Let $P,Q,R$ be polynomials and let $S(x) = P(x^3) + xQ(x^3) + x^2R(x^3)$ be a polynomial of degree $n$ whose roots $x_1,\ldots, x_n$ are distinct. Construct with the aid of the polynomials $P,Q,R$ a polynomial $T$ of degree $n$ that has the roots $x_1^3 , x_2^3 , \ldots, x_n^3.$
1. Let \( P(x), Q(x), R(x) \) be polynomials and consider the polynomial \( S(x) = P(x^3) + xQ(x^3) + x^2R(x^3) \). We are given that \( S(x) \) is a polynomial of degree \( n \) with distinct roots \( x_1, x_2, \ldots, x_n \).

2. We need to construct a polynomial \( T(x) \) of degree \( n \) that has the roots \( x_1^3, x_2^3, \ldots, x_n^3 \).

3. Let \( 1, j, j^2 \) be the three distinct complex roots of \( x^3 = 1 \), where \( j = e^{2\pi i / 3} \) and \( j^2 = e^{4\pi i / 3} \).

4. Consider the product \( S(x)S(jx)S(j^2x) \):
   \[
   S(x)S(jx)S(j^2x) = \left( P(x^3) + xQ(x^3) + x^2R(x^3) \right) \left( P((jx)^3) + jxQ((jx)^3) + j^2x^2R((jx)^3) \right) \left( P((j^2x)^3) + j^2xQ((j^2x)^3) + jx^2R((j^2x)^3) \right)
   \]

5. Simplify the expressions inside the product:
   \[
   S(jx) = P((jx)^3) + jxQ((jx)^3) + j^2x^2R((jx)^3) = P(x^3) + jxQ(x^3) + j^2x^2R(x^3)
   \]
   \[
   S(j^2x) = P((j^2x)^3) + j^2xQ((j^2x)^3) + jx^2R((j^2x)^3) = P(x^3) + j^2xQ(x^3) + jx^2R(x^3)
   \]

6. Therefore, the product becomes:
   \[
   S(x)S(jx)S(j^2x) = \left( P(x^3) + xQ(x^3) + x^2R(x^3) \right) \left( P(x^3) + jxQ(x^3) + j^2x^2R(x^3) \right) \left( P(x^3) + j^2xQ(x^3) + jx^2R(x^3) \right)
   \]

7. Notice that the product \( S(x)S(jx)S(j^2x) \) can be written as:
   \[
   S(x)S(jx)S(j^2x) = a^3 \prod_{i=1}^n (x - x_i)(jx - x_i)(j^2x - x_i)
   \]
   \[
   = a^3 \prod_{i=1}^n (x^3 - x_i^3)
   \]

8. Expanding the product on the right-hand side, we get:
   \[
   a^3 \prod_{i=1}^n (x^3 - x_i^3) = \left( P(x^3) \right)^3 + x^3 \left( Q(x^3) \right)^3 + x^6 \left( R(x^3) \right)^3 - 6x^3 P(x^3) Q(x^3) R(x^3)
   \]

9. Therefore, we can choose the polynomial \( T(x) \) as:
   \[
   T(x) = P(x)^3 + xQ(x)^3 + x^2R(x)^3 - 6xP(x)Q(x)R(x)
   \]

10. It is straightforward to verify that the degree of this polynomial is \( n \).

The final answer is \( \boxed{T(x) = P(x)^3 + xQ(x)^3 + x^2R(x)^3 - 6xP(x)Q(x)R(x)} \).