Question

Identify one of the factors of $(a^2-b^2)(c^2-d^2)-4abcd$

• A. $ac-bd+bc+ad$
• B. $ac-bd+bc-ad$
• C. Can't be determined
• D. None of these

Answer 1

The correct option is A $ac-bd+bc+ad$

We factorize the given expression $(a^2-b^2)(c^2-d^2)-4abcd$ as shown below:

$(a^2-b^2)(c^2-d^2)-4abcd=a^2{c}^2-a^2{d}^2-b^2{c}^2+b^2{d}^2-2abcd-2abcd=a^2{c}^2+b^2{d}^2-2abcd-a^2{d}^2-b^2{c}^2-2abcd=\left[\right(ac{)}^2+(bd{)}^2-2\times ac\times bd]-\left[\right(bc{)}^2+(ad{)}^2+2\times bc\times ad]=(ac-bd{)}^2-(bc+ad{)}^2(\because (x+y{)}^2=x^2+y^2+2xy,(x-y{)}^2=x^2+y^2-2xy)=(ac-bd+bc+ad)(ac-bd-bc-ad)(\because x^2-y^2=(x+y\left)\right(x-y\left)\right)$

Hence, $(ac-bd+bc+ad)$ is one of the factor of $(a^2-b^2)(c^2-d^2)-4abcd$.