Question

Verify that \[{x^3} + {y^3} + {z^3} - 3xyz = \dfrac{1}{2}\left( {x + y + z} \right)\left[ {{{\left( {x - y} \right)}^2} + \left( {y - z} \right) + {{\left( {z - x} \right)}^2}} \right]\]

Answer 1

Hint: To solve this question, use cubic identity formula given as:
 \[{a^3} + {b^3} + {c^3} - 3abc = \left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - ab - bc - ca} \right)\]
If the left-hand side of an equation is equal to the right side for any values of the variable, then that equation is an identity which is used in algebraic expansions and factorizations. In standard identities, the values are equal to each other regardless of what values are substituted for the variables. In the identity polynomial variables having degree 3 is called cubic identity.

Complete step-by-step answer:
Here, in the question we need to compare the given function with the standard function and elaborate the function following the identity discussed above to get the result.
Given equation is: \[{x^3} + {y^3} + {z^3} - 3xyz = \dfrac{1}{2}\left( {x + y + z} \right)\left[ {{{\left( {x - y} \right)}^2} + \left( {y - z} \right) + {{\left( {z - x} \right)}^2}} \right]\]
Solving the Right Hand Side of the give equation as:
\[RHS = \dfrac{1}{2}\left( {x + y + z} \right)\left[ {{{\left( {x - y} \right)}^2} + \left( {y - z} \right) + {{\left( {z - x} \right)}^2}} \right]\]
Use the algebraic identity\[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\], to approach further in the equation
\[
  R.H.S = \dfrac{1}{2}\left( {x + y + z} \right)\left[ {{{\left( {x - y} \right)}^2} + \left( {y - z} \right) + {{\left( {z - x} \right)}^2}} \right] \\
   = \dfrac{1}{2}\left( {x + y + z} \right)\left[ {\left( {{x^2} + {y^2} - 2xy} \right) + \left( {{y^2} + {z^2} - 2yz} \right) + \left( {{z^2} + {x^2} - 2zx} \right)} \right] \\
   = \dfrac{1}{2}\left( {x + y + z} \right)\left[ {2{x^2} + 2{y^2} + 2{z^2} - 2xy - 2yz - 2zx} \right] \\
   = \dfrac{1}{2}\left( {x + y + z} \right)2\left[ {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right] \\
   = \left( {x + y + z} \right)\left[ {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right] \\
 \]
We know that, \[{a^3} + {b^3} + {c^3} - 3abc = \left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - ab - bc - ca} \right)\]
Hence we can say that,
\[{x^3} + {y^3} + {z^3} - 3xyz = \left( {x + y + z} \right)\left[ {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right]\]
\[
   = {x^3} + {y^3} + {z^3} - 3xyz \\
   = L.H.S \\
 \]
Since R.H.S=L.H.S, hence we can say the equation is verified

Note: The three algebraic identities in maths are
Identity 1: \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\]
Identity 2: \[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\]
Identity 3: \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
Students can check whether a given equation is the identity or not by transforming either one side of an equation in such a way that they become equal to the other side of the equation.