If $x+y+z=0$, then the value of $x^{3}+y^{3}+z^{3}$ will be:-

Question

Accepted Answer

The correct option is 4:
To solve this problem, we need to recall a fundamental algebraic identity related to the sum of cubes.

**Key Concept:** The identity for the sum of three cubes is:
$$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$$

We are given the condition that $x+y+z=0$. Let's substitute this condition into the identity.

**Step-by-step Explanation:**1.  **Start with the algebraic identity:**
    $$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$$

2.  **Apply the given condition:** Since $x+y+z=0$, we substitute $0$ into the right side of the equation:
    $$x^3 + y^3 + z^3 - 3xyz = (0)(x^2 + y^2 + z^2 - xy - yz - zx)$$

3.  **Simplify the right side:** Any quantity multiplied by zero is zero.
    $$x^3 + y^3 + z^3 - 3xyz = 0$$

4.  **Isolate $x^3 + y^3 + z^3$:** Add $3xyz$ to both sides of the equation:
    $$x^3 + y^3 + z^3 = 3xyz$$Therefore, if $x+y+z=0$, the value of $x^3+y^3+z^3$ is $3xyz$.

The incorrect option is 1: This option suggests the answer is $9xyz$. This is incorrect as it implies an extraneous multiplication by 3.
The incorrect option is 2: This option suggests the answer is $0$. This would only be true if $3xyz=0$, which is not necessarily the case. This is a common misconception, perhaps confusing it with $x+y+z=0$.
The incorrect option is 3: This option suggests the answer is $-3xyz$. This is incorrect as it indicates a sign error in applying the identity or in moving terms across the equality.