Determine if the polynomial f(x) = $2x^5 + 5x^4 + 4x^3 + 7x^2 + 7x + 2$ is reducible to $\mathbb Q $

Question

Determine if the polynomial $f (x) = 2x ^ 5 + 5x ^ 4 + 4x ^ 3 + 7x ^ 2 + 7x + 2 $ is reducible in $ \mathbb Q $. In that case, write it as a product of irreducible polynomials.

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Using Eisenstein's criterion we have $ a_0 = a_5 = 2 $ so there is no prime that divides $a_0$ and does not divide $a_5$, so $f(x)$ is not irreducible.

Since all coefficients are integers, we apply the rational zeros theorem

The trailing coefficient is $2$ so $p$ can assume the following values: $1,-1, 2, -2$.

The leading coefficient is $2$ so $q$ can assume the following values: $1,-1, 2, -2$.

So $\frac{p}{q}$ can assume: $\frac{1}{1}$, $-\frac{1}{1}$, $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{2}{1}$, $-\frac{2}{1}$, $\frac{2}{2}$, $-\frac{2}{2}$

remove duplicates:$\ 1, -1, 2, -2, \frac{1}{2}, -\frac{1}{2}$

If $a$ is a root of the polynomial $P(x)$, then the remainder from the division of $P(x)$ by $x−a$ should equal $0$. What happens in the case of $-\frac{1}{2}$ and we have: divide $2x^4+x^3+2x^2+3x+1$ by $x + \frac{1}{2}$.

The quotient is $2x^3+2x+2$, and the remainder is $0$.

Therefore we have $2x^4+x^3+2x^2+3x+1=(x+\frac{1}{2})(2x^3+2x+2)$

And

$2x^5+5x^4+4x^3+7x^2+7x+2=(x+2)(2x+1)(x^3+x+1)$

That's what I did, any comment would be useful.

Thanks in advance.

Answer 1

You seem to be fully factoring the polynomial over $\Bbb{Q}$, which is a fair bit more work than simply determining irreducibility. From the fact that it has a root at $x = -1/2$, and therefore a factor of $x + 1/2$, you can already conclude that the polynomial is indeed reducible.

What you can't do, however, is use Eisenstein's criterion to show reducibility. Eisenstein's criterion is a sufficient, but not necessary condition for irreducibility over $\Bbb{Q}$. For example, $x^2 + x + 1$ fails Eisenstein's criterion, but is still irreducible over $\Bbb{Q}$.

Finally, there's a fair number of little issues with the write up, most of which can be chalked up to hastily typesetting your working. For example:

The leading coefficient is 2 so p can assume the following values: +1,-1, 2, -2.

should be

The leading coefficient is 2 so q can assume the following values: +1,-1, 2, -2.

More concerning is the following:

remove duplicates: +1, -1, $\frac{1}{2}$, $-\frac{1}{2}$

You seem to have misplaced $2$ and $-2$. That said, you did factor out $x + 2$ (though your working makes it unclear when that was), so I'm guessing these possible roots weren't completely forgotten.