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
         Factoring quadratic trinomials
         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)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
Factoring algebraic expressions
Factoring algebraic expression by finding (determining) the greatest common factor
Examples:   a)  3x - 6y = 3 (x - 2y),     b)  xy - y = y (x - y),     c)  a - a = a (1 - a),
d)  x3 -3x+ x = x (x2 - 3x +1),    e)  x(a + b) - (a + b) = (a + b) (x - 1),
f)   a(x - 3y) - x + 3 = 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 = 1- 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 [y - (2x)2] = 5y(y - 2x)(y + 2x),
          c)   9x- (x + 2)2 = [3x - (x + 2)] [3x + (x + 2)] = (2x -2) (4x + 2) = 4(x -1) (2x +1).
Factoring quadratic trinomials
A quadratic trinomial  ax2 + bx + can be factorized as
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
That means, to factorize a quadratic trinomial we should find such a pair of numbers x1 and x2 whose sum equals b/a and whose product equals c/a.
Therefore, when the constant term c is negative, then the signs of x1 and x2 will be different but when c is positive, their signs will be the same.
Examples:   a)  x2 - 3x -10 = x2 + (-5 + 2)x + (-5)(+2) = x2 - 5x + 2x -10 =
                         = x (x - 5) + 2 (x - 5) = (x - 5) (x + 2),
b)  2x2 - 7x + 3 = 2 (x2 - 7/2x + 3/2) =  2(x2 - 1/2x - 3x + 3/2) =
                        = 2[x(x - 1/2) - 3 (x - 1/2)] = 2 (x - 1/2)(x - 3) = (2x - 1) (x - 3),
c)  3x2 - x - 2 = 3(x2 - 1/3x - 2/3) =  3(x2 + 2/3x - x - 2/3) =
                        = 3[x(x + 2/3) - (x + 2/3)] = 3(x + 2/3)(x - 1) = (3x + 2)(x - 1).
The 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.
Intermediate algebra contents
Copyright 2004 - 2020, Nabla Ltd.  All rights reserved.