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6.1 Algebraic Expressions III

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6.1 Algebraic Expressions III

6.1.1 Expansion
1. The product of an algebraic term and an algebraic expression:
  • a(b + c) = ab + ac
  •  a(b – c) = ab – ac
2. The product of an algebraic expression and another algebraic expression:
  • (a + b) (c + d)  = ac + ad + bc + bd
  • (a + b)2= a2 + 2ab + b2
  • (a – b)2= a2 – 2ab + b2
  • (a + b) (a – b) = a2 – b2

6.1.2 Factorization
1. Factorize algebraic expressions:
  •  ab + ac = a(b + c)
  • a2– b2 = (a + b) (a – b)
  • a2+ 2ab + b2 = (a + b)2
  • ac + ad + bc + bd = (a + b) (c + d)  
2. Algebraic fractions are fractions where both the numerator and the denominator or either the numerator or the denominator are algebraic terms or algebraic expressions.
Example:
3 b , a 7 , a + b a , b a − b , a − b c + d


3(a) Simplification of algebraic fractions by using common factors:
• 1 4 b c 3 12 b d = c 3 d • b m + b n e m + e n = b ( m + n ) e ( m + n ) = b e

3(b) Simplification of algebraic fractions by using difference of two squares:
a 2 − b 2 a n + b n = ( a + b ) ( a − b ) n ( a + b ) = a − b n
 

6.1.3 Addition and Subtraction of Algebraic Fractions
1. If they have a common denominator:
a m + b m = a + b m

2. If they do not have a common denominator:
a m + b n = a n + b m n m


6.1.4 Multiplication and Division of Algebraic Fractions
1. Without simplification:
• a m × b n = a b m n • a m ÷ b n = a m × n b = a n b m  

2. With simplification:
• a c m × b m d = a b c d • a c m ÷ b d m = a c m × d m b = a d b c