<s>[INST] A cubic factor of a nine degree polynomial The polynomial $x^3−3x^2+4x−1$ is a factor of $x^9+px^6+qx^3+r$. Find $p+q+r$. How do I start with the problem?
 [/INST] As David wrote, the division must be done very carefully. To avoid it, we could use a real brute force approach writing $$\Delta=(x^9+px^6+qx^3+r)-(x^3−3x^2+4x−1)\sum_{i=0}^6 a_ix^i=0$$ This should give $$\Delta=(a_0+r)+(a_1-4 a_0) x+(3 a_0-4 a_1+a_2) x^2+ (-a_0+3 a_1-4
   a_2+a_3+q)x^3+(-a_1+3 a_2-4 a_3+a_4) x^4+(-a_2+3 a_3-4 a_4+a_5) x^5+
   (-a_3+3 a_4-4 a_5+a_6+p)x^6+(-a_4+3 a_5-4 a_6) x^7+(3 a_6-a_5) x^8+(1-a_6)
   x^9=0$$ Now, we cancel the coefficient of each power of $x$ starting from the highest. This will give $a_6=1,a_5=3,a_4=5,a_3=p+4,a_2=3p-5,a_1=5p-26,a_0=-r$. After this quite laborious task, what is left is $$\Delta= (4 p+q+r-54)x^3+(-17 p-3 r+99)x^2 + (5 p+4 r-26)x=0$$ and again we cancel the coefficients to get three linear equations for the three unknowns $p,q,r$ and the solutions as already given in the previous answers.
</s>