<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] Polynomial $x^7+x^2+1$ has higher term coefficient $1$ and also constant term coefficient $1$.
Thus if we divide it by a polynomial of degree $5$ of the form $(x^5+\cdots+1)$ as $A,B,C,D$ proposition are, it will be $x^2+ax+1$, only the central coefficient is unknown, the other two are forced to $1$.
If you multiply this polynomial by $B$ or $C$ $\begin{cases}(x^2+ax+1)(x^5+x^4+1)\\(x^2+ax+1)(x^5+x^4+x^3+x^2+1)\end{cases}$
you can notice that there will be an unremovable (and not desired) term in $x^4$, thus these possibilities should be excluded.
So you have to decide between $A$ and $D$ $\begin{cases}(x^2+ax+1)(x^5-x^4+x^2-x+1)\\(x^2+ax+1)(x^5-x^4+x^2+x+1)\end{cases}$
In both case we have $(a-1)x^6$ therefore $a=1$ to get rid of it.
The term in $x^1$ is also easy to get $A:(a-1)x=0$ and $D:(a+1)x=2x$, and since there should be no term in $x$, then $D$ should be excluded.
You just need to verify that $A$ works via a final expansion, which is indeed the case.
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