Question

Ramesh is trying to simplify the expression $(p + q)^3 - (p - q)^3 - 6q(p^2 - q^2)$ and if $q = 1$. You helped him and the solution arrived was:

• A. 4
• B. 6
• C. 8
• D. 10

Answer 1

Answer: (C)

8

Answer 2

The given expression is $(p + q)^3 - (p - q)^3 - 6q(p^2 - q^2)$.
First, expand the cubes:
$(p + q)^3 = p^3 + 3p^2q + 3pq^2 + q^3$
$(p - q)^3 = p^3 - 3p^2q + 3pq^2 - q^3$
Subtracting the second expression from the first:
$(p^3 + 3p^2q + 3pq^2 + q^3) - (p^3 - 3p^2q + 3pq^2 - q^3) = 6p^2q + 2q^3$
Next, simplify $-6q(p^2 - q^2)$:
$-6q(p^2 - q^2) = -6qp^2 + 6q^3$
Combine all terms:
$6p^2q + 2q^3 - 6qp^2 + 6q^3 = 8q^3$
Given $q = 1$:
$8(1)^3 = 8$
Thus, the answer is $\boxed{8}$.