ALGEBRAIC FACTORING - Daily Math Guide

## 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)= a+ 2(3a) +32

= a2 + 6a + 9

2.       (a - b)2 = a- 2ab + b2

example:

(x-3)= 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) =

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