We present some results concerning the unirationality of the algebraic variety ${}_f$ given by the equation $N_{K/k}(X₁+\alpha X₂+\alpha ²X₃)=f\left(t\right)$, where k is a number field, K=k(α), α is a root of an irreducible polynomial h(x) = x³ + ax + b ∈ k[x] and f ∈ k[t]. We are mainly interested in the case of pure cubic extensions, i.e. a = 0 and b ∈ k∖k³. We prove that if deg f = 4 and ${}_f$ contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then ${}_f$ is k-unirational. A similar result is proved for a broad family of quintic polynomials f satisfying...