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

Step 1: Use the identity for a cube difference.
For any x and y, \( (x+y)^3 - (x-y)^3 = 2y(3x^2 + y^2) \). Here \( x = p, y = q \), so \( (p+q)^3 - (p-q)^3 = 2q(3p^2 + q^2) = 6p^2q + 2q^3 \).

Step 2: Subtract the last term.
\( 6p^2q + 2q^3 - 6q(p^2 - q^2) = 6p^2q + 2q^3 - 6p^2q + 6q^3 = 8q^3 \).

Step 3: Substitute q = 1.
\( 8(1)^3 = 8 \). \[ \boxed{8} \]