We prove that every JB∗-triple E (in particular, every C∗-
algebra) satisfying the Daugavet property also satisfies the
stronger polynomial Daugavet property, that is, every weakly
compact polynomial P : E −→ E satisfies the Daugavet
equation IdX +P = 1 + P . The analogous conclusion
also holds for the alternative Daugavet property