augustin_p wrote: Determine the polynomial P(X) satisfying simoultaneously the conditions: a) The remainder obtained when dividing P(X) to the polynomial X^3 −2 is equal to the fourth power of quotient. b) P(−2) + P(2) = −34. I suppose you mean $P(x)\in\mathbb R[x]$. If so : First property implies that fourth power of quotient has a degree $<3$ (degree of $x^3-2$) and so has degree zero. So $P(x)=a(x^3-2)+a^4$ And $P(2)+P(-2)=-34$ is $2a^4-4a=-34$, with no real root. And so $\boxed{\text{No such polynomial}}$