Question

Simplify — (2a + b)³ - (2a - b)³

Answer 1

Simplify — (2a + b)³ - (2a - b)³

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❖ Using the formulas —

$\dag \; \underline{\boxed{\sf{\purple{ (a + b) = a^3 + 3 a^2 b + 3 a b^2 + b^3 }}}}$

$\dag \; \underline{\boxed{\sf{\purple{ (a - b) = a^3 - 3 a^2 b + 3 a b^2 - b^3 }}}}$

   

⬩ Consider:

$\sf{\orange{ (2a + b)^2 =}}$

$\implies \sf{ (2a)^3 + 3 \times (2a)^2 \times b + 3 \times 2a \times (b)^2 + (b)^3 }$

$\implies \sf{ 8a^3 + 3 \times 4a^2 \times b + 3 \times 2a \times b^2 + b^3 }$

$\implies \underline{\underline{\sf{ \; 8a^3 + 12 a^2 b + 6 ab^2 + b^3 \; }}}$

   

⬩ Similarly:

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

   

⬩ Consider:

$\sf{(2a + b)^3 - (2a - b)^3}$

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

$\implies \sf{ \cancel{8a^3} + 12 a^2 b + \canncel{6ab^2} + b^3 - \cancel{8a^3} + 12a^2b - \cancel{6ab^2} + b^3 }$

$\implies \sf{ 12 a^2 b + b^3 + 12 a^2 b +b^3 }$

$\implies \underline{\underline{\sf{ \; 24a^2b + 2ab^3 \; }}}$

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Related Algebraic Identities :

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• (a + b) (a - b) = a² - b²

• (a + b)³ = a³ + 3a²b + 3ab² + b³

• (a - b)³ = a³ - 3a²b + 3ab² + b³

• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac

• (x + a) (x + b) = x² + (a + b)x + ab

• a³ + b³ = (a + b) (a² - ab + b²)

• a³ - b³ = (a - b) (a² + ab + b²)

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Apologies for the mistakes <3

Answer 2

The answer is in the adjoining picture...