Algebra Rules Review Sheet
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23REVIEW OF ALGEBRA
Here we review the basic rules and procedures of algebra that you need to know in order
to be successful in calculus.
ARITHMETIC OPERATIONS
The real numbers have the following properties:
a
b
b
a
ab
ba
(Commutative Law)
a
b
c
a
b
c
ab c
a bc
(Associative Law)
a b
c
ab
ac
(Distributive law)
In particular, putting
a
1
in the Distributive Law, we get
b
c
1 b
c
1 b
1 c
and so
b
c
b
c
EXAMPLE 1
2
2
(a)
3xy
4x
3
4 x
y
12x
y
2
(b)
2t 7x
2tx
11
14tx
4t
x
22t
(c)
4
3 x
2
4
3x
6
10
3x
If we use the Distributive Law three times, we get
a
b c
d
a
b c
a
b d
ac
bc
ad
bd
This says that we multiply two factors by multiplying each term in one factor by each term
in the other factor and adding the products. Schematically, we have
a
b c
d
In the case where
c
a
and
d
b
, we have
2
2
2
a
b
a
ba
ab
b
or
2
2
2
1
a
b
a
2ab
b
Similarly, we obtain
2
2
2
a
b
a
2ab
b
2
EXAMPLE 2
2
2
(a)
2x
1 3x
5
6x
3x
10x
5
6x
7x
5
2
2
(b)
x
6
x
12x
36
2
(c)
3 x
1 4x
3
2 x
6
3 4x
x
3
2x
12
2
2
12x
3x
9
2x
12
12x
5x
21
1
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