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 \( \boxed{8} \).

Answer 2

We can check that the expression is always \( 8q^3 \), independent of p, by trying two different values of p and seeing that both give the same result.

1. Option A (4): Taking \( p = 1, q = 1 \): \( (2)^3 - (0)^3 - 6(1)(1 - 1) = 8 - 0 - 0 = 8 \), not 4.
2. Option B (6): This value does not appear for any p we try, so it does not fit.
3. Option C (8): Taking \( p = 2, q = 1 \): \( (3)^3 - (1)^3 - 6(1)(4 - 1) = 27 - 1 - 18 = 8 \). This matches the first trial too, giving 8 both times.
4. Option D (10): This value also does not appear for any p we try.

Trying two different values of p, 1 and 2, both give 8, showing the answer does not depend on p and always equals 8.

Let's summarize:

• Testing the expression at \( p = 1 \) and \( p = 2 \) both give 8.
• This confirms p cancels out of the expression entirely.

Therefore, the correct answer is C (8).