The document discusses two methods for expanding binomial expressions: Pascal's triangle and the binomial theorem. Pascal's triangle uses a recursive method to provide the coefficients for expanding binomials, but is only practical for smaller values of n. The binomial theorem provides an explicit formula for expanding binomials of the form (a + b)n using factorials and combinations. It works better than Pascal's triangle when n is large. Examples are provided to demonstrate expanding binomials like (3 - xy)4 and (x - 2)6 using both methods.

1. 11.6 The Binomial Theorem
Matthew 7:7 "Ask, and it shall be given you; seek, and ye
shall find; knock, and it shall be opened unto you."

2. How do we expand ( a + b ) ?
n

3. How do we expand ( a + b ) ?
n
We will explore two methods:

4. How do we expand ( a + b ) ?
n
We will explore two methods:
1) Using Pascal’s Triangle (a recursive method)

5. How do we expand ( a + b ) ?
n
We will explore two methods:
1) Using Pascal’s Triangle (a recursive method)
2) Using the Binomial Theorem (an explicit
method that uses combinatorics)

6. Pascal’s Triangle

7. Pascal’s Triangle
Recall from last year ...

8. Pascal’s Triangle
Provides coefficients for expansion of a binomial

9. Pascal’s Triangle
Provides coefficients for expansion of a binomial
0
(a + b) = 1
1
(a + b) = 1a + 1b
2 2 2
(a + b) = 1a + 2ab + 1b
3 3 2 2 3
(a + b) = 1a + 3a b + 3ab + 1b
4 4 3 2 2 3 4
(a + b) = 1a + 4a b + 6a b + 4ab + 1b
5 5 4 3 2 2 3 4 5
(a + b) = 1a + 5a b + 10a b + 10a b + 5ab + 1b

10. Pascal’s Triangle
Provides coefficients for expansion of a binomial
0
(a + b) = 1
1
(a + b) = 1a + 1b
2 2 2
(a + b) = 1a + 2ab + 1b
3 3 2 2 3
(a + b) = 1a + 3a b + 3ab + 1b
4 4 3 2 2 3 4
(a + b) = 1a + 4a b + 6a b + 4ab + 1b
5 5 4 3 2 2 3 4 5
(a + b) = 1a + 5a b + 10a b + 10a b + 5ab + 1b
Also notice the pattern of the exponents!

11. Pascal’s Triangle
Provides coefficients for expansion of a binomial
0
(a + b) = 1
1
(a + b) = 1a + 1b
2 2 2
(a + b) = 1a + 2ab + 1b
3 3 2 2 3
(a + b) = 1a + 3a b + 3ab + 1b
4 4 3 2 2 3 4
(a + b) = 1a + 4a b + 6a b + 4ab + 1b
5 5 4 3 2 2 3 4 5
(a + b) = 1a + 5a b + 10a b + 10a b + 5ab + 1b
Also notice the pattern of the exponents!
Let’s expand ( a + b ) .
6
(do on the board)

12. Let’s expand ( 3 − xy )
4

13. Let’s expand ( 3 − xy )
4
First, write out ( a + b ) in expanded form.
4

14. Let’s expand ( 3 − xy )
4
First, write out ( a + b ) in expanded form.
4
4 3 2 2 3 4
a + 4a b + 6a b + 4ab + b

15. Let’s expand ( 3 − xy )
4
First, write out ( a + b ) in expanded form.
4
4 3 2 2 3 4
a + 4a b + 6a b + 4ab + b
Then substitute 3 for a and -xy for b

16. Let’s expand ( 3 − xy )
4
First, write out ( a + b ) in expanded form.
4
4 3 2 2 3 4
a + 4a b + 6a b + 4ab + b
Then substitute 3 for a and -xy for b
4 3 2 2 3 4
( 3) + 4 ( 3) ( −xy ) + 6 ( 3) ( −xy ) + 4 ( 3) ( −xy ) + ( −xy )

17. Let’s expand ( 3 − xy )
4
First, write out ( a + b ) in expanded form.
4
4 3 2 2 3 4
a + 4a b + 6a b + 4ab + b
Then substitute 3 for a and -xy for b
4 3 2 2 3 4
( 3) + 4 ( 3) ( −xy ) + 6 ( 3) ( −xy ) + 4 ( 3) ( −xy ) + ( −xy )
and simplify

18. Let’s expand ( 3 − xy )
4
First, write out ( a + b ) in expanded form.
4
4 3 2 2 3 4
a + 4a b + 6a b + 4ab + b
Then substitute 3 for a and -xy for b
4 3 2 2 3 4
( 3) + 4 ( 3) ( −xy ) + 6 ( 3) ( −xy ) + 4 ( 3) ( −xy ) + ( −xy )
and simplify
2 2 3 3 4 4
81− 108xy + 54x y − 12x y + x y

19. Pascal’s Triangle works great when n is
small in ( a + b )
n
The Binomial Theorem is better when n
is large. Let’s take a look at that now.

20. The Binomial Theorem

21. The Binomial Theorem
a, b ∈ ° ; n ∈ positive integers
n ⎛ n ⎞ n ⎛ n ⎞ n−1 ⎛ n ⎞ n−2 2 ⎛ n ⎞ n−1 ⎛ n ⎞ n
(a + b) = ⎜ ⎟ a + ⎜ ⎟ a b + ⎜ ⎟ a b +K + ⎜ ⎟ ab + ⎜ n ⎟ b
⎝ 0 ⎠ ⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ n − 1⎠ ⎝ ⎠

22. The Binomial Theorem
a, b ∈ ° ; n ∈ positive integers
n ⎛ n ⎞ n ⎛ n ⎞ n−1 ⎛ n ⎞ n−2 2 ⎛ n ⎞ n−1 ⎛ n ⎞ n
(a + b) = ⎜ ⎟ a + ⎜ ⎟ a b + ⎜ ⎟ a b +K + ⎜ ⎟ ab + ⎜ n ⎟ b
⎝ 0 ⎠ ⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ n − 1⎠ ⎝ ⎠
Let’s review combinations ... (next slide)

23. Combinations
⎛ n ⎞ n!
⎜ r ⎟ = r!( n − r )!
⎝ ⎠

24. Combinations
⎛ n ⎞ n!
⎜ r ⎟ = r!( n − r )!
⎝ ⎠
Example: ⎛ 5 ⎞ 5!
⎜ 2 ⎟ = 2!( 5 − 2 )!
⎝ ⎠
5 ⋅ 4 ⋅ 3⋅ 2 ⋅1
=
( 2 ⋅1)( 3⋅ 2 ⋅1)
5⋅4
=
2
= 10

25. Combinations
⎛ n ⎞ n!
⎜ r ⎟ = r!( n − r )!
⎝ ⎠
Example: ⎛ 5 ⎞ 5!
⎜ 2 ⎟ = 2!( 5 − 2 )!
⎝ ⎠
5 ⋅ 4 ⋅ 3⋅ 2 ⋅1
=
( 2 ⋅1)( 3⋅ 2 ⋅1)
5⋅4
=
2
= 10
Let’s review how your calculator can do this ...

26. ⎛ 5 ⎞
To do ⎜ ⎟ enter 5 nCr 2
⎝ 2 ⎠

27. ⎛ 5 ⎞
To do ⎜ ⎟ enter 5 nCr 2
⎝ 2 ⎠
For all combinations, you can use your calculator.
In your work, just show the combination notation,
but not the nCr notation.

28. 4
Use the Binomial Theorem to expand ( a + b )

29. 4
Use the Binomial Theorem to expand ( a + b )
⎛ 4 ⎞ 4 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 2 2 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 4
⎜ 0 ⎟ a + ⎜ 1⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3 ⎟ ab + ⎜ 4 ⎟ b
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

30. 4
Use the Binomial Theorem to expand ( a + b )
⎛ 4 ⎞ 4 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 2 2 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 4
⎜ 0 ⎟ a + ⎜ 1⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3 ⎟ ab + ⎜ 4 ⎟ b
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
4 3 2 2 3 4
a + 4a b + 6a b + 4ab + b

31. 4
Use the Binomial Theorem to expand ( a + b )
⎛ 4 ⎞ 4 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 2 2 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 4
⎜ 0 ⎟ a + ⎜ 1⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3 ⎟ ab + ⎜ 4 ⎟ b
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
4 3 2 2 3 4
a + 4a b + 6a b + 4ab + b
n
For more complex binomials, first expand ( a + b )
and then substitute in for a and b ... just like we
did with the Pascal’s Triangle method.

32. 6
Use the Binomial Theorem to expand ( x −2 )

33. 6
Use the Binomial Theorem to expand ( x −2 )
⎛ 6 ⎞ 6 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 4 2 ⎛ 6 ⎞ 3 3 ⎛ 6 ⎞ 2 4 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 6
⎜ 0 ⎟ a + ⎜ 1 ⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3⎟ a b + ⎜ 4 ⎟ a b + ⎜ 5 ⎟ ab + ⎜ 6 ⎟ b
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

34. 6
Use the Binomial Theorem to expand ( x −2 )
⎛ 6 ⎞ 6 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 4 2 ⎛ 6 ⎞ 3 3 ⎛ 6 ⎞ 2 4 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 6
⎜ 0 ⎟ a + ⎜ 1 ⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3⎟ a b + ⎜ 4 ⎟ a b + ⎜ 5 ⎟ ab + ⎜ 6 ⎟ b
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
a= x b = −2

35. 6
Use the Binomial Theorem to expand ( x −2 )
⎛ 6 ⎞ 6 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 4 2 ⎛ 6 ⎞ 3 3 ⎛ 6 ⎞ 2 4 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 6
⎜ 0 ⎟ a + ⎜ 1 ⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3⎟ a b + ⎜ 4 ⎟ a b + ⎜ 5 ⎟ ab + ⎜ 6 ⎟ b
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
a= x b = −2
6 5 4 2 3 3 2 4
1 x + 6 x ( −2 ) + 15 x ( −2 ) + 20 x ( −2 ) + 15 x ( −2 )
5 6
+6 x ( −2 ) + 1( −2 )

36. 6
Use the Binomial Theorem to expand ( x −2 )
⎛ 6 ⎞ 6 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 4 2 ⎛ 6 ⎞ 3 3 ⎛ 6 ⎞ 2 4 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 6
⎜ 0 ⎟ a + ⎜ 1 ⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3⎟ a b + ⎜ 4 ⎟ a b + ⎜ 5 ⎟ ab + ⎜ 6 ⎟ b
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
a= x b = −2
6 5 4 2 3 3 2 4
1 x + 6 x ( −2 ) + 15 x ( −2 ) + 20 x ( −2 ) + 15 x ( −2 )
5 6
+6 x ( −2 ) + 1( −2 )
5 3 1
3 2
x − 12x + 60x − 160x + 240x − 192x + 64
2 2 2

37. HW #11
“I am a little pencil in the hand of a writing God who is
sending a love letter to the world.”
Mother Teresa