Question

If ${x^3} + {y^3} + {z^3} = 3xyz$ then the relation between \[x,y\] and $z$ is
(A) $x + y + z = 0$
(B) $x = y = z$
(C) ${\text{Either }}x + y + z = 0{\text{ or }}x = y = z$
(D) $x + y + z = 0{\text{ and }}x \ne y \ne z$

Answer 1

Hint: The given question is about the relationship of various variables and polynomials. On the basis of their degrees and variables. Now in this question, we have given our equation of polynomial and we have to find out the relationship between the polynomials used in that polynomial, which is obtained by using the formulas of polynomials.

Complete answer:
We are given that ${x^3} + {y^3} + {z^3} = 3xyz$ then we have to find out the relation between $x,y,z.$. There are various types of formulas or Identities. We are dealing with solving the polynomials.
For this polynomial, we have the formula
${x^3} + {y^3} + {z^3} - 3xyz = (x + y + z){\text{ }}({x^2} + {y^2} + {z^2} - xy - yz - zx){\text{ }}...............{\text{(1)}}$
Since this is the polynomial of degree therefore it is a cubic polynomial. We had used this identity only because our question was about $\left( {{x^3} + {y^3} + {z^3}} \right)$ and in this identity, the same tem is there. Now in option (A) $\left( {x + y + z} \right) = 0$
If we substitute this condition in equation (1), we will get
${x^3} + {y^3} + {z^3} - 3xyz = (0)({x^2} + {y^2} + {z^2} - xy - yz - zx)$
Now in the above equation, when $0{\text{ (Zero)}}$ is multiplied by the rest term $({x^2} + {y^2} + {z^2} - xy - yz - zx)$ then the whole term in RHS becomes zero. Because zero on multiplication with any numbers results in zero. Therefore the right hand side becomes zero.
${x^3} + {y^3} + {z^3} - 3xyz = 0$
Taking the term $3xyz$ from left hand side to right hand side and negative term in left hand side becomes right the term in right hand side. Therefore we get
 ${x^3} + {y^3} + {z^3} = 3xyz$
Which is the condition given in condition. It means option (A) is correct which means If ${x^3} + {y^3} + {z^3} = 3xyz$
That means the relation between $x,y$ and $z$ is $\left( {x + y + z = 0} \right)$ that means ${x^3} + {y^3} + {z^3} = 3xyz$

Note: There are many identities used for solving polynomials. We had taken only this because the question as well as the identity both are cubic polynomials. Cubic polynomials can be deduced to be the product of linear and quadratic polynomials and further quadratic polynomials are solved to get the solution of cubic polynomials.