James and Reshma were talking about algebraic identities they learnt in school.

James: $(a - b)^2(a + b) = (a^2 - 2ab + b^2)(a + b)$
Reshma: I have a different idea.

$$(a - b)^2(a + b) = (a - b)[(a - b)(a + b)]$$
$$= (a - b)(a^2 - b^2)$$

I will find this product to get the answer.
According to you, who is correct and why? Try to combine more such identities and find new results.
[Page No. 82]

Both methods are correct because multiplication is associative and commutative, so the factors can be grouped in any order.

James expands $(a - b)^2(a + b) = (a^2 - 2ab + b^2)(a + b)$, which simplifies to $a^3 - a^2b - ab^2 + b^3$.

Reshma first uses the identity $(a - b)(a + b) = a^2 - b^2$, and then multiplies by $(a - b)$:

$$(a - b)(a^2 - b^2) = a^3 - a^2b - ab^2 + b^3.$$

Thus, both methods give the same result, $a^3 - a^2b - ab^2 + b^3$.

Reshma's method is shorter because it applies the difference-of-squares identity before expanding.