I'm trying to solve this task:
Polynomial $p(x)$ with integer coefficients can be represented like this: $p(x) = (x – x_1)(x – x_2)(x – x_3)q(x) + 1$ where $q(x)$ is some polynomial with integer coefficients and $x_1, x_2, x_3$ – different whole numbers. Can $p(x)$ have integer roots?
My observations:
$$
p(x_1) = p(x_2) = p(x_3) = 1
$$
Also $p(x) = 1$ for any $x$ – root of $q(x)$.
Assuming $p(x_0) = 0$:
$$
q(x_0) = -\frac{1}{(x_0 – x_1)(x_0 – x_2)(x_0 – x_3)}
$$
I'm not really sure what to do next. Is Vieta's Formula useful here?
Could somebody please give a hint?
Thanks in advance.
Best Answer
You were almost there!
Continuing from where you left off, from the equation $$ q(x_0) = -\frac{1}{(x_0 - x_1)(x_0 - x_2)(x_0 - x_3)} $$ it follows that $$ -\frac{1}{(x_0 - x_1)(x_0 - x_2)(x_0 - x_3)} $$ is an integer, hence $$ (x_0 - x_1)(x_0 - x_2)(x_0 - x_3)\in\{\pm 1\} $$ but then $$ x_0 - x_1,x_0 - x_2,x_0 - x_3\in\{\pm 1\} $$ contradiction, since the distinctness of $x_1,x_2,x_3$ implies the distinctness of $x_0 - x_1,x_0 - x_2,x_0 - x_3$.