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Answer:
The formula for {a³} – {b³} is {a – b}{a² + b² + ab}
The (a³ – b³) formula is referred to as the difference of cubes formula. This particular formula is used to compute the difference between the two cubes, cubes of a and cubes of b, without actually calculating the same cubes. Additionally, the formula is used to factorize the binomials of the cubes.
Proof of the formula
To prove that (a³ – b³) = (a – b)(a² + b² + ab), it is to be proven that LHS = RHS.
Here,
LHS = (a³ – b³)
RHS = (a – b)(a² + b² + ab)
On multiplying the a and b separately with (a² + ab + b²), the following is obtained
= a(a² + ab + b²) – b(a²+ ab + b²)
= a³ + a2b + ab² – a²b – ab² – b³
= a³ + a²b – a²b + ab² – ab² – b³
= a³ – 0 – 0 – b³
= a³ – b³
Hence, LHS = RHS
The same formula can also be proven by letting the values of a = 4 and b = 2.
(4³ – 2³) = (4 – 2)(4² + 2² + 4 x 2)
LHS = (a³ – b³)
LHS = (4³ – 2³)
LHS = (64 – 8)
LHS = 56
RHS (a – b)(a² + b² + ab)
RHS = (4 – 2)(4² + 2² + 4 x 2)
RHS = (2)(16 + 4 + 8)
RHS = (2)(28)
RHS = 56
Therefore, LHS = RHS
(a³ – b³) = (a – b)(a² + b² + ab)
Hence, it is proved that the formula for ${a^3} – {b^3}$ is {a – b}{a² + b² + ab}.
