Algebraic Expressions
 
      Factoring and expanding algebraic expressions, rules for transforming algebraic expressions
      Using a variety of methods including combinations of the above to factorize algebraic expressions
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
The binomial expansion algorithm - the binomial theorem
 
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)
The sum and/or difference of any two numbers raised to the same (positive integer) power
x4 - y4 = (x - y) (x3 + x2y + xy2 + y3) = (x2 - y2) (x2 + y2
x2n - y2n = (x - y) (x2n-1 + x2n-2y + ¼  + xy2n-2 + y2n-1) = (xn - yn) (xn + yn
Factoring quadratic trinomials
ax2 + bx + c = a[x2 + (b/a)x + c/a] = a(x - x1)(x - x2) where x1 + x2 = b/a and  x1 x2 = c/a
Using a variety of methods including combinations of the above to factorize algebraic 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).
Pre-calculus contents A
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