A) Only \[(x+y)\]
B) Only \[({{x}^{3}}+{{y}^{3}}+{{z}^{3}})\]
C) Both \[(x+y)\]and \[({{x}^{3}}+{{y}^{3}}+{{z}^{3}})\]
D) None of the above
Correct Answer: C
Solution :
\[\begin{align} & \therefore \,\,\,\,\,{{x}^{4}}+x{{y}^{3}}+{{x}^{3}}y+x{{z}^{3}}+{{y}^{4}}+y{{z}^{3}} \\ & =\,\,\,\,\,({{x}^{4}}+x{{y}^{3}}+x{{z}^{3}})+({{x}^{3}}y+{{y}^{4}}+y{{z}^{3}}) \\ & =\,\,\,\,\,x({{x}^{3}}+{{y}^{3}}+{{z}^{3}})+y({{x}^{3}}+{{y}^{3}}+{{z}^{3}}) \\ & =\,\,\,\,\,(x+y)({{x}^{3}}+{{y}^{3}}+{{z}^{3}}) \\ \end{align}\] \[\therefore \] Which is divisible by both (x + y) and \[\]
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