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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
共a ⫹ b兲 ⫹ c 苷 a ⫹ 共b ⫹ c兲
a共b ⫹ c兲 苷 ab ⫹ ac

共ab兲c 苷 a共bc兲

(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
共a ⫹ b兲共c ⫹ d兲 苷 共a ⫹ b兲c ⫹ 共a ⫹ b兲d 苷 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
共a ⫹ b兲共c ⫹ d兲

Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.

In the case where c 苷 a and d 苷 b, we have
共a ⫹ b兲2 苷 a 2 ⫹ ba ⫹ ab ⫹ b 2
or
1

共a ⫹ b兲2 苷 a 2 ⫹ 2ab ⫹ b 2

Similarly, we obtain
2

共a ⫺ b兲2 苷 a 2 ⫺ 2ab ⫹ b 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
苷 12x 2 ⫺ 3x ⫺ 9 ⫺ 2x ⫺ 12 苷 12x 2 ⫺ 5x ⫺ 21

1