Question

Using the identity and proof: $ {x^3} + {y^3} + {z^3} - 3xyz = (x + y + z)({x^2} + {y^2} + {z^2} - xy - yz - xz) $ .

Answer 1

Hint: To prove this relation we have to prove that the left side of the equation is equal to the right side. In mathematics, the equation at the left side of the equal to sign is called Left Hand Side and the equation at the right side of the equal to sign is called the Right Hand Side. Thus we have to prove Left-Hand Side equal to Right-Hand Side, this can be done by expanding the right hand side of the given question.

Complete step-by-step answer:
We have to prove –
 $ {x^3} + {y^3} + {z^3} - 3xyz = (x + y + z)({x^2} + {y^2} + {z^2} - xy - yz - xz) $
Let us start by solving the Left Hand Side of the above equation.
RHS-
 $
\Rightarrow (x + y + z)({x^2} + {y^2} + {z^2} - xy - yz - xz) \\
   = x.{x^2} + x.{y^2} + x.{z^2} - x.xy - x.yz - x.xz + y.{x^2} + y.{y^2} + y.{z^2} - y.xy - y.yz - y.xz + z.{x^2} + z.{y^2} + z.{z^2} - z.xy - z.yz - z.xz \\
   = {x^3} + x{y^2} + x{z^2} - {x^2}y - xyz - {x^2}z + {x^2}y + {y^3} + y{z^2} - x{y^2} - z{y^2} - xyz + {x^2}z + z{y^2} + {z^3} - xyz - y{z^2} - x{z^2} \;
  $
There are some quantities in the above equation which are equal in magnitude but opposite in sign so they cancel out each other.
After canceling those quantities with each other, the resultant equation is –
 $ = {x^3} + {y^3} + {z^3} - 3xyz $
The Left Hand Side of the given equation is $ {x^3} + {y^3} + {z^3} - 3xyz $ .
Now, we see that the Right Hand Side is equal to the Left Hand Side.
Hence proved that $ {x^3} + {y^3} + {z^3} - 3xyz = (x + y + z)({x^2} + {y^2} + {z^2} - xy - yz - xz) $
So, the correct answer is “ $ {x^3} + {y^3} + {z^3} - 3xyz = (x + y + z)({x^2} + {y^2} + {z^2} - xy - yz - xz) $ ”.

Note: The terms in one bracket are monomial whereas in the other bracket they are binomial. While solving the right-hand side of the given identity, we have to multiply each term of one bracket with each of the terms in the other bracket. Along with the magnitudes, we have to multiply their signs too. Positive multiplied with negative gives negative and vice versa whereas positive multiplied with positive and negative multiplied with negative gives a positive result.