Name: ______________________________________________________________
Date: ________________________
Period: ______
Chapter 14: Probability & Statistics
Topic 3: Binomial Expansion
DO NOW: Expand each of the following, carefully distributing
1. (a + b)2 = (a+b)(a+b) = a2+ab+ab+b2=a2+2ab+b2
2. (a + b)3= (a+b)(a+b)(a+b)=
3.
=
=
(a+b)7 = no thank you
Don’t worry, there’s an easier way!
Things to notice:
 An expanded binomial to the n power will always have n+1 terms
 The coefficients are always Symmetrical
 The exponents of the a and b pieces count up or down, left to right.
PASCAL’S TRIANGLE
Tells us the coefficients of the entire expansion.
(a + b)0
(a + b)1
(a + b)2
(a + b)3
(a + b)4
(a + b)5
(a + b)6
(a + b)7
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
Examples: Expand
1.



5 terms because it’s to the 4th power).
Fill in the coefficients that Pascal’s Triangle gives us.
Fill in both variables on each space, then count up and down.
+
+
+
+
+
+
Then simplify whatever you can (no exponents of 1 or 0 power should remain)
+
Final answer is a nice clean polynomial of 5 terms
+
2.
BINOMIAL EXPANSION FORMULA
Tells us the entire term, one at a time.
Where n = power and r = term position


It has the same effect as Pascal’s Triangle, but without the need to create the triangle.
Especially helpful if you are asked for only one term of a binomial expansion. (you can always expand
the entire expression and select the particular term, but the formula can quickly find the needed term)
r-th term of
My Two Cents: think of each piece in its own parenthesis, and simplify one at a time…
Name: ______________________________________________________________
Examples:
1. What is the middle term in the expansion of (x + y)4?
Date: ________________________
Period: ______
….(4th power will have FIVE terms, so the THIRD term will be the MIDDLE)
Decide what term you’re looking for &
identify n, r, r-1
n=4, r=3, r-1=2
Write the formula
Substitute n, r, a and b
Simplify
Use calculator to find nCr
Multiply coefficients for final answer
2. What is the fourth term in the expansion of (2y + 1)7?
Decide what term you’re looking for &
n=7, r=4, r-1=3
identify n, r, r-1
Write the formula
Substitute n, r, a and b
Simplify
Use calculator to find nCr
Multiply coefficients for final answer
In your notebook!
3. What is the fourth term in the expansion of (2x – y)5? **WATCH THE NEGATIVE!**
4. What is the third term in the expansion of (x + 2y)5?
5. What is the third term in the expansion of (a – 3b)4?
6. What is the fourth term in the expansion of (3 + 2i)4?
7. What is the last term in the expansion of (x + 2y)5?
8. What is the middle term in the expansion of (x + y)4?
9. What is the third term in the expansion of (5 – 4i)5?
Finish for homework