6.1 Algebraic Expressions III


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(bc) = 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
  • (ab)2= a2 – 2ab + b2
  • (a + b) (ab) = a2b2


6.1.2 Factorization
1. Factorize algebraic expressions:
  •  ab + ac = a(b + c)
  • a2b2 = (a + b) (ab)
  • 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

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