In this paper we prove the following result: Let R be a prime ring with char(R)≠2,3,5\documentclass[12pt]{minimal}
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\begin{document}$${\text {char}}(R)\ne 2,3,5$$\end{document} and let T:R→R\documentclass[12pt]{minimal}
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\begin{document}$$ T :R \rightarrow R$$\end{document} be an additive mapping satisfying the relation 3T(x4)=T(x)x3+xT(x2)x+x3T(x)\documentclass[12pt]{minimal}
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\begin{document}$$ 3T(x^{4})=T(x)x^{3}+xT(x^2)x+x^{3}T(x)$$\end{document} for all x∈R\documentclass[12pt]{minimal}
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\begin{document}$$x\in R$$\end{document}. In this case T is of the form T(x)=λx\documentclass[12pt]{minimal}
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\begin{document}$$T(x)=\lambda x$$\end{document} for all x∈R\documentclass[12pt]{minimal}
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\begin{document}$$x\in R$$\end{document} and some fixed element λ∈C\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \in C$$\end{document}, where C is the extended centroid of R.
