If x + y + z = 0, then $x^3 + y^3 +z^3$ is equal to :

Q: If x + y + z = 0, then $x^3 + y^3 +z^3$ is equal to :

To solve the problem of finding the value of $x^3 + y^3 + z^3$ given that $x + y + z = 0$, we can use a well-known algebraic identity. The identity states: \[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \] Given that $x + y + z = 0$, we can substitute this into the identity: \[ x^3 + y^3 + z^3 - 3xyz = 0 \cdot (x^2 + y^2 + z^2 - xy - yz - zx) \] Since the term $(x + y + z)$ is zero, the entire right-hand side of the equation becomes zero: \[ x^3 + y^3 + z^3 - 3xyz = 0 \] This simplifies to: \[ x^3 + y^3 + z^3 = 3xyz \] Thus, the correct answer is indeed Option 3: $3xyz$. Now, let's explain why the other options are incorrect: 1. **Option 1: xyz** - This option suggests that $x^3 + y^3 + z^3 = xyz$. However, as derived from the identity, the correct expression is $x^3 + y^3 + z^3 = 3xyz$, not just $xyz$. 2. **Option 2: 2xyz** - This option implies that $x^3 + y^3 + z^3 = 2xyz$. Again, the identity clearly shows that the correct result is $3xyz$, not $2xyz$. 3. **Option 4: $x^3y^3z^3$** - This option suggests a completely different form, which is not supported by the identity used. The expression $x^3 + y^3 + z^3$ simplifies to $3xyz$ when $x + y + z = 0$, and there is no basis for it to equal $x^3y^3z^3$. In conclusion, the correct answer is Option 3: $3xyz$, as justified by the algebraic identity and the given condition $x + y + z = 0$.

Question

Accepted Answer

To solve the problem of finding the value of $x^3 + y^3 + z^3$ given that $x + y + z = 0$, we can use a well-known algebraic identity. The identity states:

$$ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) $$

Given that $x + y + z = 0$, we can substitute this into the identity:

$$ x^3 + y^3 + z^3 - 3xyz = 0 \cdot (x^2 + y^2 + z^2 - xy - yz - zx) $$

Since the term $(x + y + z)$ is zero, the entire right-hand side of the equation becomes zero:

$$ x^3 + y^3 + z^3 - 3xyz = 0 $$

This simplifies to:

$$ x^3 + y^3 + z^3 = 3xyz $$

Thus, the correct answer is indeed Option 3: $3xyz$.

Now, let's explain why the other options are incorrect:

1. **Option 1: xyz** - This option suggests that $x^3 + y^3 + z^3 = xyz$. However, as derived from the identity, the correct expression is $x^3 + y^3 + z^3 = 3xyz$, not just $xyz$.

2. **Option 2: 2xyz** - This option implies that $x^3 + y^3 + z^3 = 2xyz$. Again, the identity clearly shows that the correct result is $3xyz$, not $2xyz$.

3. **Option 4: $x^3y^3z^3$** - This option suggests a completely different form, which is not supported by the identity used. The expression $x^3 + y^3 + z^3$ simplifies to $3xyz$ when $x + y + z = 0$, and there is no basis for it to equal $x^3y^3z^3$.

In conclusion, the correct answer is Option 3: $3xyz$, as justified by the algebraic identity and the given condition $x + y + z = 0$.