Question

Factorize $ {x^3} + x - 3{x^2} - 3 $

Answer 1

Hint: In order to factorize the given equation, first we would separate the first two operands then taking out the common value, similarly, taking out common value from the other two operands, then checking if the brackets are same or not, if same then taking out a common bracket from the values and keeping rest in another bracket.

Complete step-by-step answer:
We are given the equation $ {x^3} + x - 3{x^2} - 3 $ .
Dividing the whole equation in part of two brackets, which consists of first two operands and second two operands, and they are:
 $ {x^3} + x - 3{x^2} - 3 = \left( {{x^3} + x} \right) - \left( {3{x^2} + 3} \right) $
Taking out the common values from the two brackets:
In first bracket, we can see that $ x $ is common in the value, so taking out that common, we get:
 $ \left( {{x^3} + x} \right) = x\left( {{x^2} + 1} \right) $
Similarly, in the second bracket $ - \left( {3{x^2} + 3} \right) $ we can see that $ 3 $ is common there, so taking out the common value:
 $ - \left( {3{x^2} + 3} \right) = - 3\left( {{x^2} + 1} \right) $
Since we can see that $ \left( {{x^2} + 1} \right) $ is common in both then we take only one bracket of the value and the remaining value in other bracket after combining the two separated operands, and we get:
 $
  {x^3} + x - 3{x^2} - 3 = \left( {{x^3} + x} \right) - \left( {3{x^2} + 3} \right) \\
  {x^3} + x - 3{x^2} - 3 = x\left( {{x^2} + 1} \right) - 3\left( {{x^2} + 1} \right) \\
  {x^3} + x - 3{x^2} - 3 = \left( {{x^2} + 1} \right)\left( {x - 3} \right) \;
  $
And, we get our factors for the equation and that are $ \left( {{x^2} + 1} \right) $ and $ \left( {x - 3} \right) $ .
Therefore, The factored form of $ {x^3} + x - 3{x^2} - 3 = \left( {{x^2} + 1} \right)\left( {x - 3} \right) $ .
So, the correct answer is “ $ \left( {{x^2} + 1} \right)\left( {x - 3} \right) $ .”.

Note: We could also factorize the equation by first organizing the operands in terms of highest to lowest degree, then taking common and factorizing further, would also give the same results as obtained. Let’s check for the same example $ {x^3} + x - 3{x^2} - 3 $ , organizing it in their highest to lowest degree and we get: $ {x^3} - 3{x^2} + x - 3 $ , then taking out common value $ {x^2} $ from first two values and $ 1 $ from the second two operands, and taking common brackets, we get:
 $
  {x^3} - 3{x^2} + x - 3 = {x^2}\left( {x - 3} \right) + \left( {x - 3} \right) \\
  {x^3} - 3{x^2} + x - 3 = \left( {{x^2} + 1} \right)\left( {x - 3} \right) \;
  $
Which is the same as the above obtained equation. So, it does not make any difference.