1. The Binomial Theorem
8.5

2. Binomials like (x + y)4
The coefficients of a binomial expansion are called binomial coefficients. Use the Binomial Theorem to find them.

3. The Binomial Theorem
In the expansion of (x + y)n (x + y)n = xn + nxn-1y + …+ nCrxn-ry + … + nxyn-1 + yn The coefficient of xn-ryr is nCr = The symbol is often used in place of nCr to denote binomial coefficients.

4. Example 1: Finding Binomial Coefficients
Find each binomial coefficient.
8C2
7C0
20C15

5. Example 2: Finding Binomial Coefficients
Find each binomial coefficient.
7C3
7C4
12C1
12C11

6. Pascal’s Triangle
From the FRENCH mathematician Blaise Pascal (starts with the first row called Zero row) 1

7. Example 3: Using Pascal’s Triangle
Use Pascal’s Triangle to find the binomial coefficients: 8C0, 8C1, 3C2, 8C3, 8C4, 8C5, 8C6, 8C7, 8C8

8. Example 4: Expanding a Binomial
Write the expansion for the expression (x + 1)3.
Write the expansion for the expression (x + 2)4.

9. What about (x – 1)3?
To expand binomials with differences (subtraction) alternate the signs from negative to positive.

10. Example 5: Expanding Binomial Expressions
Write the expansion for each expression.
(2x – 3)4
(x – 2y)3