1. The binomial expansions
Unit 1. The Binomial Theorem
The binomial expansions
reveal a pattern.

2. 1.1 A Binomial Expansion Pattern
The expansion of (x + y)n begins with x n and ends with y n .
The variables in the terms after x n follow the pattern x n-1y , x n-2y2 , x n-3y3 and so on to y n . With each term the exponent on x decreases by 1 and the exponent on y increases by 1.
In each term, the sum of the exponents on x and y is always n.
The coefficients of the expansion follow Pascal’s triangle.

3. 1.2 A Binomial Expansion Pattern
Pascal’s Triangle
Row

4. 1.3 Pascal’s Triangle
Each row of the triangle begins with a 1 and ends with a 1.
Each number in the triangle that is not a 1 is the sum of the two numbers directly above it (one to the right and one to the left.)
Numbering the rows of the triangle 0, 1, 2, … starting at the top, the numbers in row n are the coefficients of x n, x n-1y , x n-2y2 , x n-3y3, … y n in the expansion of (x + y)n.

5. 1.4 n-Factorial (Optional)
For any positive integer n,
and
Example Evaluate (a) 5! (b) 7!
Solution (a)
(b)

6. 1.5 Binomial Coefficients
For nonnegative integers n and r, with r < n,

7. 1.5 Binomial Coefficients
The symbols and for the binomial
coefficients are read “n choose r”
The values of are the values in the nth
row of Pascal’s triangle.
So is the first number in the third row
and is the third.

8. 1.6 Evaluating Binomial Coefficients
Example Evaluate (a) (b)
Solution
(a)
(b)

9. 1.7 The Binomial Theorem
Binomial Theorem
For any positive integers n,

10. 1.8 Applying the Binomial Theorem
Example Write the binomial expansion of
Solution Use the binomial theorem

11. 1.8 Applying the Binomial Theorem

12. 1.8 Applying the Binomial Theorem
Example Expand
Solution Use the binomial theorem with
and n = 5,

13. 1.8 Applying the Binomial Theorem
Solution

14. 1.9 rth Term of a Binomial Expansion
rth Term of the Binomial Expansion
The rth term of the binomial expansion of (x + y)n,
where n > r – 1, is

15. 1.9 Finding a Specific Term of a Binomial Expansion.
Example Find the fourth term of
Solution Using n = 10, r = 4, x = a, y = 2b in the
formula, we find the fourth term is