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).

9th Grade Math

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