Question

How do you factor the difference of two cubes \[{{x}^{3}}-216?\]

Answer 1

Hint: As here we have to factor the difference of two cubes. You can use the formula for factoring the given equation \[{{x}^{3}}-216\] is \[{{a}^{3}}-{{b}^{3}}=(a-b)\left( {{a}^{2}}+ab+{{b}^{2}} \right).\]
Here compare the given equation \[{{x}^{3}}-216\] by \[{{a}^{3}}-{{b}^{3}}\] and identify the value of \[a\]and \[b.\]

Complete step by step solution:
As we know that, we have to factor the difference of two cubes \[{{x}^{3}}-216.\]
The formula for factoring difference of \[2\]cubes is \[{{a}^{3}}-{{b}^{3}}=(a-b)({{a}^{2}}+ab+{{b}^{2}}).\]
The given equation is\[,\]
\[\Rightarrow \]\[{{x}^{3}}-216\]
Here, compare the given equation with \[{{a}^{3}}-{{b}^{3}}\]
Therefore,
\[\Rightarrow \]\[{{a}^{3}}={{x}^{3}}\]and
\[\Rightarrow \]\[{{b}^{3}}=216\]
\[\Rightarrow \]\[{{a}^{3}}={{x}^{3}}\]
Transfer cube of left side to right side and here cube will be converted into cube root.
Therefore,
\[\Rightarrow \]\[a=\sqrt[3]{{{x}^{3}}}\]
Above cube root and cube will get cancel,
\[\therefore a=x\]
and
\[\Rightarrow {{b}^{3}}=216\]
Transfer the cube from left side to right side and it will get converted into a cube root.
Therefore,
\[\Rightarrow \]\[b=\sqrt[3]{216}\]
\[\Rightarrow \]\[b=6\]
Substitute the value of \[a=r,b=6\] into the formula of \[\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right).\]
Therefore the equation will be,
\[\Rightarrow \]\[\left( x-6 \right)\left( {{x}^{2}}+x\times 6+{{6}^{2}} \right)\]
\[\Rightarrow \]\[\left( x-6 \right)\left( {{x}^{2}}+6x+36 \right)\]

Hence \[\left( x-6 \right)\left( {{x}^{2}}+6x+36 \right)\] is the factored form of \[{{x}^{3}}-216.\]

Note: Here remember that you have of to use the formula of
\[{{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\]
And choose \['a'\] and \['b'\] term from the given equation.
Apply the formula of difference of two cubes and in this way you will get the factored form of the given equation.