Question

Show that the product of ${a^3} + {b^3} + {c^3} - 3abc$ and ${x^3} + {y^3} + {z^3} - 3xyz$ can be put into the form ${A^3} + {B^3} + {C^3} - 3ABC$.

Answer 1

Hint: Do the product of ${a^3} + {b^3} + {c^3} - 3abc$and ${x^3} + {y^3} + {z^3} - 3xyz$, and assume that product as P and then solve the question.

Complete step by step answer:
We have been given two equations, ${a^3} + {b^3} + {c^3} - 3abc - (1)$ and ${x^3} + {y^3} + {z^3} - 3xyz - (2)$
Multiplying equation (1) and equation (2), we get-
$P = ({a^3} + {b^3} + {c^3} - 3abc) \times ({x^3} + {y^3} + {z^3} - 3xyz)$
Multiplying all the terms with each other we get-
$
  P = ({a^3} + {b^3} + {c^3} - 3abc) \times ({x^3} + {y^3} + {z^3} - 3xyz) \\
   = {a^3}{x^3} + {a^3}{y^3} + {a^3}{z^3} + {b^3}{x^3} + {b^3}{y^3} + {b^3}{z^3} + {c^3}{x^3} + {c^3}{y^3} + {c^3}{z^3} + 9abcxyz - 3{a^3}xyz - 3{b^3}xyz - 3{c^3}xyz - 3abc{x^3} - 3abc{y^3} - 3abc{z^3} \\
$
Now, we can also write the above equation using the formula, ${a^3} + {b^3} + {c^3} - 3abc = {(a + b + c)^3}$, we get-
\[
  P = \left[ {{{\left( {ax} \right)}^3} + {{\left( {bx} \right)}^3} + {{\left( {cx} \right)}^3} - 3abc{x^3}} \right] + \left[ {{{\left( {ay} \right)}^3} + {{\left( {by} \right)}^3} + {{\left( {cy} \right)}^3} - 3abc{y^3}} \right] + \left[ {{{\left( {az} \right)}^3} + {{\left( {bz} \right)}^3} + {{\left( {cz} \right)}^3} - 3abc{z^3}} \right] - 3xyz\left[ {{a^3} + {b^3} + {c^3} - 3abc} \right] \\
   = {\left( {\left( {a + b + c} \right)x} \right)^3} + {\left( {\left( {a + b + c} \right)y} \right)^3} + {\left( {\left( {a + b + c} \right)z} \right)^3} - 3{(a + b + c)^3}xyz \\
 \]Now we can write,
$
  A = \left( {\left( {a + b + c} \right)x} \right) \\
  B = \left( {\left( {a + b + c} \right)y} \right) \\
  C = \left( {\left( {a + b + c} \right)z} \right) \\
 $
Therefore, the equation becomes,
\[ \Rightarrow P = {A^3} + {B^3} + {C^3} - 3ABC\]

Hence the product of the given equations can be put in the form of,
${A^3} + {B^3} + {C^3} - 3ABC$.

Note: Whenever such types of questions appear, always be careful while multiplying the given equations as the multiplication is very lengthy and it will result in a long equation containing different order terms. Also, always assume the product to be some variable, as done in the solution, the product is assumed to be P.