Algebraic Expressions
     
     
      Sum and difference of cubes
         Using a variety of methods including combinations of the above to factorize expressions
      Factoring and expanding algebraic expressions, rules for transforming algebraic expressions
Sum and difference of cubes
Examples:   a)  x3 + 8 = x3 + 23 = (x + 2) (x2 - 2x + 22),
                        since  (x + 2)(x2 - 2x + 4) = x3 - 2x2 + 4x + 2x2 - 4x + 8 = x3 + 8,
b) 8a3 -125 = (2a)3 - 53 = (2a - 5) [(2a)2 + (2a)5 + 52] = (2a - 5)(4a2 + 10a + 25),
  since  (2a - 5)(4a2 + 10a + 25) = 8a3 + 20a2 + 50a - 20a2 - 50a -125 = 8a3 -125.
Using a variety of methods including combinations of the above to factorize expressions
Examples:   a)  x2 - 2xy + y2 + 2y - 2x = (x - y)2 - 2(x - y) = (x - y)(x - y - 2),
b)  x2 - y2 + xz - yz = (x - y)(x + y) + z(x - y) = (x - y)(x + y + z),
c)  4x- 4xy  + y2  - z2 = (2x - y)2   - z2 = (2x - y - z)(2x - y + z),
d)  a- 7a + 6 = a- a - 6a + 6 = a(a2 -1) - 6(a -1) = (a -1)[a(a + 1) - 6] = (a -1)(a2 + a - 6) =
                             = (a -1)(a2 + 3a - 2a - 6) = (a -1)[a(a + 3) - 2(a + 3)] = (a -1)(a + 3)(a - 2).
Factoring and expanding algebraic expressions, rules for transforming algebraic expressions
Expanding algebraic expressions
The square of a binomial, a perfect square trinomial
  (a + b)2 = a2 + 2ab + b2,
  (a - b)2a2 - 2ab + b2,
The square of a trinomial
  (a - b + c)2 = a2 + b2 + c2 - 2ab + 2ac - 2bc,
The cube of a binomial
  (a + b)3 = a3 + 3a2b + 3ab2 + b3,
  (a - b)3 = a3 - 3a2b + 3ab2 - b3,
Factoring algebraic expressions
Difference of two squares
  x2 - y2 = (x - y) (x + y),
Sum and difference of cubes
  x3 - y3 = (x - y) (x2 + xy + y2),
  x3 + y3 = (x + y) (x2 - xy + y2).
Beginning Algebra Contents B
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