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

The Correct Option is C

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 \( \boxed{8} \).