6.8 B The Binomial Theorem
Honors Algebra 2
Lesson 6.8A: The Binomial Theorem
Objectives:
 To evaluate a binomial coefficient
 To expand a binomial raised to a power
VOCABULARY n! (read: “n factorial”) is the product of all positive integers less than or equal to n. ____________________________
Complete.
0!= 1
1! = 1 • 0! = _____
2! = 2 • 1 • 0! = _____
3! = 3 • 2 • 1 • 0! = _____
4! = ____________________
5! = _________________________
6! = ______________________________
7! = ___________________________________
8! = ________________________________________
9! = _____________________________________________
BINOMIAL COEFFICIENTS
A more efficient way to determine the coefficients in a binomial raised to a relatively large power is to write them in terms of factorials.
Definition of a Binomial Coefficient n r
For nonnegative integers n and r, with n ≥ r, the expression n r
(read “n above r”) is called a binomial coefficient and is defined by n r
= n!
r! (n-r)!
Mary McConaughy Page 1 4/13/2020

6.8 B The Binomial Theorem
Honors Algebra 2
NOTE: The symbol nCr is often used in place of n r
to denote binomial coefficients.
EXAMPLE 1 Evaluating Binomial Coefficients
Evaluate: a.
6
 
b.
3
 
c.
9
 
d.
4
 
In each case, apply the definition of the binomial coefficient. a.
6
 
=
6!
2! (6-2)!
= ______________ = b.
3
 
= c.
9
 
= d.
4
 
=
THE BINOMIAL THEOREM
A Formula for Expanding Binomials: The Binomial Theorem
For any positive integer n,
(a + b) n = n
0
a n + n a n-1 b + n a n-2 b 2 + n a n-3 b 3 +… + n n
b n
EXAMPLE 2 Using the Binomial Theorem
Expand: (x + 2) 4 Note: a = x; b = 2, and n = 4
_______ + _______ + _______ + _______ + _______
Solution
(x +2) 4 X 4 8x 3 24x 2
Mary McConaughy Page 2 4/13/2020

6.8 B The Binomial Theorem
Honors Algebra 2
Technology
Graphing calculators can compute binomial coefficients. For example to find
6
 
, many calculators require the sequence 6 nCr
2 ENTER. Use your calculator to verify the other evaluations in example 1.
NOTE: The combination key (nCr) is located under the math probability menu. Enter the number of objects, n, first; then the combination key; then the number of objects to take at one time, r.
FINAL CHECKS FOR UNDERSTANDING
1.
Describe the pattern on the exponents on a in the expansion (a + b) n .
2.
Describe the pattern on the exponents on b in the expansion of (a + b) n .
3.
What is true about the sum of the exponents on a and b in any term of the expansion of (a + b) n ?
4.
How do you determine how many terms there are in a binomial expansion?
5.
What is Pascal’s triangle? How do you find the numbers in any row of the triangle?
6.
Explain how to evaluate n r
. Provide an example with your explanation.
7.
Explain how to use the Binomial Theorem to expand a binomial. Provide an example with your explanation.
8.
Are situations in which it is easier to use Pascal’s triangle than binomial coefficients? Describe these situations.
Mary McConaughy Page 3 4/13/2020

6.8 B The Binomial Theorem
Honors Algebra 2
Mary McConaughy Page 4 4/13/2020