<s>[INST] factor $x^9 + 243x^3 + 729$ Factor the polynomial $$x^{9} + 243x^{3} + 729$$
it might be helpful to see it like this $$x^{9}+3^{5}x^{3}+3^{6}$$
I would imagine this being done without a calculator, but I don't see how to do it.
 [/INST] Observe that $x^9+243x^3+729$ = $(x^3+9)^3 - 27x^6$ = $(x^3+9)^3-(3x^2)^3$. 
Now using the formula for difference of cubes we obtain:
$(x^3+9)^3-(3x^2)^3$ = $(x^3+9-3x^2)((x^3+9)^2+(x^3+9)3x^2+(3x^2)^2)$ . 
Doing the calculations on the second parenthesis and reordering the terms we get that:
$x^9+243x^3+729 = (x^3 - 3 x^2 + 9) (x^6 + 3 x^5 + 9 x^4 + 18 x^3 + 27 x^2 + 81)$
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