Question

Factors for $a^2{b}^2+c^2{d}^2-a^2{c}^2-b^2{d}^2$ are:

• A. $(a+d)(a-d)(b+c)(b-c)$
• B. $(a^2-b^2)$
• C. $(a^2-b^2)(b^2-c^2)$
• D. $(a^2-b^2)(b^2+c^2)$

Answer 1

The correct option is A

$(a+d)(a-d)(b+c)(b-c)$

Consider the given expression,

$a^2{b}^2+c^2{d}^2-a^2{c}^2-b^2{d}^2$......(i)

Rearranging the terms of expression (i), we get
$=(a^2{b}^2-a^2{c}^2)+(c^2{d}^2-b^2{d}^2)$.....(ii)

Taking $a^2$ common from first two terms and $-d^2$ from last two terms of expression (ii), we get
$=a^2(b^2-c^2)-d^2(b^2-c^2)$....(iii)

Taking $(b^2-c^2)$ common from expression (iii), we get
$=(b^2-c^2)(a^2-d^2)$.....(iv)

Using the identity $a^2-b^2=(a+b)(a-b)$ expression (iv) becomes,
$=(b+c)(b-c)(a+d)(a-d)$

Therefore, the factors of $a^2{b}^2+c^2{d}^2-a^2{c}^2-b^2{d}^2$ are $(a+d)(a-d)(b+c)$ and $(b-c)$.

i.e., $a^2{b}^2+c^2{d}^2-a^2{c}^2-b^2{d}^2$$=(b+c)(b-c)(a+d)(a-d)$