In order to  understand  factors  you

must  first  understand  the  use  of

brackets. Take for example:

     4 x 2 + 4 x 5 = 8 + 20 = 28



This can be written  as  4(2+5) = 28.

    The brackets ( ) enclose a number

or  numbers each of which are multip-

lied   by  the  number   outside  the

brackets  in  this  case  4  as shown

above.

^z

Note:

It is  not  necessary  to  write  the

multiplication symbol x between 4 and

the brackets eg 4 x (2+5) = 28.

    This  is  understood  by  context

just as you  do not  write + in front

of a positive number.



    When you  are  dealing  with  the

same   quantities inside the  bracket

you add ~subtract ~multiply or divide

^z

the  quantities  inside  the brackets

FIRST !  and  then  the  result   is

multiplied by the number outside  the

brackets~ie always solve the brackets

first.



      Eg: 4(9 x 3 + 4)  =

          4( 27 + 4)    =

          4(   31  )    = 124



      Eg: 2(15 v 3 + 6) =

          2(   5   + 6) =

          2(     11   ) = 22

^z

^q

5(27+12-3)

3(7-1-13)

4(7-3+8)

4(12)

^t

Rewrite 5 x 27 + 5 x 12 -5 x 3 using

bracket notation

blank



^t

Rewrite 3 x 7 - 3 x 1 -3 x 13  using

bracket notation

blank



^t

Rewrite 4 x 7 - 3 x 4 + 4 x 8  using

bracket notation

blank



^t

which can be shortened to give

blank

^z

^q

12(7+5-5)

84

2(7-2-8)

-6

14(7-4+6)

126

^t

Rewrite 12 x 7 + 5 x 12 -5 x 12 using

bracket notation  blank

^t

which equals blank



^t

Rewrite  2 x 7 - 2 x 2 -2 x 8  using

bracket notation  blank

^t

which equals blank



^t

Rewrite 14 x 7 - 14 x 4 + 14 x 2 x 3

using bracket notation blank

^t

which equals blank

^z

^q

4(7+6+4)

7(y+10+2)

5a(9)

3(9b+8)

^t

4 x 7 + 4 x 6 + 4 x 4 can be

written as blank



^t

7y +7 x 10 + 14 can be written as

blank



^t

Notice you can split 14 up into 7x2

for the purposes of factorisation.



5a + 25a + 15a   = blank



^t

27b + 3 + 7 x 3  = blank



^z

^q

416

63

9

-108

100

10

93

^t

4(9 x 12  +  3  -  1 x 7) =   blank



^t

3(12 v 4   +   2 x 9 )    =   blank



^t

3(3)                      =   blank



^t

6(4 + 5   x   12 - 14)    =   blank



^t

5(5 x 5   -   15 v 3)     =   blank



^t

5 x 2(4  +  6 x 2 - 15)   =   blank



^t

3(7 x 7  -  48 v 3  -  2) =   blank

^z

^q

4

20

56

9

115

3

-8

^t

2(28 v 2  -  3 x 2  -  6) =   blank



^t

5(24 v 3  -  7  +  6 v 2) =   blank



^t

7(3  +  12  -  56 v 8)    =   blank



^t

3(4  +  5  -  12 x 2)     =   blank



^t

5(4 x 3  +  21 v 3  +  4) =   blank



^t

+(15  -  6  -  9 v 3)     =   blank



^t

,(4+  -  5+ x 4  -  14+)  =   blank

^z

Brackets may be nested eg:



       4(3(3 + 6)) =

       4(3(  9  )) =

       4(   27   ) = 108



In this case you work on the  numbers

in  the innermost  brackets first and

then work your way outwards.

      eg: (3 + 6) = 9

         3(  9  ) = 27

         4( 27 )  = 108



^z

^q

innermost

5

3

7(5x3)

105

^t

Let's try an example of the use of

nested brackets.

What does  7( (15 v 3)(7 - 4) ) =



We start from the blank

brackets and work our way outwards.



^t

(15 v 3) = blank     and

^t

(7 - 4)  = blank



^t

Thus the expression can be written

in a simpler form as blank



^t

Therefore 7( (15 v 3)(7 - 4)) = blank



^z

^q

4-x

160-40x

^t

Here's another one

      4( (2 + 8)(4 - x) ) =



Again working our way outwards we try

and simplify the expression.

         (2 + 8) = 10

       10(blank ) = 40 -10x



^t

     4( 40- 10x ) = blank



Therefore 4( (2 + 8)(4 - x) ) =

                    160 - 40x

^z

^q

(2x3)

8

6(8(4-x))

6(32-8x)

192-48x

^t

Now try these :

      6( (48 v (2 x 3) )(4 - x) ) =



The innermost expression is blank



^t

48 v 6 = blank



^t

Therefore the expression can be

rewritten as blank



^t

Which simplifies to give blank



^t

Therefore 6( (48 v (2 x 3) )(4 - x) )

          = blank

^z

^q

(4+8)v(2x3)

12v6

6((2)(4-x)

6(8-2x)

48-12x

^t

 6( ( (4 + 8) v (2 x 3) )(4 - x) ) =



The innermost expressions are

blank



^t

These simplify to give blank



^t

Therefore the expression can be

rewritten as blank



^t

Which simplifies to give blank



^t

Therefore

6( ( (4 + 8) v (2 x 3) )(4 - x) )

         = blank

^z

^q

(9v3) & (4x+3)

15x+9

2x(15x+9)

18x

^t

 2x( x( 9 v 3 ) + 3( 4x + 3 ) )



The innermost expressions are

blank



^t

Therefore the expression inside the

first set of brackets is blank



^t

Therefore the expression can be

rewritten as blank



^t

Therefore

2x( x( 9 v 3 ) + 3( 4x + 3 ) )

         = 30x} + blank

^z



Notice  that  every time  you  open a

bracket you must close it.



    Eg:

       4(3x(3 +6)) =

        ^  ^    ^^

        |  |    ||-------------\

        |  |    \-------?      |

        |  \ second set Y      |

        |                      |

        \--     first set    --/

^z

A  factor  of a  given  number  is  a

number which divides evenly into  the

given number.



  4  is a factor of  12  because it

  divides evenly into 12.



  8 is a factor of 64 because it

  divides evenly into 64 to give

  you 8



^z

A factor  of an algebraic expression

on  the other  hand need  not divide

evenly  into it  but  by multiplying

the factors  of an  expression toge-

ther  we can obtain the  expression.



       3(4 + 2x) = 12 + 6x

      factors -------> expression

              <-------



How can we derive  the  factors of an

expression ?

^z

    If the  entire  expression can be

divided   by  a  number   or  simpler

expression  this number or expression

is called a common factor.

For example:

            12 + 6x is an  expression

which can be divided by  the number 3

(12 + 6x)  v  3  =  (4 + 2x).

    It  can  also  be  divided by the

simpler expression  (4 + 2x)

(12 + 6x)  v  (4 + 2x) = 3.

    Therefore (4 + 2x)  and 3 are the

factors  of  the  expression 12 + 6x.

^z

Naturally this  may not seem  obvious

or apparent  to you  at first ~ so we

will establish a systematic technique

which will allow us  to find  factors

for even the most complex of express-

ions.

    We will use a  systematic step by

step approach called an  algorithm to

solve  for  the  factors of an expre-

ssion.

^z

          FACTOR ALGORITHM

  ---------------------------------

  |1 Look at smallest coefficient |

  |  of the expression if there   |

  |  is a common factor then it   |

  |  cannot be greater than this  |

  |  number.                      |

  ---------------------------------

                 |

                 V

^z

                 |

  ---------------------------------

  |2 Does  this  number  divide   |

  |  evenly  into  every  other   |

  |  coefficient of the expression|

  |  ie is it a factor ?          |

  ---------------------------------

         |       ?         |

^z

         |                 |

        No                Yes

------------------ -----------------

|3 Can you think | |3               |

|of a number that| |                |

|will divide     | |   This is the  |

|evenly into this| |  common factor |

|number ie a     | |                |

|factor?         | |                |

------------------ ------------------

      |  ?  |

      |     ----------------

      |-                   |

^z

       |                   |

       No                 Yes

------------------ ------------------

|4 The expression| |4 Is  this  the |

|does not have a | |largest  number |

|common factor we| |that will divide|

|will see how to | |into  all  the  |

|factorise it    | |coefficients  of|

|later.          | |the expression ?|

------------------ ------------------

                       |  ?  |

        ----------------     |

        |                  ---

^z



        |                  |

       No                 Yes



------------------ ------------------

|5               | |5               |

|The largest num-| |                |

|ber  that  will | | This number is |

|divide in is the| |   the common   |

|common factor   | |     factor     |

|                | |                |

------------------ ------------------

^z

^q

divides

Multiplying

evenly

^t

A factor is a number or expression

which blank   evenly into  another

number or expression.



^t

blank       the factors of an

expression together gives you the

expression.



^t

A common factor of an expression is

one which divides blank  into the

expression.



^z

^q

2

common

Dividing

x-4

factors

^t

Lets use the factor algorithm to

find the factors of the following

expressions :    2x -8



The smallest coefficient that

divides evenly into the expression

is blank.



^t

Therefore 2 is the blank  factor.



^t

blank    the expression by the

common factor gives us

^t

blank .

^t

Therefore the blank   of the

expression are 2 and x - 4

^z

^q

3

3x

3x(x+3y-5z)

3x

(x+3y-5z)

^t

Find the factors of 3x} + 9xy - 15xz



The smallest coefficient of the

expression is blank.



^t

However not only will 3 divide

evenly into the expression but so

will blank



^t

Therefore 3x is a common factor.

So we can  rewrite the expression

as blank



^t

Thus the factors of 3x} + 9xy -15xz

are blank

^t

and blank

^z

^q

a

+ c

j

(-1-3p)

5a

(1-3b+4c)

^t

What are the factors of ab} + ac



              blank and

^t

              b} blank



^t

jp - j - 4jp

              blank and

^t

              blank



^t

5a -15ab +20ac

              blank and

^t

              blank

^z

^q

r

-3s+5

ab

(5a+10b+1)

(x+y)

(5+z)

(5a-4b)

(4x+3y)

^t

rs} + 5r -3rs

              blank and

^t

              (s} blank  )



^t

5a}b + 10ab} + ab

              blank and

^t

              blank



^t

5(x + y) + z(x + y)

              blank   and

^t

              blank



^t

4x(5a -4b) + 3y(5a -4b)

              blank     and

^t

              blank

^z

Sometimes an expression does not have

a common factor.In this case we split

the  expression up  into two  or more

parts and factorise them individually.

For example :

             xy + xz + ay + az

We split this up into

   xy + xz             ay + az



has the common     has the common

factor x           factor a

  x(y + z)             a(y + z)

^z

x(y + z) + a(y + z) = xy + xz + ay +az



(y + z) is now a common factor of the

expression written in this form.



(x + a)(y + z) = xy + xz + ay + az



Thus   (x + a)   and   (y + z)  are

the factors of the expression.

^z

Find the factors of

                    ab - ac - db + dc

First split  it  into  two parts  and

extract  the  common factor from both

parts.

       a(b - c)  and  d(-b + c)

Since   (b - c) = -(-b + c)

These factors can be written as :

       a(b - c)  and -d(b - c)

Therefore  (a - d) & (b - c)  are the

factors of the expression.

^z

^q

ab+ac

db+dc

a(b+c)

d(b+c)

(a+d)

(b+c)

^t

Let's try a few examples of these

types of problems



            ab + ac + db + dc



Splitting it into two parts we get

blank     and

^t

blank



^t

Factorising these parts individually

we get blank     and

^t

blank



^t

Therefore the factors are blank

^t

and blank

^z

^q

4a+4

ab+a

4(a+1)

b(a+1)

(4+b)

(a+1)

^t



            4a + 4 + ab + b



Splitting it into two parts we get

blank     and

^t

blank



^t

Factorising these parts individually

we get blank     and

^t

blank



^t

Therefore the factors are blank

^t

and blank

^z

^q

5+15y

-x-3xy

5(1+3y)

-x(1+3y)

(5-x)

(1+3y)

^t



            5 - x + 15y - 3xy



Splitting it into two parts we get

blank     and

^t

blank



^t

Factorising these parts individually

we get blank     and

^t

blank



^t

Therefore the factors are blank

^t

and blank

^z

^q

-3dc+3bd

ab-ac

3d(-c+b)

a(b-c)

(3d+a)

(b-c)

^t



          ab - 3dc - ac + 3bd



Splitting it into two parts we get

blank      and

^t

blank



^t

Factorising these parts individually

we get blank      and

^t

blank



^t

Therefore the factors are blank

^t

and blank

^z

Next we come to a very important  use

for factors:

 -  Solving Quadratic Equations  -



An expression of the form

   ax} + bx + c

{ Where a is not equal to 0 }

is called a quadratic expression.



    Quadratic  expressions   may   be

written in  many  different  forms  to

allow you to factorise them.

^z

There are  three  steps to  solving a

quadratic expression.



    Step 1: Multiply  the   constant

    { c } by the coefficient  of the

    x} term and call this number the

    guide number.



    Step 2: Find the factors  of the

    guide  number  which  when added

    together give the coefficient of

    the x term.

^z

    Step 3: Rewrite  the expression

    using   these  factors  as  the

    coefficients of x.



    Now   factorise  the   expression

    using using the methods  you have

    already seen.

^z

As this  section will be dealt  with

in  another  program  we  will  just

illustrate one example here :

     x} + x - 8 = 4



The constant here  is not -8 we must

first  bring  4  over  to  the other

side  of the equation  to  determine

the constant :

     x} + x - 12 = 0

     The  constant = -12  the  guide

number = -12.

     The factors of the guide number

which  when  added together give the

^z

coefficient of  the x term are 4 and

-3.

     There is no easy way of getting

these its just a matter of trial and

error and experience.

     Lets just examine  one  way  of

doing it.

     -12 has the factors

      4 ~ -3         3  ~ -4

      2 ~ -6        -6  ~  2

      1 ~ -12       -12 ~  1

Now which pair of factors when added

together give the coefficient of the

x term  1 ?

     Thats right  4 and -3.

^z

Rewriting the equation



         x} + 4x - 3x -12 = 0

      x(x + 4)  -3(x + 4) = 0



Therefore the factors are :



           (x + 4)(x - 3) = 0




