Introduction
Expanding binomial expressions can sometimes be
tedious and time-consuming, but the expansion of
binomials raised to a power follows patterns.
Understanding these patterns not only helps with
determining the binomial expansion, but also with finding
single terms in a binomial.
1
2.2.3: The Binomial Theorem
Key Concepts
• When a binomial is raised to a power, (a + b)n, there is
a pattern in both the powers of the terms and the
coefficients of the terms; a and b can be numbers,
variables, or a combination of numbers and variables.
• The powers of the terms of the expanded form of
(a + b)n follow the pattern anb0, an – 1b1, an – 2b2, … ,
a2bn – 2, a1bn – 1, a0bn.
• The power of a is n – k, where k is the term number.
Note that the first term is term number 0, or k = 0. The
power of b is k.
2
2.2.3: The Binomial Theorem
Key Concepts, continued
• For example, if n = 1, (a + b)1, the powers of the terms
are a1 and b1. If n = 2, (a + b)2, the powers of the terms
are a2, a1b1, and b2. If n = 3, (a + b)3, the powers of the
terms are a3, a2b1, a1b2, and b3.
• Pascal’s Triangle is a triangle displaying a pattern of
numbers in which the terms in additional rows are found
by adding pairs of terms in the previous rows, so that
any given term is the sum of the two terms directly
above it.
3
2.2.3: The Binomial Theorem
Key Concepts, continued
Pascal’s Triangle
4
2.2.3: The Binomial Theorem
Key Concepts, continued
• Notice that the number 1 is the top number of the
triangle, and is also the first and last number of each row.
• The top row, containing only the number 1, is called
“row 0.” The next row is “row 1” and so on.
• The first term in each row is “term 0.” The next term is
“term 1,” and so on.
• The pattern in Pascal’s Triangle can be used to find the
coefficients of terms in the expanded form of (a + b)n.
The coefficients of the terms depend on the power of
the binomial, n.
5
2.2.3: The Binomial Theorem
Key Concepts, continued
• The coefficients of (a + b)n can be found in the nth row
of Pascal’s Triangle.
• For example, if n = 1, the coefficients of the terms are
the terms in the row 1 of the triangle: 1 and 1. If n = 2,
the coefficients of the terms are the terms in row 2 of
the triangle: 1, 2, and 1. If n = 3, the coefficients of the
terms are the terms in row 3 of the triangle: 1, 3, 3,
and 1.
• The pattern of the powers and coefficients can be
used to describe the pattern for finding any power of a
binomial.
2.2.3: The Binomial Theorem
6
Key Concepts, continued
• The Binomial Theorem states that (a + b)n can be
expanded using the following expression:
7
2.2.3: The Binomial Theorem
Key Concepts, continued
• Recall that the symbol ! represents a factorial, which
is the product of an integer and all preceding positive
integers;
. For example,
5! = 5 • 4 • 3 • 2 • 1. By definition, 0! = 1. Factorials
can be performed on most calculators.
• The symbol å stands for sigma, a Greek letter used
to represent the summation of values.
8
2.2.3: The Binomial Theorem
Key Concepts, continued
• The values for
n!
( n - k )! k !
can be found using the rows
of Pascal’s Triangle. Each expanded expression
contains n + 1 terms.
• For example, if n = 1, use row 1 of Pascal’s Triangle to
determine the coefficients, and then follow the pattern
of the powers using the Binomial Theorem: (a + b)1 =
1a1 + 1b1. If n = 2, use row 2 of Pascal’s Triangle and
the pattern of the powers: (a + b)2 = 1a2 + 2a1b1 + 1b2.
If n = 3, use row 3 of Pascal’s Triangle and the pattern
of the powers: (a + b)3 = 1a3 + 3a2b1 + 3a1b2 + 1b3.
2.2.3: The Binomial Theorem
9
Key Concepts, continued
• To expand a binomial expression raised to a power,
replace a, b, and n in (a + b)n with the actual values.
• For example, to expand (2x + 3)2, use the expansion
(a + b)2 = 1a2 + 2a1b1 + 1b2. Replace a with 2x and b
with 3: (2x + 3)2 = (2x)2 + 2(2x)(3) + (3)2 = 4x2 + 12x + 9.
• Binomial expansion as it relates to Pascal’s Triangle is
displayed in the table on the following slide.
(Coefficients and powers of 1 are not shown, but are
implied.)
10
2.2.3: The Binomial Theorem
Key Concepts, continued
Pascal’s Triangle
Row
(a + b)0 = 1
1
0
(a + b)1 = a + b
1 1
1
(a + b)2 = a2 + 2ab + b2
1 2 1
2
1 3 3 1
3
1 4 6 4 1
4
1 5 10 10 5 1
5
Binomial expansion
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
11
2.2.3: The Binomial Theorem
Key Concepts, continued
• Pascal’s Triangle can also be used to find the number
of combinations, or unique groups of objects, from a
larger group.
• The row number of the triangle is the total number of
objects in the group, and the term number in the row
is the number of objects being selected from the
group. The coefficient for that term number is the
number of combinations.
12
2.2.3: The Binomial Theorem
Common Errors/Misconceptions
• incorrectly replacing a and b with the actual binomial
terms
• incorrectly calculating powers of numbers and
variables
• misidentifying the coefficients from Pascal’s Triangle
13
2.2.3: The Binomial Theorem
Guided Practice
Example 1
Use the Binomial Theorem to expand (6x + 2y)3.
14
2.2.3: The Binomial Theorem
Guided Practice: Example 1, continued
1. Create Pascal’s Triangle to the
appropriate row.
Note that the first line of the triangle is “row 0.” The
expression (6x + 2y) is raised to the third power, so
the first four rows of the triangle are needed; that is,
rows 0–3.
15
2.2.3: The Binomial Theorem
Guided Practice: Example 1, continued
Find the terms for each new row by adding pairs of
terms in the row above.
16
2.2.3: The Binomial Theorem
Guided Practice: Example 1, continued
2. Identify the row of Pascal’s Triangle with
the coefficients of the expanded
expression.
The power of the binomial is 3, so the coefficients will
come from row 3 of Pascal’s Triangle.
17
2.2.3: The Binomial Theorem
Guided Practice: Example 1, continued
The coefficients of the terms in row 3 are 1, 3, 3, and 1.
18
2.2.3: The Binomial Theorem
Guided Practice: Example 1, continued
3. Write the expanded expression with the
coefficients and powers of each term.
The powers of the terms will follow the pattern a3, a2b1,
a 1b 2, b 3.
Replace a with 6x and b with 2y.
The coefficients of the terms are from row 3 of Pascal’s
Triangle.
(6x)3 + 3(6x)2(2y) + 3(6x)(2y)2 + (2y)3
19
2.2.3: The Binomial Theorem
Guided Practice: Example 1, continued
4. Evaluate each term.
Use the order of operations to evaluate each term, first
evaluating any exponents, then finding the products.
(6x)3 + 3(6x)2(2y) + 3(6x)(2y)2 + (2y)3
=
216x3
=
216x3
+
3(36x2)(2y)
+
216x2y
+
+
3(6x)(4y2)
72xy2
+
8y3
+
8y3
Expression
from the
previous step
Evaluate the
exponents.
Find the
products.
20
2.2.3: The Binomial Theorem
Guided Practice: Example 1, continued
The expression (6x + 2y)3, when expanded,
is 216x 3 + 216x 2y + 72xy 2 + 8y 3.
✔
2.2.3: The Binomial Theorem
21
Guided Practice: Example 1, continued
22
2.2.3: The Binomial Theorem
Guided Practice
Example 4
Kamali has 7 unique bracelets, and is trying to decide
which 3 to wear on a date. Use Pascal’s Triangle to find
the number of different combinations of 3 bracelets that
Kamali could choose from the 7 she owns.
23
2.2.3: The Binomial Theorem
Guided Practice: Example 4, continued
1. Create Pascal’s Triangle to the
appropriate row.
Note that the first line of the triangle is “row 0,” so the
first eight rows of the triangle are needed.
Find the terms for each new row by adding pairs of
terms in the row above.
24
2.2.3: The Binomial Theorem
Guided Practice: Example 4, continued
25
2.2.3: The Binomial Theorem
Guided Practice: Example 4, continued
2. Find term 3 in the row.
The first term in each row is named “term 0,” so term
3 is actually the fourth term in the row.
26
2.2.3: The Binomial Theorem
Guided Practice: Example 4, continued
Term 3 from row 7 of Pascal’s Triangle is 35.
There are 35 different combinations of 3 bracelets
that Kamali could choose to wear out of the 7
bracelets she owns.
✔
27
2.2.3: The Binomial Theorem
Guided Practice: Example 4, continued
28
2.2.3: The Binomial Theorem