Question

How do expand and simplify \[\left( {{x}^{2}}-1 \right)\left( x-1 \right)\]?

Answer 1

Hint: For the given question we have been asked to expand and simplify the algebraic expression \[\left( {{x}^{2}}-1 \right)\left( x-1 \right)\]. So, to solve this question we first have to multiply each and every term on the right factor that is the second bracket factor with each term on the left factor that is with \[{{x}^{2}}\] and \[-1\]. After doing the multiplication process we will simplify and get the required answer to the given question.

Complete step by step solution:
Firstly, we will multiply the second factor with both the terms in the first factor. So, we get the expression reduced to as follows.
\[\Rightarrow \left( {{x}^{2}}-1 \right)\left( x-1 \right)\]
\[\Rightarrow {{x}^{2}}\left( x-1 \right)-1\left( x-1 \right)\]
Now, we will simplify the above expression using the basic operation in mathematics which is multiplication. So, the expression will be reduced as follows.
\[\Rightarrow {{x}^{2}}\left( x-1 \right)-1\left( x-1 \right)\]
Here we have to do the multiplication so carefully as the integer 1 has negative sign carrying with it. So, we will take the whole \[-1\] as a single unit and continue the further process of multiplication so that it makes the simplification more accurate and easier to understand.
So, we will get the expression after doing the above process as follows.
\[\Rightarrow {{x}^{2}}\left( x \right)-{{x}^{2}}\left( 1 \right)+\left( -1 \right)\left( x \right)+\left( -1 \right)\left( -1 \right)\]
Now, we will expand the above expression. So, the expression is simplified as follows.
\[\Rightarrow {{x}^{3}}-{{x}^{2}}-x+1\]
Therefore, the solution to the given question will be \[ {{x}^{3}}-{{x}^{2}}-x+1\].

Note:
Students must be careful in doing the calculations. Students must have good knowledge in the basic concept of mathematical operations like addition and multiplications. Students must not do simplification or calculation mistakes like if we not consider the negative sign for each and every term in multiplication process our solution will be \[ {{x}^{2}}\left( x \right)-{{x}^{2}}-x-1\] \[\Rightarrow {{x}^{3}}-{{x}^{2}}-x-1\] which is wrong. So, we must be careful in doing the calculations of this sort.