Divide (x^2011+x^1783-3x^1707+2x^341+3x^2-3) by (x^3-x)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.`

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