How do you factor $8{{c}^{3}}-8c$ ?

Hint: In this question, we have to find the factors of an algebraic expression. So, we will use the basic mathematical rules and the algebraic identity to get the required solution. We start solving this problem by taking common c from the given algebraic expression. Then, we see that 8 is common in both the terms, thus we will again take common 8 from the two terms which lies within the brackets. After the necessary calculations, we will apply the algebraic identity $\left( {{a}^{2}}-{{b}^{2}} \right)=\left( a+b \right)\left( a-b \right)$ in the expression, to get the required solution to the problem.

Complete step by step solution:
According to the question, we have to find the factors of an algebraic expression.
So, we will apply the basic mathematical rules to get the required result.
The expression given to us is $8{{c}^{3}}-8c$ ---------------- (1)
Now, we start solving our problem by taking the common c from the equation (1), we get
$\Rightarrow c\left( 8{{c}^{2}}-8 \right)$
Now, we see that in the brackets we have 8 common in both the terms. Therefore, we will take 8 common from the two terms which lies within the brackets, we get
\[\Rightarrow c\left( 8\left( {{c}^{2}}-1 \right) \right)\]
So, we will solve further the above expression, we get
$\Rightarrow 8c\left( {{c}^{2}}-1 \right)$
In the last, we will apply the algebraic identity $\left( {{a}^{2}}-{{b}^{2}} \right)=\left( a+b \right)\left( a-b \right)$ in the brackets of the above expression, we get
$\Rightarrow 8c\left( c+1 \right)\left( c-1 \right)$ which is the required solution.
Therefore, for the algebraic expression $8{{c}^{3}}-8c$ , its factors are equal to $8c\left( c+1 \right)\left( c-1 \right)$.

Note: While solving this equation, do step-by-step calculations to avoid confusion and mathematical errors. One of the alternative methods to solve this problem is to take common 8c from the given expression and then apply the algebraic identity $\left( {{a}^{2}}-{{b}^{2}} \right)=\left( a+b \right)\left( a-b \right)$ to get the required result for the problem. Also, you can check your answer by using the distributive property in the solution, which is equal to the expression given in the problem.