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Question

Factorise: $${ a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }-3abc$$


Solution

To factorise: $${ a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }-3abc$$

From the standard identity,

$${ a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)+3abc$$
$$\Rightarrow { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$

Hence, the factorised form is, 
$$(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$



Mathematics

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