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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).
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