ALGEBRAIC FACTORING

ALGEBRAIC FACTORING

Factoring is just like dismantling or separating the common part(s) of the expression from the expression itself, or from a series of expressions.  

Illustration:

The "a" is the common factor of ax and ay

The common factor of two or more terms is a numerical coefficient or variable(s) that appear both in each term.

Below were the illustrations of the monomial factor.

    1.       (ax + ay) = a (x + y)

    2.       (3ab – 6ax) = (3ab-2.3ab) = 3a (b -2x )

    3.       (2x²y - 4xy+6y²) = (2.x.x.y - 2.2.x.y + 2.3.y.y) = 2y(x² -2x + y)

FACTORING THE DIFFERENCE OF TWO SQUARES:

a² – b² = (a - b)(a + b) or (a + b)(a - b)

example:

x²-4=x²-2²=(x+2(x-2)

FACTORING THE TRINOMIAL PERFECT SQUARE:

1.       (a + b)² = a² + 2ab + b² 

example:

(a+3)² = a² + 2(3a) +3²

      = a² + 6a + 9

2.       (a - b)² = a² - 2ab + b² 

example:

(x-3)² = x² -2(3x)+ 3²

      = x² - 6x + 9

FACTORING THE TRINOMIAL OF OTHER FORMS:

1.      a² + a (x+ y) + x y  = ( a + x )(a + y)  

 

example:

b²+5b+6 = b²+ b(3+2)+2(3)=(b+2)(b+3)

2.     acx² + (ad + b c )x + b d = ( ax + b )( cx + d )  

example:

6x² + 19x + 15 = 2x(3x) + 2x(5)+ 3(3x) + 3(5)

               = 6x² + 10x + 9x + 15

               =(2x+3)(3x+5)

FACTORING THE SUM and DIFFERENCE OF TWO CUBES:

1.       a³ + b³ = ( a + b )( a³ –a b + b³ )

example:

x³ + 2³ = (x + 2)(x³ - 2x + 2³)

        = (x + 2)(x³ - 2x + 8)

2.       a³ – b³ = ( a – b )( a³ + a b + b³ )

example:

x³ - 3³ = (x - 3)(x³ + 3x + 3³ )

        

        = (x - 3)(x³ - 3x + 27)

EXERCISES:

Factor the following:

     1.(2x – 2y) =

     2.(2x + 4y) =

     3.(3x – 6y + 9z) =

     4.( 2a + 3b – c)² = 

     5. -2( 4a – 8b) =

     6.(2x² y - 4xy+6y²) =

     7.(4ax² y – 2axy+12ay²) =

     8.(a² – 4) =

     9.(a² – 4)² =

    10.(9x² – 4y²) =

    11.(a+2bc) ² =

    12.(a +b - c)² =

    13.( m³ + n² )² =

    14.( x³ – 8 ) =

    15.( 27m³ + 8n³) =