Factorization of Expressions of the Form a^3 - b^3

Here we will learn the process of Factorization of Expressions of the Form a^3 - b^3.

We know that (a - b)^3 = a^3 - b^3 - 3ab(a - b), and so

a^3 - b^3 = (a - b)^3 + 3ab(a - b) = (a - b){(a - b)^2 + 3ab}


Therefore, a3 - b3 = (a - b)(a2 + ab + b2)





Solved Examples on Factorization of Expressions of the Form a^3 - b^3

1. Factorize: 64m^6 – n^6

Solution:

Here, given expression = 64m^6 – n^6

                                  = 2^6 ∙ m^6 – n^6

                                  = (2^3m^3)^2 – (n^3)^2

                                  = (2^3m^3 + n^3)(2^3m^3 – n^3)


Now, 2^3m^3 + n^3 = (2m)^3 + n^3

                                = (2m + n){(2m)^2 – 2m ∙ n + n^2}

                                = (2m + n)(4m^2 – 2mn + n^2).


Again, 2^3m^3 – n^3 = (2m)^3 - n^3

                                  = (2m - n){(2m)^2 + 2m ∙ n + n^2}

                                  = (2m - n)(4m^2 + 2mn + n^2).

Therefore, given expression = (2m + n)(4m^2 – 2mn + n^2) ∙ (2m - n)(4m^2 + 2mn + n^2)

                                         = (2m + n)(2m - n)(4m^2 – 2mn + n^2) (4m^2 + 2mn + n^2).

 

2. Factorize: 8x^3 - 27

Solution:

Here, given expression = 8x^3 - 27

                                  = (2x)^3 - 3^3

                                  = (2x - 3){(2x)^2 + 2x ∙ 3 + 3^2}

                                  = (2x - 3)(4x^2 + 6x + 9)


3. Factorize: 64x^6 – y^6

Solution:

Here, given expression = 64x^6 – y^6

                                  = (4x^2)^3 – (y^2)^3

                                  = (4x^2 – y^2){(4x^2)^2 + 4x^2  ∙ y^2 + (y^2)^2}

                                  = {(2x)^2 – y^2}(16x^4 + 4x^2y^2 + y^4)

                                  = (2x + y)(2x – y)(16x^4 + 8x^2y^2 + y^4 – 4x^2y^2)

                                  = (2x + y)(2x – y){(4x^2)^2 + 2 ∙ (4x^2)y^2 + (y^2)^2 – 4x^2y^2}

                                  = (2x + y)(2x – y){(4x^2 + y^2)^2 – (2xy)^2}

                                  = (2x + y)(2x – y)(4x^2 + y^2 + 2xy)(4x^2 + y^2 – 2xy)

 

Alternative Method:

Given expression = 64x^6 – y^6

                          = (8x^3)^2 – (y^3)^2

                          = (8x^3 + y^3) (8x^3 - y^3)

                          = {(2x)^3 + y^3}{(2x)^3 – y^3}

                          = (2x + y){(2x)^2 – 2x ∙ y + y^2} ∙ (2x – y){(2x)^2 + 2x ∙ y + y^2}

                          = (2x + y)(2x – y)(4x^2 + y^2 + 2xy)(4x^2 + y^2 – 2xy)


Factorization of expressions reducible to a^3 ± b^3 form

Factorize: x^3 + 3x^2y + 3xy^3 + 2y^3.

Solution:

Given expression = x^3 + 3x^2y + 3xy^3 + 2y^3

                          = x^3 + 3x^2y + 3xy^3 + y^3 + y^3

                          = (x + y)^3 + y^3, [Which is of the form a^3 + b^3]

                          = {(x+ y) + y}{(x + y)^2 – (x + y)y + y^2}

                          = (x + 2y)(x^2 + 2xy + y^2 – xy – y^2 + y^2)

                          = (x + 2y)(x^2 +xy + y^2).








9th Grade Math

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