How do you use the remainder theorem to evaluate $ f\left( x \right) = {x^3} + {x^2} - 5x - 6 $ at $ x = 2 $ ?

Hint: We are required to find the value of a function for a certain value of the variable. This question requires us to have the knowledge of basic and simple algebraic rules and operations such as substitution, addition, multiplication, subtraction and many more like these. A thorough understanding of functions and its applications can be of great significance.

Complete step-by-step answer:
In the given question, we are required to find the value of a function for a certain value of variable. The value of a function at a certain value of variable is found by substituting the value of variable as specified in the question into the function.
So, we need to replace the variable in the function given to us in the question by the value specified.
So, the function given to us is: $ f\left( x \right) = {x^3} + {x^2} - 5x - 6 $ .
We are required to find the value of $ f\left( 2 \right) $ by replacing the value of variable x in the function.
Hence, $ f\left( x \right) = {x^3} + {x^2} - 5x - 6 $
 $ \Rightarrow f\left( x \right) = {\left( 2 \right)^3} + {\left( 2 \right)^2} - 5\left( 2 \right) - 6 $
 $ \Rightarrow f\left( x \right) = 8 + 4 - 10 - 6 $
 $ \Rightarrow f\left( x \right) = 12 - 16 $
 $ \Rightarrow f\left( x \right) = - 4 $
Hence, we get the value of the function by replacing the variable x in the original function $ f\left( x \right) = {x^3} + {x^2} - 5x - 6 $ by the specified value, that is $ -4 $ .
So, the correct answer is “-4”.

Note: Such questions that require just simple change of variable can be solved easily by keeping in mind the algebraic rules such as substitution and transposition. Substitution of a variable involves putting a certain value in place of the variable. That specified value may be a certain number or even any other variable.