Japan
Suppose ^@ x - y = 1.^@ Find the value of ^@ x^4 - xy^3 - x^{ 3 }y - 3x^{ 2 }y + 3xy^{ 2 } + y^4 . ^@
Answer:
^@ 1 ^@
- Given ^@ x - y = 1 ^@
We need to find the value of ^@ x^4 - xy^3 - x^{ 3 }y - 3x^{ 2 }y + 3xy^{ 2 } + y^4 . ^@ - ^@ \begin{align} & x^4 - xy^3 - x^{ 3 }y - 3x^{ 2 }y + 3xy^{ 2 } + y^4 \\ = & (x^4 - x^3 y) - (xy^3 - y^4) -(3x^2 y - 3xy^2 ) && [\text{Rewriting the expression}] \\ = & x^3 (x - y) - y^3 (x - y) - 3xy(x - y) \\ = & (x - y)[x^3 - y^3 - 3xy] \\ = & (x - y)\left[(x - y)(x^2 + xy + y^2) - 3xy \right] && [\text{Using } a^3 - b^3 = (a - b)(a^2 + ab + b^2)] \\ = & (x^2 + xy + y^2) - 3xy && [\text{Given } x - y = 1] \\ = & x^2 - 2xy + y^2 \\ = & (x - y)^2 && [\text{Using } (a - b)^2 = a^2 - 2ab + b^2] \\ = & 1 && [\text{Given } x - y = 1] \end{align} ^@
- Hence, the value of the given expression is ^@ 1 ^@.
