1. ©2001 by R. Villar All Rights Reserved
9.5 The Binomial Theorem
©2001 by R. Villar
All Rights Reserved

2. The Binomial Theorem
Recall that a binomial has two terms...
(x + y)
The Binomial Theorem gives us a quick method to expand binomials raised to powers such as… (x + y)0 (x + y)1 (x + y)2 (x + y)3
Study the following… 1
1 1
Row 0
This triangle is called Pascal’s
Triangle (named after mathematician
Blaise Pascal).
Row 1
Row 2
Notice that row 5 comes from adding up
row 4’s adjacent numbers.
(The first row is named row 0).
Row 3
Row 4
Row 5
What will row 6 be?
Row 6
This pattern will help us find the coefficients when we expand
binomials...

3. (x + y)0 = (x + y)1 = (x + y)2 = (x + y)3 = (x + y)4 = (x + y)5 =
Study the following...
(x + y)0 = (x + y)1 = (x + y)2 = (x + y)3 = (x + y)4 = (x + y)5 = 1
x + y
x2 + 2xy + y2
x3 + 3x2y + 3xy2 + y3
x4 + 4x3y + 6x2y2 + 4xy3 + y4
x5 + 5x4y +10x3y2 +10x2y3+5xy4 + y5
(x + y)6 =
x6+6x5y+15x4y2+20x3y3+15x2y4+6xy5 + y6
Notice that for (x + y)n there are n + 1 terms…
Notice that the binomial coefficients follow Pascal’s Triangle…
Notice that the first term is xn and the last term is yn …
Notice that the sum of the exponents of each term is n…
Use this pattern to expand (x + y)6...

4. The binomial expansion of (x + y)n is
The Binomial Theorem:
The binomial expansion of (x + y)n is
(x + y)n = xn + nxn–1 y + … n! xn-m ym +…+ nxyn–1 + yn (n – m)!m! The coefficient xn – my m is denoted by
This can be used to find any coefficient for the expansion of any binomial...
Example. Find the binomial coefficients for :
Row 6 of Pascal’s Triangle looks like this:
Here is another way to find the value of this coefficient
= 6! !2!
= 6 • 5 • 4! ! • 2 • 1 =

5. Example: Find the binomial coefficients for :
= ! !8!
= 12 • 11 • 10 • 9 • 8! • 3 • 2 • 1 • 8! =
= 495

6. Row 6 of Pascal’s Triangle looks like this: 1 6 15 20 15 6 1
Example: Use Pascal’s Triangle to find the binomial coefficients for the expansion of (x + y)8 .
Row 6 of Pascal’s Triangle looks like this:
Complete the pattern for rows 7 and 8...

7. Example: Use the binomial Theorem to expand (x – 3)4
Row 4 of Pascal’s Triangle looks like this:
which translates to:
x4 + 4x3y + 6x2y2 + 4xy3 + y4
Replace the y’s with –3...
x4 + 4x3(–3) + 6x2(–3)2 + 4x(–3)3 + (–3)4
Simplify...
x4 – 12x3+ 54x2 – 108x + 81

8. Example: Use the binomial Theorem to expand (2x + 5)5
Row 5 of Pascal’s Triangle looks like this:
which translates to:
x5 + 5x4y + 10x3y x2y3 + 5xy4 + y5
Replace the x’s with 2x and the y’s with +5...
(2x)5 +5(2x)4(5)+10(2x)3(5)2 +10(2x)2(5)3 +5(2x)(5)4 +(5)5
Simplify…
32x5 +5(16x4)(5)+10(8x3)(25) +10(4x2)(125) +5(2x)(625)
32x x x x x