Question

What is factor of \[{m^3} - {m^2} - m + 4\]

Answer 1

Hint: This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. Also, we need to know the basic algebraic formulae to make easy calculations. Also, we need to know how to find the common term between the two terms. We need to know how to perform arithmetic operations with the involvement of different sign terms.

Complete step by step solution:
The given expression is shown below,
 \[{m^3} - {m^2} - m + 4 \to \left( 1 \right)\]
The above equation can also be written as,
 \[{m^3} - {m^2} - m + 1 + 3 \to \left( 2 \right)\] (By changing the term \[4 = 3 + 1\] we can make easy calculation)
In the above equation, we can find \[{m^2}\] is commonly present in the first two terms of the above equation. So, we get
 \[ \Rightarrow \] \[{m^2}\left( {m - 1} \right) - m + 1 + 3\]
In the above equation, we can find \[ - 1\] is commonly present in the third and fourth terms of the above equation. So, we get
 \[ \Rightarrow \] \[{m^2}\left( {m - 1} \right) - 1\left( {m - 1} \right) + 3\]
In the above equation, we have the common term \[m - 1\] , so we get
 \[ \Rightarrow \] \[\left( {m - 1} \right)\left( {{m^2} - 1} \right) + 3\] \[ \to \left( 3 \right)\]
We know that,
 \[\left( {{m^2} - 1} \right) = \left( {{m^2} - {1^2}} \right)\]
The above expression is in the form of,
 \[\left( {{a^2} - {b^2}} \right)\]
By using the algebraic formula,
 \[\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)\]
So, we get
 \[\left( {{m^2} - {1^2}} \right) = \left( {{m^2} - 1} \right) = \left( {m - 1} \right)\left( {m + 1} \right)\]
Let’s substitute the above equation in the equation \[\left( 3 \right)\] , we get
 \[\left( 3 \right) \to \left( {m - 1} \right)\left( {{m^2} - 1} \right) + 3\]
 \[ \Rightarrow \left( {m - 1} \right)\left( {m - 1} \right)\left( {m + 1} \right) + 3\]
We know that \[\left( {m - 1} \right)\left( {m - 1} \right) = {\left( {m - 1} \right)^2}\] , so the above equation can also be written as,
 \[ \Rightarrow {\left( {m - 1} \right)^2}\left( {m + 1} \right) + 3\]
So, the final answer is,
 \[{m^3} - {m^2} - m + 4 = {\left( {m - 1} \right)^2}\left( {m + 1} \right) + 3\]
So, the correct answer is “$ {\left( {m - 1} \right)^2}\left( {m + 1} \right) + 3 $”.

Note: Note that to solve these types of questions we would compare the given equation with basic algebraic formulae and we try to make or find the common term between two or more terms to make easy calculations. Also, these types of questions describe the arithmetic operations like addition/ subtraction/ multiplication/ division. Also, note that if the same term is involved in multiplication we can write the two-term as a single term by Put Square with the term \[\left( {n \cdot n = {n^2}} \right)\] .