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According to Casas Alvero conjecture, if a one variable polynomial of degree$n$ over a field of characteristic 0 is not prime with each of the $n-1$ firstderivees, then it is of the form $c (X-r)^{n}$. Let $p$ be a prime number andan integer $e$, the conjecture is showed to be true for polynomials of degree$p^{e}, 2p^{e}, 3p^{e} (p neq 2) and 4p^{e} (p neq 3,5, 7) $. In this work weshow that the conjecture is true for polynomials of degree $5p^{e} (p neq2,3,7,11,131,193,599,3541,8009) $ . It also corrects an error in Draisma andJong (2011) for the polynomials of degree $4p^{e}$  Selon la conjecture deCasas Alvero, si un polyn\^{o}me \'{a} une variable de degr\'{e} $n$ sur uncorps commutatif de caract\'{e}ristique 0 est non premier avec chacune de ses$n-1$ premi\'{e}res d\'{e}riv\'{e}s, alors il est de forme $c(X-r)^{n}$. Soient$p$ un nombre premier et $e$ un entier, la conjecture a \'{e}t\'{e}d\'{e}montr\'{e}e pour les polyn\^{o}mes de degr\'{e} $p^{e},2p^{e}, 3p^{e}(p\neq 2) et 4p^{e} (p\neq 3,5,7)$. Dans ce travail on montre que la conjectureest vrai pour les polyn\^{o}mes de degr\'{e} $5p^{e} (p\neq2,3,7,11,131,193,599,3541,8009)$. On corrige aussi une erreur dans Draisma etJong (2011) pour les degr\'{e} $4p^{e}$

La conjecture de Casas Alvero pour les degr\'es $5p^{e}$