Question

Factorize: \[27{x^3} + {y^3} + {z^3} - 9xyz\]

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.
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.

Complete step-by-step answer:
In the given expression\[27{x^3} + {y^3} + {z^3} - 9xyz - - - (i)\], the degree of the expression is 3, hence compare the given expression with a cubic identity.
Equation (i) can be written as:
\[
  27{x^3} + {y^3} + {z^3} - 9xyz = {\left( {3x} \right)^3} + {y^3} + {z^3} - 9xyz \\
   = {\left( {3x} \right)^3} + {y^3} + {z^3} - 3 \times \left( {3x} \right) \times y \times z \\
 \]
Hence by comparing the expression with a cubic identity
\[{a^3} + {b^3} + {c^3} - 3abc = \left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - ab - bc - ca} \right)\]
It can be written as
\[
  a = 3x \\
  b = y \\
  c = z \\
 \]
Hence by putting the value of x, y, z in the cubic identity equation, we get
 \[
  {a^3} + {b^3} + {c^3} - 3abc = \left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - ab - bc - ca} \right) \\
  {\left( {3x} \right)^3} + {y^3} + {z^3} - 3 \times \left( {3x} \right) \times b \times c = \left( {3x + y + z} \right)\left( {{{\left( {3x} \right)}^2} + {y^2} + {z^2} - \left( {3x} \right)y - yz - z\left( {3x} \right)} \right) \\
   = \left( {3x + y + z} \right)\left( {9{x^2} + {y^2} + {z^2} - 3xy - yz - 3zx} \right) \\
 \]
Hence by factorizing the given expression, \[27{x^3} + {y^3} + {z^3} - 9xyz\]we get\[\left( {3x + y + z} \right)\left( {9{x^2} + {y^2} + {z^2} - 3xy - yz - 3zx} \right)\]

Note: 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.