Question

$8a {}^{3} - b {}^{3} - 12a {}^{2} b + 6ab {}^{2}$
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Answer 1

$8a {}^{3} - b {}^{3} - 12a {}^{2}b + 6ab {}^{2} \\ \\ = (2a) {}^{3} + ( - b) {}^{3} - 6ab(2a - b) \\ \\ = (2a) {}^{3} + ( - b) {}^{3} + 3(2a)( - b) {2a + ( - b)} \\ \\ = ({{{2a + ( - b)} {}}})^{3} \\ \\ = (2a - b) {}^{3}$

Some useful identities:

→ (a + b)^2 = a^2 + 2ab + b^2

→ (a – b)^2 = a^2 – 2ab + b^2

→ a^2 – b^2 = (a + b) (a – b)

→ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca

→ (a + b – c)^2 = a^2 + b^2 + c^2 + 2ab – 2bc – 2ca

→ (a – b – c)^2 = a^2 + b^2 + c^2 – 2ab + 2bc – 2ca

→ (a + b)^3 = a^3 + b^3 + 3ab(a + b)

→ (a – b)^3 = a^3 – b^3 – 3ab(a – b)

→ (a^3 + b^3) = (a + b) (a^2 – ab + b^2)

→ (a^3 – b^3) = (a – b) (a^2 + ab + b^2)