1. Pascal’s Triangle and the Binomial Theorem, then Exam!
20.0 Students know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers.

2. Pascal’s Triangle and the Binomial Theorem
Objectives
Key Words
Relate Pascal’s Triangle to the terms of a Binomial Expansion
The Binomial Theorem
Pascal’s triangle
The arrangement of 𝑛 𝐶 𝑟 in a triangular pattern in which each row corresponds to a value of n. (pg 553, you have to see it to believe it!)

3. Pascal’s Triangle
If you arrange the values of 𝑛 𝐶 𝑟 in a triangular pattern in which each row corresponds to a value of n, you get a pattern called Pascal’s triangle.
Turn to page 553.

4. For any positive integer n, the expansion of 𝑎+𝑏 𝑛 is:
The Binomial Theorem
For any positive integer n, the expansion of 𝑎+𝑏 𝑛 is:
𝑎+𝑏 𝑛 =
𝑛 𝐶 0 𝑎 𝑛 𝑏 𝑛 𝐶 1 𝑎 𝑛−1 𝑏 1 + 𝑛 𝐶 2 𝑎 𝑛−2 𝑏 2 +⋯+ 𝑛 𝐶 𝑛 𝑎 0 𝑏 𝑛
Note that each term has the form 𝑛 𝐶 𝑟 𝑎 𝑛−𝑟 𝑏 𝑟 where r is an integer from 0 to n.
Examples:
𝑎+𝑏 1
𝑎+𝑏 2
𝑎+𝑏 3

5. Example 1 Expand ( )4. b a + SOLUTION
Expand a Power of a Simple Binomial Sum
Expand (
)4. b a +
SOLUTION
In , the power is n So, the coefficients of the terms are the numbers in the 4th row of Pascal’s Triangle. = ( )4 b a +
Coefficients: 1, 4, 6, 4, 1
Powers of a: a 4, a 3, a 2, a 1, a 0
Powers of b: b0, b1, b2, b3, b4 =
1a 4b 0 + ( )4 b a
4a 3b 1
6a 2b 2
1a 0b 4
4a 1b 3 =
a 4 +
4a 3b
6a 2b 2
4ab 3
b 4 5

6. Use the binomial theorem with a x and b 5. =
Example 2
Expand a Power of a Binomial Sum
Expand (
)3. 5 x +
SOLUTION
Use the binomial theorem with a x and b = 5 x ( + 3 =
3C0x 350
3C1x 251
3C2x 152
3C3x 053 = + ( )
x 3 1
x 2 3 5
x 1 25
x 0
125 = +
x 3
15x 2
75x
125 6

7. First rewrite the difference as a sum: SOLUTION
Example 3
Expand a Power of a Binomial Difference
Expand y 2x ( – )4 .
First rewrite the difference as a sum:
SOLUTION = 2x [ + 4 (
– y ] y –
) 4 =
Then use the binomial theorem with a 2x and b –y.
4C3
4C2
4C4 = + 2x [ 4 (
– y ]
4C0
4C1 3 1 2 7

8. Example 3 = + ( ) 16x 4 1 8x 3 4 y – 4x 2 6 y 2 2x – y 3 y 4 = 16x 4 +
Expand a Power of a Binomial Difference = + ( )
16x 4 1
8x 3 4 y –
4x 2 6
y 2 2x
– y 3
y 4 =
16x 4 + –
32x 3y
24x 2y 2
8xy 3
y 4 8

9. Expand the power of the binomial sum or difference.
Checkpoint
Expand a Power of a Binomial Sum or Difference
Expand the power of the binomial sum or difference. 1. b a ( + )5
ANSWER
a 5 +
5a 4b
10a 3b 2
10a 2b 3
5ab 4
b 5 2. 2 x ( + )4
ANSWER
x 4 +
8x 3
24x 2
32x 16 3. 5 3x ( + )3
ANSWER
27x 3 +
135x 2
225x
125

10. Expand the power of the binomial sum or difference.
Checkpoint
Expand a Power of a Binomial Sum or Difference
Expand the power of the binomial sum or difference. 4. 4 p ( )3 –
p 3
ANSWER + –
12p 2
48p 64 5. – n m ( )4
m 4
ANSWER + –
4m 3n
6m 2n 2
4mn 3
n 4 6. t 3s ( – )3
27s 3 + –
27s 2t
9st 2
t 3
ANSWER

11. Conclusions Summary Assignment Exit Slip:
How can you calculate the coefficients of the terms of 𝑎+𝑏 𝑛 ?
Each term in the expansion of 𝑎+𝑏 𝑛 has the form 𝑛 𝐶 𝑟 𝑎 𝑛−𝑟 𝑏 𝑟 , where r is an integer from 0 to n.
Pg 555
#(2,6-13)
Write the assignment down, you will work on it after the you finish the exam, early.
Get ready for the exam.

12. 45 Minutes No talking – Read Rubric – Read Directions – Good Luck!
Exam on the Fundamental Counting Principle
45 Minutes No talking – Read Rubric – Read Directions – Good Luck!