Question

Identify one of the factors of \[\left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd\].
A. \[ac - bd + bc + ad\]
B. \[ac - bd + bc - ad\]
C. Can’t be determined
D. None of these

Answer 1

Hint: Here, in the given question, it is asked to identify one of the factors of the given problem\[\left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd\] out of the given options. So, to identify from the options, first we will have to find the factors. Simplify the problem, rearrange according to the requirement, and use the applicable identities for further simplification. We will have two factors, now the factors from the given options if there are any

Formula used:
\[{\left( {x + y} \right)^2} = {x^2} + {y^2} + 2xy\]
\[{\left( {x - y} \right)^2} = {x^2} + {y^2} - 2xy\]
\[{x^2} - {y^2} = \left( {x + y} \right)\left( {x - y} \right)\].

Complete step-by-step answer:
Given problem =\[\left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd\]
We are asked to identify one of the factors out of the given options. To identify, we simply have to find the factor and match the options.
To find the factors, first of all simplify the problem.
\[
  \left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd \\
   \Rightarrow {a^2}{c^2} - {a^2}{d^2} - {b^2}{c^2} + {b^2}{d^2} - 2abcd - 2abcd \\
 \]
Rearranging them, we get,
\[{a^2}{c^2} + {b^2}{d^2} - 2abcd - {a^2}{d^2} - {b^2}{c^2} - 2abcd\] \[\]
\[ \Rightarrow [{(ac)^2} + {(bd)^2} - 2 \times ac \times bd] - [{(bc)^2} + {(ad)^2} + 2 \times bc \times ad]\]
Using the identities, \[{\left( {x + y} \right)^2} = {x^2} + {y^2} + 2xy\]and\[{\left( {x - y} \right)^2} = {x^2} + {y^2} - 2xy\], we obtain,
\[{\left( {ac - bd} \right)^2} - {\left( {bc + ad} \right)^2}\]
\[\]Using the identity, \[{x^2} - {y^2} = \left( {x + y} \right)\left( {x - y} \right)\], we obtain,
\[\left( {ac - bd + bc + ad} \right)\left( {ac - bd - bc - ad} \right)\]
Now, we have two factors, and from the given options we have, \[\left( {ac - bd + bc + ad} \right)\]

So, the correct answer is “Option A”.

Note: In such types of questions, it is very important to have the knowledge about the basic formulae and identities. Also, one equally important thing is to know in which direction the solution should go. As in the given question, rearranging was important to have a different view of the solution. In most of the cases, it would be our job to make the solution worth using any formula or identity. Otherwise, one can be stuck in between the solution and may leave it halfway.