CHAPTER 4

SPECIAL PRODUCTS & FACTORING 4.1. Special Products It is an alternative method in finding the products of an algebraic expression containing special forms of the factors. These products can be easily solved by applying the Special Product Formulas. The following are the different types of special products: I.

Product of a Monomial and a Polynomial (Distributive Property)

ab  c  d  ab  ac  ad Illustrative Examples: 1. 2x4a  5b  6c  7d  8ax  10bx  12cx  14dx



2.  3x 2 y 3 6xy 4  5x 2 y 5  2x 5 y II.



=  18x 3 y 7  15x 4 y 8  6x 7 y 4

Product of the Sum and Difference of the Same Two

a  ba  b  a2  b2 Steps to follow in the product of the sum and difference of the same two : 1. Square the first term, minus 2. square the last term. Illustrative Examples: 1.

2a  5b 2a  5b  2a2  5b2

2.

7x

2

 4a2  25b 2



 

  2

y 3 z  6a3b 2 c4 7x 2 y 3 z  6a3b 2 c4  7x 2 y 3 z  6a3b 2 c4



2

 49x 4 y 6 z 2  36a6b 4 c8 3.

x

1/ 2



    y 

 y1/ 2 x1/ 2  y1/ 2  x1/ 2



2

1/ 2 2



4. 4x 3 y 2 z 4 2x 4  3y 3 8x 7 y 2 z 4  12x 3 y 5 z 4



 xy

  



 8x 7 y2 z 4  12x 3 y5 z 4 8x 7 y2 z 4  12x 3 y5 z 4  8x 7 y 2 z 4

  12x 2

 64x14 y 4 z 8  144x 6 y10 z 8 5.

6773  70  3 70  3  702  32 1

 4900  9  4891

3

y5z 4



2

III.

Square of Binomial

a  b2

a  b2

 a2  2ab  b 2

 a2  2ab  b 2

Steps to follow in square of binomial: 1. Square the first term. 2. Twice the product of the first and the last term. 3. Square the last term. Illustrative Examples: 1.

x  2y2  x 2  2x 2y  2y2

2.

3x

2

y 5  5w 4 z 3

  3x y  2

5 2

2

 x 2  4xy  4y 2





 

 2 3x 2 y 5 5w 4 z 3  5w 4 z 3



2

 9x 4 y10  30w 4 x 2 y 5 z 3  25w 8 z 6 3.

x

1/ 2



 

 y1/ 2 x1/ 2  y1/ 2  x1/ 2  y1/ 2

  x  2

1/ 2 2

 

 2  x1/ 2  y1/ 2  y1/ 2

2

 x  2x1/ 2 y1/ 2  y





4. 4x 3 y 2 z 4 2x 4  3y 3 8x 7 y 2 z 4  12x 3 y 5 z 4

  8x y z 

  



 8x 7 y 2 z 4  12x 3 y 5 z 4 8x 7 y 2 z 4  12x 3 y 5 z 4  8x 7 y 2 z 4  12x 3 y 5 z 4 7

2

4 2





 

 2 8x 7 y 2 z 4 12x 3 y 5 z 4  12x 3 y 5 z 4



2



2

 64x14 y 4 z 8  192x10 y 7 z 8  144x 6 y10 z 8 5.

IV.

542  50  42  502  2504  42

 2500  400  16  2916

Product of Two Binomials

x  ax  b  x 2  a  bx  ab

ax  bcx  d  acx 2  ad  bc x  bd

F  First x First O I

}

Outer + Inner

L  Last x Last Illustrative Examples: 1.

x  2x  3  x 2  3  2 x   23  x 2  x  6

2.

2a  5b 3a  4b  2  3a2   4  2  5  3ab  5b 4b  6a2  7ab  20b2

2

3x y  5w z 4x y  7w z   3  4 x y   3  7  5  4 w x y z  5w z  7w z 

3.

2

5

2

4

3

2

5

4

5 2

4

2

3

5

3

4

3

4

3

 12x 4 y10  w 4 x 2 y 5 z 3  35w 8 z 6 4.

5847  50  850  3  502   3  850  8 3  2500  5  50  24  2726

5.

3662  30  660  2  30  60  30  2  6  60  62  1800  420  12  2232

V.

Cube of a Binomial

a  b3

a  b3

 a3  3a2b  3ab 2  b 3

 a3  3a2b  3ab 2  b 3

Steps to follow in cube of a binomial: 1. Cube the first term 2. Three times the square of the first term times the last term 3. Three times the first term times square of the last term 4. Cube the last term Illustrative Examples: 1.

2a  5b3  2a3  32a2 5b  32a5b2  5b3  8a3  60a2b  150ab 2  125b 3

2.

x  2y3  x 3  3x 2 2y  3x 2y2  2y3

3.

3b j  4y z   3b j 2

3

4 3

2

3



 3 3b 2 j

 x 3  6x 2 y  12xy 2  8y 3

 4y z   3 3b j4y z   4y z  2

3

4

2

3

4 2

3

4 3

 27b6 j 3  108b 4 j 2 y 3 z 4  144b 2 jy6 z 8  64y 9 z12

  2x y 

    2x  32x y  6xy   32x y 6xy   6xy 

4. 2xyx  3y  2x 2 y  6xy 2 2

3



2

2

 2x 2 y  6xy 2 2x 2 y  6xy 2 2

2

2

2 2

2

2

y  6xy 2



3

2 3

 8x 6 y 3  72x 5 y 4  216x 4 y 5  216x 3 y 6 5.

133  10  33  103  3 102 3  3 1032  33  2197

3

 1000  900  270  27

VI.

Product of Binomial and Trinomial Leading to the Sum or Difference of Two Cubes

a  ba2  ab  b2   a3  b3

a  ba2  ab  b2   a3  b3

Steps to follow: 1. Cube the first term 2. Cube the last term Illustrative Examples: 1.

2a  5b4a2  10ab  25b2   2a3  5b3

2.

3x

2

 8a3  125b 3



 

y 5  5w 4 z 3 9x 4 y10  15w 4 x 2 y 5 z 3  25w 8 z 6  3x 2 y 5

  5w z  3

4

3 3

 27x 6 y15  125w12 z 9 3.

3b j  4y z 9b j 2

3

4

4 2

 

  3

 12b 2 jy3 z 4  16y 6 z 8  3b 2 j  4y 3 z 4



3

 27b6 j 3  64y 9 z12



   2x y   6xy 



4. 2xyx  3y  4x 4 y2  12x 3 y3  36x 2 y 4  2x 2 y  6xy2 4x 4 y 2  12x 3 y 3  36x 2 y 4 3

2

2 3

 8x 6 y 3  216x 3 y 6

    c d  2cd c d  2cd c d  4c d  16c d   c d  4c d  c d  4c d  16c d   c d   4c d 

5. cdc  2d c2 d  2cd2 c8 d4  4c6 d6  16c4 d8 2

4

2

2

2

2

4

2

8

4

8

6

4

6

6

4

6

8

4

4

8

2 3

2

4 3

 c12d6  64c6 d12 VII.

Square of a Trinomial

a  b  c2

 a2  b 2  c2  2ab  2ac  2bc

Steps to follow in square of a trinomial: 1. Square the following: 1st, 2nd and 3rd 2. Twice the product of the following: a. 1st and 2nd b. 1st and 3rd c. 2nd and 3rd

4



Illustrative Examples: 1. x  2y  3z   x   2y   3z   2  x  2y  2  x  3z  2  2y  3z 2

2

2

2

 x 2  4y 2  9z 2  4xy  6xz  12yz 2. 2a  5b  4c

2

 2a   5b  4c  22a 5b  22a4c  2 5b4c 2

2

2

 4a2  25b 2  16c2  20ab  16ac  40bc

    6ax  3ay  15az 6ax  3ay  15az   6ax  3ay  15az   6ax    3ay    15az   26ax  3ay   26ax  15az   2 3ay  15az 

3. 3a 2x 2  y 3  5z 4 6ax 2  3ay 3  15az 4 2

3

4

2 2

2

3 2

3

3

4 2

4

2

2

4 2

3

3

2

4

4

 36a2 x 4  9a2 y 6  225a2 z 8  36a2 x 2 y 3  180a2 x 2 z 4  90a2 y 3 z 4

x y  2a  x y  4a x y  4a   x y  4a x y  4a x y  4a x y  4a   x y  4a x y  4a   x y    4a x y   4a   2x y  4a x y   2x y 4a   2 4a x y 4a 

4.

2 2

3

6

2

2

6

2 2

6

3

3

2

4

2

2

2

3

2

3

6

4

2

2

3

4 2

6

4

2

6

2

3

2

2

6

2

4 2

3

4

4

 x12 y 4  16a4 x 6 y 2  16a8  8a2 x 9 y 3  8a4 x 6 y 2  32a6 x 3 y  x12 y 4  24a4 x 6 y 2  16a8  8a2 x 9 y 3  32a6 x 3 y 5.

542  50  3  12  502  32  12  2  50  3  2  50 1 2  3 1  2500  9  1 300  100  6

 2916

5

4.2. Factoring Factoring is the reverse process of special product formulas. A polynomial with integral coefficient is completely factored if: 1. the coefficients are relatively prime, that is, it has no common factor except 1, and 2. it cannot be expressed as the product of two polynomials of lower degree and apply any of the following types of factoring listed below.

Factoring

Common Monomial Factor

Number of

Two (2)

Difference of Two Squares

Three (3)

Perfect Square Trinomial

Sum or Difference of Two Cubes

Four (4) or More

Non - Perfect Square Trinomial

Factoring by Grouping

Note: To check the validity of your answer, use the special product formulas. Basic Types of Factoring I.

Common Monomial Factor

ab  ac  ad  ab  c  d Illustrative Examples: 1. 8ax  10bx  12cx  14dx = 2x4a  5b  6c  7d 2.



 18x 3 y 7  15x 4 y 8  6x 7 y 4 =  3x 3 y 4 6y 3  5xy 4  2x 4





3. 25a4b 3 c7  40a6b 2 c9  30a8b 5 c4  5a4b 2 c4 5bc 3  8a2 c5  6a4b 3

6



II.

Difference of Two Squares

a2  b2  a  ba  b Steps to follow in factoring the difference of two squares: 1. Extract the square roots of the first and the second ; 2. express the sum and the difference of the roots as factors. Illustrative Examples: 1. 4a2  25b 2  2a  5b  2a  5b 2a  5b 2

2

  7x

  6a b c  y z  6a b c  7x y z  6a b c  64x y z  144x y z  16x y z 4x  9y   16x y z 2x  3y  2x  3y  2x y  32xy  2xy x  16y   2xy x  4y x  4y   2xy x  4y x  2y x  2y  2

2. 49x 4 y 6 z 2  36a6b 4 c8  7x 2 y 3 z

3.

14

4

8

6

10

2

3

8

6

6

4.

5

5

4

III.

2

2

4

3

4

8

4

8

8

4

2

3

4 2

2

3

3

2

4

6

4

3

2

2

4

2

3

2

2

Sum or Difference of Two Cubes



a3  b3  a  b a2  ab  b2





a3  b3  a  b a2  ab  b2



Steps to follow in factoring the sum or difference of two cubes: 1. To obtain the first factor, extract the cube roots of the first and the second of the given expression. 2. To find the second factor, a. from the first factor; square its first term, b. take the opposite sign of the product of the first and second , and c. square its second term. Illustrative Examples:

   5w z   5w z 9x

1. 8a3  125b3  2a  5b  2a  5b 4a2  10ab  25b2 3

3

  3x

2. 27x 6 y15  125w12 z 9  3x 2 y 5 2

y5

3

4

4

7

3



3 3 4

y10  15w 4 x 2 y 5 z 3  25w 8 z 6



   4y z   3b j  4y z 9b j  12b jy z  16y z  375x y z  1029x y  3x y 125x z  343y   3x y 5x z   7y    3x y 5x z  7y 25x z  35x y z  49y  2a b  2ab  2aba  b   2aba  b a  b   2ab a  ba  ab  b a  ba  ab  b 

3. 27b6 j 3  64y 9 z12  3b 2 j

3

2

4.

5.

8

3 12

7

2

7

3

12

6

4 3

3

4

2

3

2

3

4 2

3

6 12

2

6

4

6

9

4

3

3

2

IV.

2

3

2

4

3

2

8

8

3

4 3

2

2

3

4

3 3

6

3

2

2

Perfect Square Trinomial

a2  2ab  b2  a  b

a2  2ab  b 2  a  b

2

2

Steps to follow in factoring perfect square trinomial: 1. Extract the square roots of the first and the third of the given expression ( that are perfect squares). 2. Raise the whole expression by 2 or squared, 3. Check the middle term of the given expression (term that is not a perfect square) by solving twice the product of the 1st and 2nd obtained in step 1. Illustrative Examples: 1.

x 2  4x  4  x  2

2

Checking: Middle term = 2 times the product of the first & second

4x  2 x 2 4x  4x 2.

x 2  4xy  4y 2  x  2y 

2

Checking: Middle term = 2 times the product of the first & second

 4xy  2 x  2y   4xy   4xy

8



3. 9x 4 y10  30w 4 x 2 y 5 z 3  25w 8 z 6  3x 2 y 5  5w 4 z 3



2

Checking: Middle term = 2 times the product of the first & second





30w 4 x 2 y 5 z 3  2 3x 2 y 5 5w 4 z 3



30w4 x 2 y5 z 3  30w4 x 2 y5 z 3





4. 2ab 2  24abc  72ac 2  2a b 2  12bc  36c2  2a b  6c 5.

V.

2

x  y2  6 x  y  9  x  y  32  x  y  32 Non - Perfect Square Trinomial

x 2  a  bx  ab  x  ax  b

acx 2  ad  bc x  bd  ax  bcx  d

Illustrative Examples: 1. x 2  5x  4  x  4x  1 Possible Factors

Product a b

Sum a  b

Remark

2&2

22  4

22  5

rejected

4&1

4 1  4

4  1 5

accepted

2. x 2  x  6 

x  3x  2

Possible Factors

Product a b

Sum a  b

Remark

–6 & 1

 6 1 6

 6  1 1

rejected

–3 & 2

 3  2  6

3  2 1

rejected

3 & –2

3  2  6

3   2  1

accepted

3. 6a2  7ab  20b2  2a  5b 3a  4b Possible Factors

Sum of the Product of Outer & Inner

6a  5b a  4b 2a  5b 3a  4b 2a  5b 3a  4b

6 4  51ab  7ab 24   53ab  7ab 2 4  53ab  7ab

9

Remark rejected rejected accepted





4. 12x 4 y10  w 4 x 2 y 5 z 3  35w 8 z 6  3x 2 y 5  5w 4 z 3 4x 2 y 5  7w 4 z 3

Sum of the Product of Outer & Inner =

Possible Factors

2x y 3x y 3x y

 4x y 4x y

Middle Term

2

5

 5w 4 z 3 6x 2 y 5  7w 4 z 3

2

5

 5w 4 z 3

2

5

 5w 4 z 3

2

5

 7w 4 z 3

2

5

 7w 4 z 3

 2  7  56w x y z  3 7   54w x y z  3  7  54w x y z 2

5

3

 w 4 x 2 y 5 z 3

rejected

4

2

5

3

 w 4 x 2 y 5 z 3

rejected

4

2



 x

 x

Remark

4

5. 5x10 y  10x 6 y 5  15x 2 y 9  5x 2 y x 8  2x 4 y 4  3y 8

 y x



5 3

  w4 x 2 y 5 z 3

accepted



 5x 2 y x 4  y 4 x 4  3y 4  5x 2

2

 y2

2

 y2



4

 3y 4





 5x 2 y x  y x  y  x 2  y 2 x 4  3y 4 VI.



Factoring by Grouping

ax  ay  bx  by  a  bx  y Proof:

ax  ay  bx  by  ax  ay   bx  by   a x  y   b x  y   x  y a  b  a  bx  y  Illustrative Examples:

     x 4a  5b   3y 4a  5b   4a  5b x  3y  b x  27b y  c x  27c y  b x  c x   27b y  27c y   x b  c   27y b  c   b  c x  27y   b  c b  c x  3y x  3xy  9y   b  cb  cb  c x  3y x  3xy  9y 

1. 4a3 x 2  5b 2 x 2  12a3 y 3  15b 2 y 3  4a3 x 2  5b 2 x 2  12a3 y 3  15b 2 y 3 2

2.

4

3

2

3

4

3

2

4

3

3

4

4

2

3

3

2

3

4

4

2

2

2

3

3

4

4

4

2

2

2

3

4

4

3

3

10

2

3

4

3

2

2

2

3

4

3





3. x 2  6xy  9y 2  4  x 2  6xy  9y 2  4  x  3y   2 2

2

 x  3y   2x  3y   2  x  3y  2x  3y  2





4. 25  4x 2  20xy  25y 2  25  4x 2  20xy  25y 2  5  2x  5y 

 5  2x  5y5  2x  5y  5  2x  5y5  2x  5y

11

2

2