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)  4x2  - 4xy  + y2  - z2 = (2x - y)2   - z2 = (2x - y - z)(2x - y + z), d)  a3  - 7a + 6 = a3  - 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)2 =  a2 - 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