1. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Vocabulary
A way to find coefficients in binomial expansions (a + b)2 where n is a positive integer.
Pascal’s triangle
n = 0 (0th row) 1
1 ___ 1
1 ___ ___ 1
1 ___ ___ ___ 1
n = 1 (1st row)
n = 2 (2nd row)
n = 3 (3rd row)
n = 4 (4th row)
The first and last numbers in each row are ___. Beginning with the second row, every other number is formed by ________ the two numbers immediately above the number. 1
adding

2. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Vocabulary
Binomial expansion

3. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Example 1
Use Pascal’s triangle
Use the fourth row of Pascal’s triangle to find the numbers in the fifth and sixth rows of Pascal’s triangle.
Solution
1 ___ ___ ___ ___ 1
1 ___ ___ ___ ___ ___ 1
n = 4 (4th row)
n = 5 (5th row)
n = 6 (6th row)

4. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Checkpoint. Complete the following exercises.
Find the numbers in the eighth row of Pascal’s triangle.
1 ___ ___ ___ ___ ___ ___ 1
1 ___ ___ ___ ___ ___ ___ ___ 1

5. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Example 2
Expand a power of a binomial sum
Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion of (x + 5)4.
Solution
The binomial coefficients from the fourth row of Pascal's triangle are ____, ____. ____, ____, and ____. So the expansion is as follows.

6. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Checkpoint. Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion.

7. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Checkpoint. Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion.

8. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Example 3
Expand a power of a binomial difference
Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion of (x - 6)3.
Solution
The binomial coefficients from the third row of Pascal's triangle are ____, ____. ____, and ____. So the expansion is as follows.

9. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Checkpoint. Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion.

10. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Checkpoint. Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion.

11. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Example 4
Expand a power of a binomial sum
Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion of (5 + 2x)4.
Solution
The binomial coefficients from the fourth row of Pascal's triangle are ____, ____. ____, ____, and ____. So the expansion is as follows.

12. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Checkpoint. Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion.

13. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Example 5
Find a coefficient in an expansion
Find the coefficient of x3 in (4x + 3)4.
Solution
The binomial coefficients from the fourth row of Pascal's triangle are ____, ____. ____, ____, and ____. So the expansion is as follows.
The coefficients of the x3–term is (___)(___)3(___) = _____.

14. Use the Binomial Theorem
2.4
Use the Binomial Theorem
Checkpoint. Complete the following exercise.
Find the coefficient of x2 in the expansion of (7 - x)5.

15. Use the Binomial Theorem
2.4
Use the Binomial Theorem
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