Question

One of the factors of \[\left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd\] is _______________.
A) \[\left( {ac - bd + bc + ad} \right)\]
B) \[ac + bd + bc + ad\]
C) Cannot be determined
D) \[ac + bd - bc - ad\]

Answer 1

Hint:
Here, we are required to find one of the factors of the given expression. Hence, we will first of all, open the brackets and then use various identities to solve this expression further. At the end, we will be able to find two factors of this given expression. On comparing any one of them with the given options, we will find the required factor of the given expression.

Formula Used:
We will use the following formulas
1) \[{\left( {a \pm b} \right)^2} = \left( {{a^2} \pm 2ab + {b^2}} \right)\]
2) \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\]

Complete Step by Step Solution:
Given expression is:
\[\left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd\]
Now, first of all we will open the brackets by multiplying each term of the first bracket by all the terms of the second bracket respectively.
\[ \Rightarrow \left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd = {a^2}\left( {{c^2} - {d^2}} \right) - {b^2}\left( {{c^2} - {d^2}} \right) - 4abcd\]
\[ \Rightarrow \left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd = {a^2}{c^2} - {a^2}{d^2} - {b^2}{c^2} + {b^2}{d^2} - 4abcd\]
Now, we will try to make the identity \[\left( {{a^2} - 2ab + {b^2}} \right)\] wherever possible.
\[ \Rightarrow \left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd = \left[ {{{\left( {ac} \right)}^2} - 2abcd + {{\left( {bd} \right)}^2}} \right] - \left[ {{{\left( {ad} \right)}^2} + {{\left( {bd} \right)}^2} + 2abcd} \right]\]
Hence, using the identity, \[{\left( {a \pm b} \right)^2} = \left( {{a^2} \pm 2ab + {b^2}} \right)\], we can write the above expression as:
\[ \Rightarrow \left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd = {\left( {ac - bd} \right)^2} - {\left( {ad + bc} \right)^2}\]
Here, we will use the formula \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\].
Therefore, the above expression becomes:
\[ \Rightarrow \left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd = \left( {ac - bd - ad - bc} \right)\left( {ac - bd + ad + bc} \right)\]
Hence, we can say that the factors of \[\left( {{a^2} - {b^2}} \right)\left( {{c^2} - {d^2}} \right) - 4abcd\] are:
\[\left( {ac - bd - ad - bc} \right)\] and \[\left( {ac - bd + ad + bc} \right)\]
Therefore, one of the factors of the given expression is \[\left( {ac - bd + bc + ad} \right)\]

Hence, option A is the correct answer.

Note:
Factors of a given number are the numbers, which when multiplied gives the original number. When we are given any expression involving sum or difference of its terms, then, expressing those terms as a product of different terms is known as factorization. The greatest factor that is common to all the terms is known as the highest common factor or HCF whereas; a prime expression cannot be factorized because it is already in its simplest form.