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Unformatted text preview: 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: (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 s a 2 b d 2 a 2 2 2 ab 1 b 2 2 s a 1 b d 2 a 2 1 2 ab 1 b 2 1 s a 1 b d 2 a 2 1 ba 1 ab 1 b 2 d b c a s a 1 b ds c 1 d d s a 1 b ds c 1 d d s a 1 b d c 1 s a 1 b d d ac 1 bc 1 ad 1 bd 4 2 3 s x 2 2 d 4 2 3 x 1 6 10 2 3 x 2 t s 7 x 1 2 tx 2 11 d 14 tx 1 4 t 2 x 2 22 t s 3 xy ds 2 4 x d 3 s 2 4 d x 2 y 2 12 x 2 y 2 s b 1 c d 2 b 2 c 2 s b 1 c d s 2 1 ds b 1 c d s 2 1 d b 1 s 2 1 d c a 2 1 a s b 1 c d ab 1 ac s ab d c a s bc d s a 1 b d 1 c a 1 s b 1 c d ab ba a 1 b b 1 a 2 ■ REVIEW OF ALGEBRA EXAMPLE 2 (a) (b) (c) 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: a b c d a b 3 d c ad bc 2 a b 2 a b a 2 b a b ? c d ac bd a b 1 c d ad 1 bc bd a b c 1 a b 1 c a b 1 a c a 1 c b a b 1 c b a b 1 c b 1 b 3 a 1 1 b 3 c 1 b s a 1 c d a 1 c b 12 x 2 2 5 x 2 21 12 x 2 2 3 x 2 9 2 2 x 2 12 3 s x 2 1 ds 4 x 1 3 d 2 2 s x 1 6 d 3 s 4 x 2 2 x 2 3 d 2 2 x 2 12 s x 1 6 d 2 x 2 1 12 x 1 36 s 2 x 1 1 ds 3 x 2 5 d 6 x 2 1 3 x 2 10 x 2 5 6 x 2 2 7 x 2 5 REVIEW OF ALGEBRA ◆ 3 EXAMPLE 3 (a) (b) (c) (d) Factoring We have used the Distributive Law to expand certain algebraic expressions. We some times need to reverse this process (again using the Distributive Law) by factoring an expression as a product of simpler ones. The easiest situation occurs when the expres sion has a common factor as follows: To factor a quadratic of the form we note that so we need to choose numbers so that and . EXAMPLE 4 Factor . SOLUTION The two integers that add to give and multiply to give are and . Therefore EXAMPLE 5 Factor . SOLUTION Even though the coefficient of is not , we can still look for factors of the form and , where . Experimentation reveals that Some special quadratics can be factored by using Equations 1 or 2 (from right to left) or by using the formula for a difference of squares: a 2 2 b 2 s a 2 b ds a 1 b d 3 2 x 2 2 7 x 2 4 s 2 x 1 1 ds x 2 4 d rs 2 4 x 1 s 2 x 1 r 1 x 2 2 x 2 2 7 x 2 4 x 2 1 5 x 2 24 s x 2...
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 Winter '08
 JaberAbdualrahman
 Calculus, Algebra, Quadratic equation, Elementary algebra, Mathematics in medieval Islam, Sx

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