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## 6e reviewofalgebra.pdf

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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:
a⫹b苷b⫹a
ab 苷 ba

a共b ⫹ c兲 苷 ab ⫹ ac

(Commutative Law)
(Associative Law)
(Distributive law)

In particular, putting a 苷 ⫺1 in the Distributive Law, we get
⫺共b ⫹ c兲 苷 共⫺1兲共b ⫹ c兲 苷 共⫺1兲b ⫹ 共⫺1兲c
and so
⫺共b ⫹ c兲 苷 ⫺b ⫺ c
EXAMPLE 1

(a) 共3xy兲共⫺4x兲 苷 3共⫺4兲x 2y 苷 ⫺12x 2y
(b) 2t共7x ⫹ 2tx ⫺ 11兲 苷 14tx ⫹ 4t 2x ⫺ 22t
(c) 4 ⫺ 3共x ⫺ 2兲 苷 4 ⫺ 3x ⫹ 6 苷 10 ⫺ 3x
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 c 苷 a and d 苷 b, we have

or
1

Similarly, we obtain
2

EXAMPLE 2

(a) 共2x ⫹ 1兲共3x ⫺ 5兲 苷 6x 2 ⫹ 3x ⫺ 10x ⫺ 5 苷 6x 2 ⫺ 7x ⫺ 5
(b) 共x ⫹ 6兲2 苷 x 2 ⫹ 12x ⫹ 36
(c) 3共x ⫺ 1兲共4x ⫹ 3兲 ⫺ 2共x ⫹ 6兲 苷 3共4x 2 ⫺ x ⫺ 3兲 ⫺ 2x ⫺ 12

1