1. Section 8.5
The Binomial Theorem

2. The Binomial Theorem In this section you will learn two techniques for
expanding a binomial when raised to a power.
The first method is called expanding by using
Pascal’s Triangle.
Read about Pascal’s Triangle and Expanding a
Binomial at

3. The Binomial Theorem Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1
1 1
Can you figure out the pattern?
What would be the next row?

4. The Binomial Theorem
Each row of Pascal’s Triangle is the coefficients for the
Terms when the binomial is expanded.
(x + y)0
= 1
(x + y)1
= 1x + 1y
(x + y)2
= 1x2 + 2xy + 1y2
(x + y)3
= 1x3 + 3x2y + 3xy2 + 1y3
See the Triangle?

5. The Binomial Theorem When expanding a binomial remember: (x + y ) 4
The first term of the binomial will decrease in powers.
The second term of the binomial will increase in powers.
(x + y ) 4
_x4 +_ x3y + _x2y2 + _xy3 +_ y4
In Pascal’s Triangle find the row whose second number
is the power to be expanded. Here it is 4. Now place the
numbers in the row in front of the terms.
First
Term
Second
Term
= 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4
The operation between
the two terms goes with
the second term.
Now simplify where possible.
= x4 + 4x3y + 6x2y2 + 4xy3 + y4

6. The Binomial Theorem Expand (x – 3y)3.
x is the first term and negative 3y is the second term
(x – 3y)3
_x3 + _x2(–3y) + _x(–3y)2 + _(–3y)3
= 1x3 + 3x2(–3y) + 3x(–3y)2 + 1(–3y)3
= 1x3 +3x2(–3y) + 3x(9y2) + 1(– 27y3)
= x3 – 9x2y + 27xy2 – 27y3

7. The Binomial Theorem
The second method is expanding by the Binomial Theorem.
Before beginning one needs to know the notation nCr.
Example: Find the value of 5C3
Can you find how to do 5C3 on
the calculator?

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

9. The Binomial Theorem Expand (x + y)3 using the Binomial Theorem.
Notice in line 1 that n stays constant and r decreases by 1 each time. Also, notice the
sum of the exponents is the same as the exponent of the binomial being expanded.

10. The Binomial Expression
Expand (2x – y)5 using the Binomial Theorem.

11. The Binomial Theorem
Sometimes only a certain term of the binomial expansion
needs to be found. In that case remember n stays constant
and the term to be found uses the formula r + 1.
Find the fourth term of (x +2y)6.
Solution:
Here n = 6 and since r + 1 = 4, then r = 3.

12. The Binomial Theorem What you should know:
How to apply Pascal’s Triangle to expand a binomial.
2. How to apply the Binomial Theorem to expand a binomial.
3. How to find a certain term of an expanded binomial.