Case 3: when the sum is positive and product is negative

 

In this case, we look for the two pairs that we can subtract to get the sum. The smaller of the two factors will take a negative sign.

 

Example

Factorize the following

     1.   y2 + y - 20

     2.   6c2 + 13c – 8

     3.   15 + 4m – 3m2

 

Solution

      1.   y2 + y – 20

            a = 1 b = 1 c = -20

            product = - 20

            sum = 1

20 = 1, 2, 4, 5, 10, 20      

(1 x 20)        (2 x 10)        (4 x 5)

 

4 and 5 will subtract to give 1. 4 will take the negative sign

 

factors are – 4 and 5.

 

y2 + y – 20

y2 – 4y + 5y – 20

(y2 – 4y) + (5y – 20)

y(y – 4) + 5(y – 4)

(y – 4)( y + 5)

 

      2.   6c2 + 13c – 8

            a = 6 b = 13           c = - 8

            product = - 48

            sum = 13

48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

(1 x 48)        (2 x 24)        (3 x 16)        (4 x 12)        (6 x 8)

 

Factors are – 3 and 16

6c2 + 13c – 8

6c2 – 3c + 16c – 8

(6c2 – 3c) + (16c – 8)

3c(2c – 1) + 8(2c – 1)

(2c – 1)(3c + 8)

 

      3.  15 + 4m – 3m2

            Product = - 45

            Sum = 4

45 = 1, 3, 5, 9, 15, 45

(1 x 45)        (3 x 15)        (5 x 9)

 

Factors are – 5 and 9  

 

15 + 4m – 3m2

15 – 5m + 9m – 3m2

(15 – 5m) + (9m – 3m2)

5(3 – m) + 3m(3 – m)

(3 – m)(5 + 3m)

 

            Exercise C

Factorize the following

      1.   y2 + 2y – 35

      2.   5p2 + 9p – 18

      3.   3k2 + 14k – 24

      4.   55 + 6t – t2

      5.   6 + 17p – 3p2

      6.   6d2 + 5d – 25

      7.   2a2 + 15a – 19

      8.   m2 + 2m – 8

 

Case 4: when both the sum and the product are negative.

This is just like case 3, only that the negative sign goes to the bigger factor.

 

            Example

Factorize the following

      1.   k2 – k – 6

      2.   5α2 – 8α – 21

      3.   12 – 5x – 3x2

 

Solution

      1.   k2 – k – 6

 

Product = - 6

Sum = -1

6 = 1, 2, 3, 6

(1 x 6)           (2 x 3)

 

2 and 3 will subtract to give 1. And 3 (the larger factor) will take the negative sign

 

Factors are 2 and – 3

k2 – k – 6

k2 +2k – 3k – 6

(k2 +2k) – (3k + 6)

k(k + 2) – 3(k + 2)

(k + 2)(k – 3)

 

      2.  2 – 8α – 21

Solution

Product = -105

Sum = -8

105 = 1, 2, 3, 5, 7, 15, 21, 35, 75, 105

(1 x 105)     (2 x 75)        (3 x 35)        (5 x 21)        (7 x 15)

Factors are 7 and – 15

2 – 8α – 21

2 + 7α – 15α – 21

(5α2 + 7α) – (15α + 21)

α(5α + 7) – 3(5α + 7)

(5α + 7)( α – 3)

 

      3.   12 – 5x – 3x2

 

Solution

Product = - 36

Sum = - 5

36 = 1, 2, 3, 4, 6, 9, 12, 18, 36

(1 x 36)        (2 x 18)        (3 x 12)        (4 x 9)          

 

Factors are 4 and – 9

12 – 5x – 3x2

12 + 4x – 9x – 3x2

(12 + 4x) – (9x + 3x2)

4(3 + x) – 3x(3 + x)

(3 + x)(4 – 3x)

           

 

            Exercise D

Factorize the following

      1.   2m2 – 15m + 7

      2.   5y2 – 11y – 12

      3.   p2 – p – 6

      4.   20 + x - x2

      5.   3y2 – 4y – 15

      6.   k2 – 3k – 18

      7.   3t2 – 7t – 12

      8.   2d2 – d – 6

 

 

Case 5: when there are two unknowns in the expression as in the examples below.

 

Factorize the following

      1.   3a2 – 11ab + 6b2

      2.   8c2 – 7cd – d2

 

    Solution

 

      1.   3a2 – 11ab + 6b2

            a = 3              b = - 11       c = 6

            product = 18

            sum = - 11

18 = 1, 2, 3, 6, 9, 18

(1 x 18)        (2 x 9)           (3 x 6)

            Factors are – 2 and – 9

3a2 – 2ab – 9ab + 6b2

(3a2 – 2ab) – (9ab – 6b2)

a(3a – 2b) – 3b(3a – 2b)

(3a – 2b)(a – 3b)

 

The process involved is still the same as the ones we have done so far.

 

 

      2.   8c2 – 7cd – d2

a = - 1                       b = -7            c = 8

product = - 8

sum = - 7

8 = 1, 2, 4, 8

(1 x 8)           (2 x 4)

Factors are 1 and – 8

8c2 – 7cd – d2

8c2 + cd – 8cd – d2
(8c2 + cd) – (8cd + d2)
c(8c + d) – d(8c + d)

(8c + d)(c – d)

 

            Exercise E

Factorize the following

      1.   x2 – 2xy + y2

      2.   5p2 – 23pq + 12q2

      3.   2m2 – 11mn + 15n2

      4.   j2 – jk – 6k

           

Case 6: When there is a common factor to all the terms in the expression.

            We first factorize the common factor before proceeding to factorize.

 

Factorize the following:

1.        6x2 – 9x – 27

2.        60 + 20v – 5v2

 

            Solution

1.       6x2 – 9x – 27

          3 is a common factor

            3(2x2 – 3x - 9)

                        Factorize 2x2 – 3x – 9

                                    a = 2              b = -3            c = -9

                                    product = - 18

                                    sum = -3

                                    18 = 1, 2, 3, 6, 9, 18

                                    (1 x18)         (2x9)            (3 x 6)

                                    Factors are 3 and – 6

                        2x2 – 3x - 9

                        2x2 + 3x – 6x – 9

                        (2x2 + 3x) – (6x + 9)

                        x(2x + 3) – 3(2x +3)

(2x +3)(x – 3)

Now place the 3 in front of the factorized expression

            3(2x +3)(x – 3)

 

2.      60 + 20v – 5v2 

          5 is a common factor

            5 (12 + 4v – v2)

            Factorize 12 + 4v – v2

                        a = -1            b = 4             c = 12

                        product = - 12

                        sum = 4

                        12 = 1, 2, 3, 4, 6, 12

                        (1 x 12)        (2 x 6)           (3 x 4)

                        Factors are – 2 and 6

                        12 + 4v – v2

                        12 – 2v + 6v – v2 

                        (12 – 2v) + (6v – v2)

                        2(6 – v) + v (6 – v)

                         (6 – v)(2 +v)

            5(6 – v)(2 +v)

 

                   Exercise F

Factorize

1.        9t2 – 15t – 6

2.        10a – 8a2 +12

3.        45 + 12d – 9d2

4.        8x2 – 4x – 60

 

Case 7: When c is absent.

            We factorize directly by looking for what is common.

 

Factorize the following:

      1.   5r2 – 35r

      2.   3p – 4p2

      3.   16t – 12t2

 

              Solution

1.      5r2 – 35r

            5r is common

            5r (r – 7)

 

2.      3p – 4p2

          p (3 – 4p)

 

3.       16t – 12t2

          4t (4 – 3t)

           

          Exercise G

Factorize the following

1.        k2 – 3k

2.        72y – 48y2

3.        6e2 – 15e

4.         18y2 – 40y

5.        12k2 – 21k

6.         7u – 8u2

 

 

Case 8: When b is absent and c is a -ve number.

             In this case use difference of two squares to factorize the expressions.

 

            Difference of two squares

Difference of two squares is used to factorize expressions such as a2 – b2, x2 – 25, 25p2 – 49 etc. Take note of the minus between the two expressions.

To factorize such expressions e.g. a2 – b2 we apply difference of two squares. This will give (a – b)(a + b).

x2 – y2 = (x – y)(x + y)

k2 – 4

= k2 – 22

= (k – 2)(k + 2)

 

c2 – 1

= c2 – 12

= (c – 1)(c + 1)

 

Note: we can only apply difference of two squares when c is a negative number. we can not apply difference of two squares to expressions such as a2 + b2, x2 + 25, 25p2 + 49 etc.

 

            Examples

Factorize the following

      1.   k2 – 36

      2.   81a2b2 – 400c2

      3.   36p2 – 16q2

 

                     Solution     

      1.   k2 – 36

            k2 – 62

            (k – 6)(k + 6)

 

      2.   81a2b2 – 400c2

            92 a2b2 – 202c2

            (9ab)2 – (20c)2

            (9ab – 20c)(9ab – 20c)

 

      3.   3c2 – 12

            3 is common, so factorize first

            3(c2 – 4)

            3(c2 – 22)

            3(c – 2)(c + 2)

 

         4.   36p2 – 16q2

Always factorize when there are common factors before applying difference of two squares.

            4(9p2 – 4q2)

            4(32p2 – 22q2)

            4(3p – 2q)(3p – 2q)

 

                        Exercise H

Factorize the following

      1.   x2 – 49

      2.   m2 – 169

      3.   25p2 – 64

      4.   5 – 75a2b2

      5.   28f2 – 36k2

 

                        Exercise I

Factorize the following

      1.   15y2 – y – 6

      2.   7e2 + 24e – 14

      3.   9d2 – 5d + 6

      4.   5m2 + 18m + 9

      5.   100j2k2 – 441r2

      6.   12 + 4x – x2

      7.   15α2 – αβ – β2

      8.   b2 – 121

      9.   12 – 5x – 3x2

      10.   7 + 8u + u2

      11.   12 – 5x – 3x2

      12.   16e2 – 225

      13.   θ2 – 16θ + 15

      14.   y2 – 2yz – 8z2

      15.   24 + 7c – 5c2

      16.   b2 – 10b + 25

      17.   3g2 + 38g – 13

      18.   20 – t – t2

      19.   3 + 2f – f2

      20.   17 + 16r – r2

      21.   18 – 3d - d2

      22.   c2 – 21c + 20

      23.   27 – 12n – 4n2

      24.   7a2 – 18a + 8

      25.   5 + 8j – 4j2

      26.   8 – 7q – q2

      27.   256x2y2z2 – 144

      28.   s2 + 30s + 225

      29.   4n2 – 12n – 27

      30.   9p2 – 23p + 10

      31.   6q2 – 4q

      32.   1 – 2500t2    

      33.   15a2 – 16a – 7

      34.   12p2 – 4p – 8

 

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