<s>[INST] The polynomial $ x^7 + x^2 +1$ is divisible by (A) $ x^5 - x^4 + x^2 -x +1 \quad$ (B) $ x^5 + x^4 +1 \quad$ (C) $ x^5 + x^4 + x^2 +x +1\quad$
(D) $ x^5 - x^4 + x^2 +x +1$
My effort: Looking at the polynomial, I know that it will have only one real root, which is negative. All other 6 roots should be imaginary. And that's how it is in the four options as well. Without actually dividing each polynomials in the options, which looks a tedious way, is there a method for finding it?
 [/INST] $$x^7+x^2+1=x^7-x+x^2+x+1=$$
$$=(x^2+x+1)(x(x-1)(x^3+1)+1)=$$
$$=(x^2+x+1)(x^5-x^3+x^2-x+1).$$
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