Factorization of expressions of the Form a^3 + b^3 + c^3 – 3abc
Here we will learn the process of On Factorizations of expressions of the Form a3 + b3 + c3 – 3abc.
We have, a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – bc – ca – ab). [Verify by actual multiplication.]
Solved example on factorization of expressions of the form a3 + b3 + c3 – 3abc:
1. Factorize: x3 + y3 – 3xy + 1.
Solution:
Here, given expression = x3 + y3 – 3xy + 1.
= x3 + y3 + 13 – 3 ∙ x ∙ y ∙ 1
= (x + y + 1)(x2 + y2 + 12 – y ∙ 1 – 1 ∙ x – xy)
= (x + y + 1)(x2 + y2 – xy – x – y + 1).
2. Factorize: 8x3 + 27y3 – 90xy + 125.
Solution:
Given expression = 8x3 + 27y3 – 90xy + 125
= (2x)3 + (3y)3 + (5)3 – 3 ∙ 2x ∙ 3y ∙ 5
= (2x + 3y + 5)(4x2 + 9y2 – 6xy – 15y – 10x + 25).
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