<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] Let $f(x) = x^3-3x^2+4x-1 $ Because the $ 9^\text{th}\ $ degree polynomial only have terms of multiples of $ 3\ $. I consider that it is the root to $ f \left ( \sqrt[3]{x} \right ) = 0 $
We have
$ \begin{eqnarray}
f \left ( \sqrt[3]{x} \right ) & = & 0 \\
x - 3x^{2/3} + 4x^{1/3} - 1 & = & 0 \\
(x-1) & = & x^{1/3} \cdot (3x^{1/3} - 4) \\
(x-1)^3 & = & x( 27x - 64 - 36x^{1/3} (3x^{1/3} - 4)) \\
& = & x(27x- 64 - 36(x-1)) \\
x^3 + 6x^2 + 31x - 1 & = & 0 \\
\end{eqnarray} $
Replace $ x $ with $ x^3 $ and compare, we have 

$ p = 6, q = 31, r = -1 $

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