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}. |