Question

The factors of \[{{a}^{2}}-\dfrac{1}{{{a}^{2}}}\] are
(A) \[\left( a+\dfrac{1}{a} \right)\left( a+\dfrac{1}{a} \right)\]
(B) \[\left( a-\dfrac{1}{a} \right)\left( a-\dfrac{1}{a} \right)\]
(C) \[\left( a-\dfrac{1}{a} \right)\left( a+\dfrac{1}{a} \right)\]
(D) None of these

Answer 1

Hint: We are given an expression and we are asked to find the factors of the given expression from the given options. We can see that the given expression is of the form, \[{{a}^{2}}-{{b}^{2}}\]. So, we can use the algebraic formula here, which is, \[{{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)\]. We will look through all the given options and choose the most appropriate one, one that closely resembles the above algebraic formula.

Complete step by step answer:
According to the given question, we are given an expression and we are asked to find the factors from the given options.
The expression that we have is,
\[{{a}^{2}}-\dfrac{1}{{{a}^{2}}}\]
If we look carefully at the above expression, we can see that the expression is of the form, \[{{a}^{2}}-{{b}^{2}}\]. We know one of the algebraic formula of the similar form for the same expression, which is, \[{{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)\]. So, we will use this algebraic identity to find the factors of the given expression. We will get the factors as follows,
\[\Rightarrow \left( a-\dfrac{1}{a} \right)\left( a+\dfrac{1}{a} \right)\]

So, the correct answer is “Option C”.

Note: The other given options cannot be the factors of the given expression. If we expand the answer, we will get the expression itself and it is as follows,
\[\left( a-\dfrac{1}{a} \right)\left( a+\dfrac{1}{a} \right)\]
We will now multiply the terms and we will get the new expression as,
\[a\left( a+\dfrac{1}{a} \right)-\dfrac{1}{a}\left( a+\dfrac{1}{a} \right)\]
We get the product of the terms as follows and keeping the signs correct as well, we get,
\[\Rightarrow {{a}^{2}}+1-1-\dfrac{1}{{{a}^{2}}}\]
We will now cancel out the common terms and we have the expression as,
\[\Rightarrow {{a}^{2}}-\dfrac{1}{{{a}^{2}}}\]
which is the original expression given to us in the question.
Therefore, we have the correct answer.