The polynomial`x^2011 + x^1783 - 3x^1707 + 2x^341 + 3x^2 - 3` is divided by `x^3 - x` if both polynomials share common roots.

I recommend you to start by finding the roots of `x^3 - x` , since it is easy.

`x^3 - x = 0 =gt x*(x^2 - 1)= 0`

`` Use the formula of difference of squares for `x^2 - 1` .

`x*(x - 1)(x + 1)= 0`

`x_1 = 0; x - 1 = 0 =gt x_2 = 1; x+1 = 0 =gt x_3 = -1`

Substitute the roots `+-1` in the polynomial `x^2011 + x^1783 - 3x^1707 + 2x^341 + 3x^2 - 3.`

x = -1 => `(-1)^2011 + (-1)^1783 - 3*(-1)^1707 + 2*(-1)^341 + 3*(-1)^2 - 3` = -1 - 1 + 3 - 2 + 3 - 3 = -1 `!=` 0 => (x+1) does not divide the polynomial `x^2011 + x^1783 - 3x^1707 + 2x^341 + 3x^2 - 3.`

`x = 1 ` => `(1)^2011 + (1)^1783 - 3*(1)^1707 + 2*(1)^341 + 3*(1)^2 - 3 = 1+ 1- 3+ 2 + 3 - 3 = 1 != 0 =gt (x-1)` does not divide the polynomial `x^2011 + x^1783 - 3x^1707 + 2x^341 + 3x^2 - 3.`

Since the polynomial `x^2011 + x^1783 - 3x^1707 + 2x^341 + 3x^2 - 3` is not factorized by x, then it is not divided by x.

**Conclusion: Since x,(x-1) and (x+1) are not the divisors of the polynomial `x^2011 + x^1783 - 3x^1707 + 2x^341 + 3x^2 - 3` , then `x^3-x` does not divide the polynomial`x^2011 + x^1783 - 3x^1707 + 2x^341 + 3x^2 - 3.` **