Let R be a prime ring of characteristic different from 2, Qr\documentclass[12pt]{minimal}
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\begin{document}$$Q_r$$\end{document} be its right Martindale quotient ring and C be its extended centroid, G be a nonzero X-generalized skew derivation of R, and S be the set of the evaluations of a multilinear polynomial f(x1,…,xn)\documentclass[12pt]{minimal}
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\begin{document}$$f(x_1,\ldots ,x_n)$$\end{document} over C with n non-commuting variables. Let u,v∈R\documentclass[12pt]{minimal}
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\begin{document}$$u,v \in R$$\end{document} be such that uG(x)x+G(x)xv=0\documentclass[12pt]{minimal}
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\begin{document}$$uG(x)x+G(x)xv=0$$\end{document} for all x∈S\documentclass[12pt]{minimal}
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\begin{document}$$x\in S$$\end{document}. Then one of the following statements holds:v∈C\documentclass[12pt]{minimal}
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\begin{document}$$v\in C$$\end{document} and there exist a,b,c∈Qr\documentclass[12pt]{minimal}
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\begin{document}$$a,b,c \in Q_r$$\end{document} such that G(x)=ax+bxc\documentclass[12pt]{minimal}
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\begin{document}$$G(x)=ax+bxc$$\end{document} for any x∈R\documentclass[12pt]{minimal}
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\begin{document}$$x\in R$$\end{document} with (u+v)a=(u+v)b=0\documentclass[12pt]{minimal}
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\begin{document}$$(u+v)a=(u+v)b=0$$\end{document};f(x1,…,xn)2\documentclass[12pt]{minimal}
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\begin{document}$$f(x_1,\ldots ,x_n)^2$$\end{document} is central-valued on R and there exists a∈Qr\documentclass[12pt]{minimal}
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\begin{document}$$a \in Q_r$$\end{document} such that G(x)=ax\documentclass[12pt]{minimal}
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\begin{document}$$G(x)=ax$$\end{document} for all x∈R\documentclass[12pt]{minimal}
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\begin{document}$$x\in R$$\end{document} with ua+av=0\documentclass[12pt]{minimal}
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\begin{document}$$ua+av=0$$\end{document}.
