Algebraic Expressions

Factoring and expanding algebraic expressions, rules for transforming algebraic expressions
Factoring algebraic expressions - methods, the greatest common factor
Factoring by grouping
Perfect square trinomials - the square of a binomial
The difference of two squares
The sum and difference of cubes
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
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 + . . . + xy2n-2 + y2n-1) = (xn - yn) · (xn + yn)
 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
Factoring algebraic expressions
Factoring algebraic expression by finding (determining) the greatest common factor
 Examples: a)  3x - 6y = 3 · (x - 2y),     b)  xy - y2  = y · (x - y),     c)  a - a2  = a · (1 - a), d)  x3 -3x2  + x = x · (x2 - 3x +1),    e)  x(a + b) - (a + b) = (a + b) · (x - 1), f)   a(x - 3y) - x + 3y  = a(x - 3y) - (x - 3y) = (x - 3y) · (a - 1).
Grouping like terms, grouping and factorizing four terms
An addition sign, or plus sign, in front of the brackets leaves the sign of every term inside the brackets unchanged.
A minus sign in front of the bracket indicates that, when removing the bracket, the sign of all terms inside must be changed.
 Examples: a)  ax - bx - a + b = x(a - b) - (a - b) = (a - b) · (x - 1), b)  a - 1 - ab + b = (a - 1) - b · (a - 1) = (a - 1) · (1 - b), c)  x2 + ax - bx - ab = x(x + a) - b · (x + a) = (x + a) · (x - b), d)  5ab2 - 3a3 - 10b3 + 6a2b = 5b2(a - 2b) -3a2(a - 2b) = (a - 2b)(5b2 - 3a2).
Perfect square trinomials - the square of a binomial
 Examples: a)  1 - 4x + 4x2 = 12  - 2 · 2x + (2x)2 = (1 - 2x)2  = (1 - 2x) · (1 - 2x), b)  a5 + 6a4b + 9a3b2 = a3 · (a2  + 6ab  + 9b2 ) = a3(a + 3b)2 = a3(a + 3b)(a + 3b).
The difference of two squares
 Examples: a)  16x2 - 1 = (4x)2 - 12 = (4x -1) · (4x +1), b)  5y3 - 20x2y = 5y · (y2 - 4x2) = 5y [y2  - (2x)2] = 5y(y - 2x)(y + 2x), c)   9x2  - (x + 2)2 = [3x - (x + 2)] · [3x + (x + 2)] = (2x -2) · (4x + 2) = 4(x -1) · (2x +1).