Binomial Expansion
The Binomial Theorem
The Binomial Theorem
Here is the expansion of
(x + y)n
for
n = 0, 1,…, 5
:
(x + y)0 = 1
(x + y)1 = x + y
(x + y)2 = x
2 +2xy + y
2
(x + y)3 = x
3 +3x
2
y + 3xy
2 + y
3
(x + y)4 = x
4 +4x
3
y + 6x
2
y
2 +4xy
3 + y
4
(x + y)5 = x
5 +5x
4
y + 10x
3
y
2 +10x
2
y
3 +5xy
4 + y
5
Look familiar? The coefficients of each expansion are the entries in Row
n
of
Pascal's Triangle. Thus, the coefficient of each term
r
of the expansion of
(x + y)n
is given by
C(n, r - 1)
. The exponents of
x
descend, starting
with
n
, and the exponents of
y
ascend, starting with 0, so the
r
th
term
of the expansion of
(x + y)2
contains
x
n-(r-1)
y
r-1
.
This information can be summarized by the Binomial Theorem:
For any positive integer n , the expansion of (x + y)n is C(n, 0)x n + C(n, 1)x n-1 y + C(n, 2)x n-2 y 2 + ... + C(n, n - 1)xy n-1 + C(n, n)y n .
Each term r in the expansion of (x + y)n is given by C(n, r - 1)x n-(r-1) y r-1 .
Example: Write out the expansion of
(x + y)7
.
(x + y)7 = x
7 +7x
6
y + 21x
5
y
2 +35x
4
y
3 +35x
3
y
4 +21x
2
y
5 +7xy
6 + y
7
.
When the terms of the binomial have coefficient(s), be sure to apply the exponents to these coefficients.
Example: Write out the expansion of
(2x + 3y)4
.
| (2x + 3y)4 | = | (2x)4 +4(2x)3(3y) + 6(2x)2(3y)2 +4(2x)(3y)3 + (3y)4 | |
| = | 16x 4 +4(8x 3)(3y) + 6(4x 2)(9y 2) + 4(2x)(27y 3) + 81y 4 | ||
| = | 16x 4 +96x 3 y + 216x 2 y 2 +216xy 3 +81y 4. |
Example: Write out the expansion of
(5x - y)3
.
| (5x - y)3 | = | (5x)3 +3(5x)2(- y) + 3(5x)(- y)2 + (- y)3 | |
| = | 125x 3 +3(25x 2)(- y) + 3(5x)(y 2) + (- y 3) | ||
| = | 125x 3 -75x 2 y + 15xy 2 - y 3. |
