Question

How do you factor $8{x^3} - {\left( {2x - y} \right)^3}$ ?

Answer 1

Hint: A factor is nothing but a number or expression when multiplied with another number or expression to get the required expression or number. Here we are given an expression $8{x^3} - {\left( {2x - y} \right)^3}$
Here, we are asked to find the factors of the given expression. Since the given expression is algebraic, we need to apply the algebraic identities to find the factors. We may note that the given expression is cubic; we shall apply the respective algebraic identities.
Formula to be used:
The appropriate algebraic identities to be used to find the factors of the given expression are as follows.
\[{a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})\]
${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$

Complete answer:
The given algebraic expression is$8{x^3} - {\left( {2x - y} \right)^3}$
First, let us simplify the given expression.
$8{x^3} - {\left( {2x - y} \right)^3} = {2^3}{x^3} - {\left( {2x - y} \right)^3}$
$ \Rightarrow 8{x^3} - {\left( {2x - y} \right)^3} = {\left( {2x} \right)^3} - {\left( {2x - y} \right)^3}$
Here, we may note that the given expression is cubic; we shall apply the respective algebraic identities.
Now, we need to apply the formula\[{a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})\]
Here, we have$a = 2x$ and$b = 2x - y$
$ \Rightarrow 8{x^3} - {\left( {2x - y} \right)^3} = \left( {2x - \left( {2x - y} \right)} \right)\left( {{{\left( {2x} \right)}^2} + 2x\left( {2x - y} \right) + {{\left( {2x - y} \right)}^2}} \right)$
$ = \left( {2x - 2x + y} \right)\left( {4{x^2} + 4{x^2} - 2xy + 4{x^2} - 4xy + {y^2}} \right)$
(Here we have applied ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$)
     $ = y\left( {4{x^2} + 4{x^2} - 2xy + 4{x^2} - 4xy + {y^2}} \right)$
     $ = y\left( {12{x^2} - 6xy + {y^2}} \right)$
     $ = y\left( {12{x^2} + {y^2} - 6xy} \right)$
Therefore, we got$8{x^3} - {\left( {2x - y} \right)^3}$$ = y\left( {12{x^2} + {y^2} - 6xy} \right)$

Note:
Thus, a factor is a number or expression when multiplied with another number or expression to get the required expression or number. Also, the given expression can contain one or more than one factor.
During factorization, we need to follow the given steps.
The first step is to factor out any common terms from the given expression.
The next step is to keep going and we need to stop when the factorization cannot be done further.
We can also verify the obtained factors; to check the answer we can multiply each factor with each other, if we get the same expression then the calculated answer is correct otherwise not.