

C a b C a b

Page 6: 15-5 The Binomial Theorem Pascal’s Triangle. At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1)

Expand (a + b)5

5 0 5 1 1 5 2 25 0 5 1 5 2

5 3 3 5 4 4 5 5 55 3 5 4 5 5

C a b C a b C a b

C a b C a b C a b

5 0 4 1 3 2 2 3

1 4 0 5

1 5 10 10

5 1

a b a b a b a b

a b a b

5 4 1 3 2 2 3 4 55 10 10 5a a b a b a b ab b

Page 7: 15-5 The Binomial Theorem Pascal’s Triangle. At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1)

Expand (a -b)5

5 0 4 1 3 2 2 3

1 4 0 5

1 5 10 10

5 1

a b a b a b a b

a b a b

5 4 1 3 2 2 3 4 55 10 10 5a a b a b a b ab b

5 0 5 1 1 5 2 25 0 5 1 5 2

5 3 3 5 4 4 5 5 55 3 5 4 5 5

C a b C a b C a b

C a b C a b C a b

Page 8: 15-5 The Binomial Theorem Pascal’s Triangle. At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1)

Expand (2a + 1)5

5 0 4 1 3 2 2 3

1 4 0 5

1 2 1 5 2 1 10 2 1 10 2 1

5 2 1 1 2 1

a a a a

a a

5 4 3 232 80 80 40 10 1a a a a a

5 0 5 1 1 5 2 25 0 5 1 5 2

5 3 3 5 4 4 5 5 55 3 5 4 5 5

C a b C a b C a b

C a b C a b C a b

Page 9: 15-5 The Binomial Theorem Pascal’s Triangle. At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1)

Expand (3a-2b2)5

0 1 2 35 4 3 22 2 2 2

4 51 02 2

1 3 2 5 3 2 10 3 2 10 3 2

5 3 2 1 3 2

a b a b a b a b

a b a b

5 4 2 3 4 2 6 8 10243 810 1080 720 240 32a a b a b a b ab b

5 0 5 1 1 5 2 25 0 5 1 5 2

5 3 3 5 4 4 5 5 55 3 5 4 5 5

C a b C a b C a b

C a b C a b C a b

Page 10: 15-5 The Binomial Theorem Pascal’s Triangle. At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1)

Expand (x2 + 3)6

6 0 6 1 1 6 2 26 0 6 1 6 2

6 3 3 6 4 4 6 5 5 0 66 3 6 4 6 5 6 6

C a b C a b C a b

C a b C a b C a b C a b

6 5 4 30 1 2 32 2 2 2

2 1 04 5 62 2 2

1 3 6 3 15 3 20 3

15 3 6 3 1 3

x x x x

x x x

12 10 8 6 4 218 135 540 1215 1458 729x x x x x x

Page 11: 15-5 The Binomial Theorem Pascal’s Triangle. At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1)

Assignments

•15.5 Assignment•Page 592 6-24 even

•Review Assignment•Page 587 1-24


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