Are the roots of polynomials "with almost integer" roots irrational?

Question

Ok, this may be not the most clear title, but my question is straightforward.

Say we choose $n$ integers $\{ z_1,\dots,z_n \}$ and we construct the polynomial $$ P(x) = \prod_{i=1}^n (x-z_i). $$

Are the all the roots of the polynomial $P(x) + 1$ irrational?

Answer 1

No.

Consider $P(x)=x^2-2x=(x-0)(x-2)$.

Then $P(x)+1=x^2-2x+1=(x-1)(x-1)$.

Answer 2

No. Let $n=2, z_1=0$ and $z_2=2.$ Then $P(x)=x(x-2)=x^2-2x$, thus $P(x)+1=(x-1)^2.$