Coefficient of the variable raised to the highest power in a polynomial with integer coefficients

Question

Let $P(x) = a_nx^n+a_{n-1}x^{n-1} + ... + a_1x + a_0 $. Then, if there exists an integer $c$, which is a root of the polynomial, does it mean that the $a_n$ coefficient is equal to 1?

Answer 1

HINT: if you set $P(x)=0$ and divide by $a_n$ you get

$$x^n+\frac{a_{n-1}}{a_n}x^{n-1} + ... + \frac{a_1}{a_n}x + \frac{a_0}{a_n} =0$$

From this you can easily get why, if there exist an integer root $c$, it is not necessary that $a_n$ is $1$.