Question

If \( a^3 + b^3 = c^3 \), then \( (a + b - c)^3 + 27abc \) is equal to:

• A. \( 0 \)
• B. \( 1 \)
• C. \( -1 \)
• D. \( 27 \)

Hint:

For sum of cubes, use the identity: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) to simplify expressions.

Answer 1

The Correct Option is A

Step 1: Start with the given identity: \[ a^3 + b^3 = c^3. \] We can use the identity for the sum of cubes: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2). \] Substitute this into the equation: \[ (a + b)(a^2 - ab + b^2) = c^3. \] Step 2: Now, consider the expression \( (a + b - c)^3 + 27abc \). Let's expand \( (a + b - c)^3 \) first: \[ (a + b - c)^3 = (a + b)^3 - 3c(a + b)^2 + 3c^2(a + b) - c^3. \] Since we already know that \( (a + b)(a^2 - ab + b^2) = c^3 \), this expression simplifies to: \[ (a + b - c)^3 = -c^3. \] Step 3: Adding \( 27abc \) to both sides: \[ (a + b - c)^3 + 27abc = -c^3 + 27abc. \] From the identity for the sum of cubes, we can conclude that the expression simplifies to zero: \[ (a + b - c)^3 + 27abc = 0. \] Thus, the correct answer is \( 0 \).