Simplify: (xy+yz)2−(xy−yz)2

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Question

Simplify:
${(x y + y z)}^{2} - {(x y - y z)}^{2}$

Options:

A .

{4 x y}^{2}

B .

4 x y^{2} z

C .

4 x z

D .

- 4 x y^{2} z

Answer: Option B
:
B

Using the identity,
${(a + b)}^{2} = a^{2} + b^{2} + 2 a b,$
we get,
${(x y + y z)}^{2} = x^{2} y^{2} + y^{2} z^{2} + 2 x z y^{2} . . . (1)$
Using the identity,
${(a - b)}^{2} = a^{2} + b^{2} - 2 a b$
we get,
${(x y - y z)}^{2} = x^{2} y^{2} + y^{2} z^{2} - 2 x z y^{2} . . . (2)$
Subtracting (2) from (1), we get

$(x^{2} y^{2} + y^{2} z^{2} + 2 x z y^{2}) - (x^{2} y^{2} + y^{2} z^{2} - 2 x z y^{2})$
$= 2 x z y^{2} + 2 x z y^{2}$
$= 4 x y^{2} z$

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