Question

Find the sum of all negative integral value of x satisfying \[\left( {{x^2} - 3x + 2} \right)\left( {{x^3} - 3{x^2}} \right)\left( {4 - {x^2}} \right) \leqslant 0\]

Answer 1

Hint: The simplest method of solving this type of question is first simplifying the inequality given to its simplest form. So try to first simplify or factorize the above inequality. After this step finding the negative integral value of x will be easy and then add them to get the final answer as asked in the question.

Complete step-by-step answer:
We will first write down the above given inequality
 \[\left( {{x^2} - 3x + 2} \right)\left( {{x^3} - 3{x^2}} \right)\left( {4 - {x^2}} \right) \leqslant 0\] and try to simplify it.
 \[\left( {{x^2} - 3x + 2} \right) = \left( {x - 1} \right)\left( {x - 2} \right)\]
 \[\left( {{x^3} - 3{x^2}} \right) = {x^2}\left( {x - 3} \right)\]
 \[\left( {4 - {x^2}} \right) = - \left( {x + 2} \right)\left( {x - 2} \right)\]
Substituting all the simplified values in the inequality given to us we get
 \[
  \left( {{x^2} - 3x + 2} \right)\left( {{x^3} - 3{x^2}} \right)\left( {4 - {x^2}} \right) \leqslant 0 \\
   \Rightarrow - {x^2}\left( {x - 1} \right){\left( {x - 2} \right)^2}\left( {x - 3} \right)\left( {x + 2} \right) \leqslant 0 \\
   \Rightarrow {x^2}\left( {x - 1} \right){\left( {x - 2} \right)^2}\left( {x - 3} \right)\left( {x + 2} \right) \geqslant 0 \;
 \]
The value x takes for this above inequality to satisfy will be x=0, -2, 1, 2, 3.
The negative integral values of x will be -2.
Therefore the sum of the integral value will be = -2
the sum of all negative integral value of x satisfying \[\left( {{x^2} - 3x + 2} \right)\left( {{x^3} - 3{x^2}} \right)\left( {4 - {x^2}} \right) \leqslant 0\] will be -2 respectively.
So, the correct answer is “-2”.

Note: A generalized question to find all the integral values of x can also be asked. The basic method to solve all these questions is to satisfy the inequality given. The questions can be in any forms. Special care has to be taken while solving this kind of question when there is a denominator involved.