Review of Algebra
, 2 ■ REVIEW OF ALGEBRA
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:
abba ab ba (Commutative Law)
a b c a b c abc abc (Associative Law)
ab c ab ac (Distributive law)
In particular, putting a 1 in the Distributive Law, we get
b c 1b c 1b 1c
and so
b c b c
EXAMPLE 1
(a) 3xy4x 34x 2y 12x 2y
(b) 2t7x 2tx 11 14tx 4t 2x 22t
(c) 4 3x 2 4 3x 6 10 3x
If we use the Distributive Law three times, we get
a bc d a bc a bd 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 bc d
In the case where c a and d b, we have
a b2 a 2 ba ab b 2
or
1 a b2 a 2 2ab b 2
Similarly, we obtain
2 a b2 a 2 2ab b 2
, 2 ■ REVIEW OF ALGEBRA
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:
abba ab ba (Commutative Law)
a b c a b c abc abc (Associative Law)
ab c ab ac (Distributive law)
In particular, putting a 1 in the Distributive Law, we get
b c 1b c 1b 1c
and so
b c b c
EXAMPLE 1
(a) 3xy4x 34x 2y 12x 2y
(b) 2t7x 2tx 11 14tx 4t 2x 22t
(c) 4 3x 2 4 3x 6 10 3x
If we use the Distributive Law three times, we get
a bc d a bc a bd 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 bc d
In the case where c a and d b, we have
a b2 a 2 ba ab b 2
or
1 a b2 a 2 2ab b 2
Similarly, we obtain
2 a b2 a 2 2ab b 2
