REVIEW 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:
(Commutative Law)
(Associative Law)
(Distributive law)
In particular, putting
in the Distributive Law, we get
and so
EXAMPLE 1
(a)
(b)
(c)
If we use the Distributive Law three times, we get
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
In the case where
and
, we have
or
Similarly, we obtain
EXAMPLE 2
(a)
(b)
(c)
12x 2 5x 21
12x2 3x 9 2x 12
3x 14x 3 2x 6 34x 2 x 3 2x 12
x 62 x 2 12x 36
2x 13x 5 6x 2 3x 10x 5 6x 2 7x 5
a b2 a2 2ab b2
2
a b2 a2 2ab b2
1
a b2 a2 ba ab b2
d b
c a
a bc d
a bc d a bc a bd ac bc ad bd
4 3x 2 4 3x 6 10 3x
2t7x 2tx 11 14tx 4t 2x 22t
3xy4x 34x 2y 12x 2y
b c b c
b c 1b c 1b 1c
a 1
ab c ab ac
abc abc
a b c a b c
ab ba
a b b a
1
Thomson Brooks-Cole copyright 2007
FRACTIONS
To add two fractions with the same denominator, we use the Distributive Law:
Thus, it is true that
But remember to avoid the following common error:
|
(For instance, take
to see the error.)
To add two fractions with different denominators, we use a common denominator:
We multiply such fractions as follows:
In particular, it is true that
To divide two fractions, we invert and multiply:
EXAMPLE 3
(a)
(b)
(c)
s2t
u
ut
2
s2t 2u
2u
s2t 2
2
x 2 2x 6
x 2 x 2
3
x 1
x
x 2
3x 2 xx 1
x 1x 2
3x 6 x 2 x
x 2 x 2
x 3
x
x
x
3
x
1
3
x
a
b
c
d
a
b
d
c
ad
bc
a
b
a
b
a
b
a
b
c
d
ac
bd
a
b
c
d
ad bc
bd
a b c 1
a
b c
a
b
a
c
a c
b
a
b
c
b
a
b
c
b
1
b
a
1
b