How do you factor \[{{x}^{3}}-2x-21\]?

Hint: This question is from the topic of polynomials. In this question, first we will understand the theorem that is rational root theorem. After that, we will find out the zero or root of the equation \[{{x}^{3}}-2x-21\] by using the rational root theorem. After finding the root of the equation , then we will solve the further equation and get the factor of the equation \[{{x}^{3}}-2x-21\].

Complete step by step solution:
Let us solve this question.
In this question, we have asked to find the factor of the term or we can say the equation which is given in the question that is \[{{x}^{3}}-2x-21\].
For finding the factor, we will first find out the zero of the equation \[{{x}^{3}}-2x-21\] using the rational root theorem.
 The rational root theorem says that zeros or roots of any equation or function \[y={{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+{{a}_{2}}{{x}^{n-2}}.............+{{a}_{n-1}}x+{{a}_{n}}\] is expressed in the form of \[\dfrac{p}{q}\], where p is a factor of \[{{a}_{n}}\] and q is a factor of \[{{a}_{0}}\].
Hence, possible roots of the equation \[{{x}^{3}}-2x-21\] will in the form of \[\dfrac{p}{q}\], where p is a factor of -21 and q is a factor of 1.
So, we can say zeros or roots of the equation \[\dfrac{p}{q}\] will be \[\pm 1,\pm 3,\pm 7\]
Now, let us check what will be the exact root by putting the value of possible roots in the place of x in the equation \[{{x}^{3}}-2x-21\].
If at some value, the value of the equation is 0, then that value will be the root of the equation.
Let us check the value of equation \[{{x}^{3}}-2x-21\] at x=1.
\[{{1}^{3}}-2\times 1-21=1-2-21=-18\]
Hence, 1 is not the root of the given equation.
Let us check at -1.
\[{{\left( -1 \right)}^{3}}-2\times \left( -1 \right)-21=-1+2-21=-20\]
Hence, -1 is not the root of the given equation.
Let us check at x=3.
\[{{\left( 3 \right)}^{3}}-2\times \left( 3 \right)-21=27-6-21=0\]
So, now we can say that 3 is a root of the equation. Hence, we can say that (x-3) is a factor of equation \[{{x}^{3}}-2x-21\].
Therefore, we can write
\[{{x}^{3}}-2x-21=\left( x-3 \right)\left( {{x}^{2}}+3x+7 \right)\]
Now, we have found the factor of the equation \[{{x}^{3}}-2x-21\]. The factor is \[\left( x-3 \right)\left( {{x}^{2}}+3x+7 \right)\].

Note: As we can see that this question is from the topic of polynomials, so we should have a better knowledge in the topic of polynomials. We should know about rational root theorems. This theorem says that if we have an equation like \[y={{c}_{0}}{{x}^{n}}+{{c}_{1}}{{x}^{n-1}}+{{c}_{2}}{{x}^{n-2}}.............+{{c}_{n-1}}x+{{c}_{n}}\], then the possibilities of the root of the equation will be in the form of \[\dfrac{a}{b}\], where ‘a’ is the factor of the term \[{{c}_{n}}\] and ‘b’ is the factor of the term \[{{c}_{0}}\]. And always remember that if it has a root such as \[{{x}_{1}}\], then \[\left( x-{{x}_{1}} \right)\] is a factor of the equation.