Explain how you can prove the difference of two cubes' identity. a³ - b³ = (a - b)(a² + ab + b²)

Solution:

The algebraic identity a³ - b³ = (a - b)(a² + ab + b²) is the two cubes' identity.

This can be proved by considering right hand side term.

(a - b)(a² + ab + b²) = a(a² + ab + b²) - b(a² + ab + b² )

=(a - b)(a² + ab + b²) = a³ + a²b + ab² - a²b - ab² - b³ [by taking the common factor (a-b) out]

=a³ + a²b - a²b + ab²- ab² - b³ [by bringing the like terms together and cancelling the possible terms like a²b and ab² in the right hand side term]

= (a - b)(a² + ab + b²) = a³ - b³ = L.H.S

Hence the equality is satisfied.

Explain how you can prove the difference of two cubes' identity. a³ - b³ = (a - b)(a² + ab + b²)

Summary:

Therefore, a³ - b³ = (a - b)(a² + ab + b²)

If you consider an example of a = 2 and b = 3 and consider LHS a³ - b³ = 23 - 33 = 8 - 27 = -19 and RHS (a - b)(a² + ab + b²) = (2 - 3)(2² + (2)(3) + 3²) = (-1)(4 + 6 + 9) = (-1)(19) = -19.

Therefore, this example proves the the equality a³ - b³ = (a - b)(a² + ab + b²) is true for any real number.