How do you factor $9{x^3} - x$?

Hint: In the given question we have $x$ in common, so take common outside and can write the equation as $x(9{x^2} - 1)$. Now, $9{x^2} - 1$ is in the form of difference of square, so by making use of general form of difference of square given by: ${a^2} - {b^2} = (a - b)(a + b)$. Here ${a^2} = {(3x)^2}$ and ${b^2} = {1^2}$ . So by simplifying using the formula we can arrive at the solution.

Complete step by step answer:
Here in the given question, they have asked to find the factor of $9{x^3} - x$. First, we check for common terms in the given expression to make the simplification easier.
In the given expression, $9{x^3} - x$ , we have $x$ common in both the terms so we can write $9{x^3} - x$ this as $ \Rightarrow x(9{x^2} - 1)$.
Now by looking at the expression we got in the above step, it is in the form of the difference of squares. That is $9{x^2} - 1$ is in the form of difference of square, so by making use of general form of difference of square given by: ${a^2} - {b^2} = (a - b)(a + b)$.
Now by comparing the general form with $9{x^2} - 1$ we can get ${a^2} = {(3x)^2}$ and ${b^2} = {1^2}$ . Therefore we can write as ${(3x)^2} - {1^2}$ , which gives $a = 3x$ and $b = 1$.
Now by substituting this in the above equation, we get
${(3x)^2} - {1^2} = (3x - 1)(3x + 1)$ which are the factors of $9{x^2} - 1$.
Now in order to find the factors of $x$ we can equate the equation to zero that is $x(9{x^2} - 1) = 0$. Therefore, we can write as $x(3x - 1)(3x + 1) = 0$.
Now to find the factors separately we need to equate the expression to zero, as below
 $x = 0$
 $3x - 1 = 0$
 $3x + 1 = 0$
 $ \Rightarrow x = \dfrac{1}{3}$
$ \Rightarrow x = - \dfrac{1}{3}$
 $ \Rightarrow x = 0.333$
$ \Rightarrow x = - 0.333$

Hence the factors of $9{x^3} - x$ are $0$, $0.333$ and $ - 0.333$.

Note: Whenever we have this type of problem, first take out the common terms so that we can make use of any general formula for simplification. Here we are having a difference of square method to simplify taking the correct value for $a$ and $b$ so that you will get the correct answer.