1. 4.2 Pascal’s Triangle and the Binomial Theorem

2. Consider the binomial expansions again…

3. specifically… Consider the x2a term There are 3 ways to get that term.
That is, there are ways to get that term.

4. Pascal’s Triangle using
Value of n 1 2 3 4 5
r = 0
r = 1
r = 2
r = 3
r = 4
r = 5

5. Binomial Theorem
The coefficients of the form are called binomial coefficients.

6. Expand and simplify using the binomial theorem
(x + y)6
(2x – 1)4

7. Expand and simplify using the binomial theorem
(3x – 2y)5

8. Example 2 Using the binomial theorem, rewrite
1 + 10x2 + 40x4 + 80x6 + 80x8 + 32x10
in the form (a + b)n.
n = 5 (6 terms)

9. Pascal’s Identity

10. General Term of Binomial Expansion
The general in the expansion of (a + b)n is

11. Example 3 Use Pascal’s Identity to write an expression for n = 47

12. Example 4 Consider the expansion of What is the constant term? or
We want an-rbr = x0
8 – 3r = 0
r must be a whole number, so there is no constant term!