Question

Factorize the following expression:
${a^3} - {b^3} - {c^3} - 3abc$

Answer 1

Hint: In this particular question use the concept of standard identity which is given as, ${x^3} + {y^3} + {z^3} - 3xyz = \left( {x + y + z} \right)\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right)$, so use this concept to reach the solution of the question.

Complete step-by-step answer:
Given expression:
${a^3} - {b^3} - {c^3} - 3abc$
Now we have to factorize this
Now as we know that ${\left( { - 1} \right)^3} = - 1,\left( { - 1} \right)\left( { - 1} \right) = 1$, so from this property the above expression is also written as
$ \Rightarrow {a^3} + {\left( { - b} \right)^3} + {\left( { - c} \right)^3} - 3a\left( { - b} \right)\left( { - c} \right)$...................... (1)
Now as we know the standard identity which is given as
${x^3} + {y^3} + {z^3} - 3xyz = \left( {x + y + z} \right)\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right)$................... (2)
Now compare equation (1) and (2) we have,
$ \Rightarrow $x = a, y = -b, and z = -c
Now substitute this value in equation (2) we have,
$ \Rightarrow {a^3} + {\left( { - b} \right)^3} + {\left( { - c} \right)^3} - 3a\left( { - b} \right)\left( { - c} \right) = \left[ {a + \left( { - b} \right) + \left( { - c} \right)} \right]\left[ {{a^2} + {{\left( { - b} \right)}^2} + {{\left( { - c} \right)}^2} - \left( {a \times - b} \right) - \left( { - b \times - c} \right) - \left( { - c \times a} \right)} \right]$
Now simplify according to property that, ${\left( { - 1} \right)^3} = - 1,{\left( { - 1} \right)^2} = 1$, (-1) (1) = -1, and (-1) (-1) = 1, so the above equation becomes
$ \Rightarrow {a^3} - {b^3} - {c^3} - 3abc = \left[ {a - b - c} \right]\left[ {{a^2} + {b^2} + {c^2} + ab - bc + ca} \right]$
So these are the required factors of the given expression.
So this is the required answer.

Note: Whenever we face such types of questions the key concept we have to remember is the standard identity which is stated above, this identity is the basis of the solution, without knowing this identity we cannot factorize the given expression quickly and effectively and always recall that, ${\left( { - 1} \right)^3} = - 1,{\left( { - 1} \right)^2} = 1$, (-1) (1) = -1, and (-1) (-1) = 1.