Find the G.C.D. of 4x^4 + y^4,2x^3 - xy^2 - y | vedclass

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Question

(A) Let the expressions be $E_1 = 4x^4 + y^4$,$E_2 = 2x^3 - xy^2 - y^3$,and $E_3 = 2x^2 + 2xy + y^2$.For $E_1 = 4x^4 + y^4$,we can complete the square

Vedclass · Accepted Answer

$2x^2 + 2xy + y^2$

Vedclass · Answer

(A) Let the expressions be $E_1 = 4x^4 + y^4$,$E_2 = 2x^3 - xy^2 - y^3$,and $E_3 = 2x^2 + 2xy + y^2$.For $E_1 = 4x^4 + y^4$,we can complete the square: $4x^4 + y^4 = (2x^2 + y^2)^2 - 4x^2y^2 = (2x^2 + y^2 - 2xy)(2x^2 + y^2 + 2xy)$.For $E_2 = 2x^3 - xy^2 - y^3$,we can rewrite it as $2x^3 - 2y^3 + y^3 - xy^2 - y^3 = 2(x^3 - y^3) - y^2(x - y) = 2(x - y)(x^2 + xy + y^2) - y^2(x - y) = (x - y)(2x^2 + 2xy + 2y^2 - y^2) = (x - y)(2x^2 + 2xy + y^2)$.For $E_3 = 2x^2 + 2xy + y^2$,it is already in its simplest form.The common factor present in all three expressions is $(2x^2 + 2xy + y^2)$.Therefore,the $G.C.D.$ is $2x^2 + 2xy + y^2$.

Find the $G.C.D.$ of $4x^4 + y^4$,$2x^3 - xy^2 - y^3$,and $2x^2 + 2xy + y^2$.

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