Question : $\left(4 x^3 y-6 x^2 y^2+4 x y^3-y^4\right)$ can be expressed as:
Option 1: $(x+y)^4-x^4$
Option 2: $(x+y)^4-y^4$
Option 3: $(x-y)^4-x^4$
Option 4: $x^4-(x-y)^4$
Correct Answer: $x^4-(x-y)^4$
Solution :
$\left(4 x^3 y-6 x^2 y^2+4 x y^3-y^4\right)$
$=2xy^3 + 2xy^3 - 2x^2y^2 - 4x^2y^2 + 4x^3y- y^4 $
$=2xy^3 - 2x^2y^2 - y^4 + 4x^3y + 2xy^3 - 4x^2y^2$
$=-y^2(-2xy + 2x^2 + y^2) + 2xy (2x^2 + y^2 - 2xy)$
$=(2xy - y^2) (2x^2 + y^2 - 2xy)$
$=(x^2 - x^2 - y^2 + 2xy) (x^2 + x^2 + y^2 - 2xy)$
$=[x^2 - (x - y)^2] [x^2 + (x - y)^2]$
$=x^4 - (x - y)^4$
Hence, the correct answer is $x^4 - (x - y)^4$.
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