Algebra Rules Review Sheet Page 6
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6
REVIEW OF ALGEBRA
THE BINOMIAL THEOREM
Recall the binomial expression from Equation 1:
2
2
2
a
b
a
2ab
b
If we multiply both sides by
a
b
and simplify, we get the binomial expansion
3
3
2
2
3
a
b
a
3a
b
3ab
b
8
Repeating this procedure, we get
4
4
3
2
2
3
4
a
b
a
4a
b
6a
b
4ab
b
In general, we have the following formula.
The Binomial Theorem
If is a positive integer, then
k
9
k k
1
k
k
k 1
k 2
2
a
b
a
ka
b
a
b
1 2
k k
1 k
2
k 3
3
a
b
1 2 3
k k
1
k
n
1
k n
n
a
b
1 2 3
n
k 1
k
kab
b
5
EXAMPLE 13
Expand
x
2
.
SOLUTION
Using the Binomial Theorem with
a
x
,
b
2
,
k
5
, we have
5 4
5 4 3
5
5
4
3
2
2
3
4
5
x
2
x
5x
2
x
2
x
2
5x
2
2
1 2
1 2 3
5
4
3
2
x
10x
40x
80x
80x
32
RADICALS
The most commonly occurring radicals are square roots. The symbol
s1
means “the posi-
tive square root of.” Thus
2
x
sa
means
x
a
and
x
0
2
Since
a
x
0
, the symbol
sa
makes sense only when
a
0
. Here are two rules for
working with square roots:
a
sa
10
sab
sa sb
b
sb
However, there is no similar rule for the square root of a sum. In fact, you should remem-
ber to avoid the following common error:
|
sa
b
sa
sb
(For instance, take
a
9
and
b
16
to see the error.)
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