Polynomial Remainder Theorem

If the Coefficients of the Polynomial

$$d_nx^n+d_{n-1}x^{n-1}+\ldots +d_0=0$$ (1)

are specified to be Integers, then integral Roots must have a Numerator which is a factor of $d_0$ and a Denominator which is a factor of $d_n$ (with either sign possible). This follows since a Polynomial of Order $n$ with $k$ integral Roots can be expressed as

$$(a_1x+b_1)(a_2x+b_2)\cdots(a_kx+b_k)(c_{n-k}x^{n-k}+\ldots+c_0) = 0,$$ (2)

where the Roots are $x_1 = - b_1/a_1$, $x_2 = -b_2/a_2$, ..., and $x_k = -b_k/a_k$. Factoring out the $a_i$s,

$$a_1a_2\cdots a_k\left({x-{b_1\over a_1}}\right)\left({x-{b_2... ...left({x-{b_k\over a_k}}\right)(c_{n-k}x^{n-k}+\ldots+c_0) = 0.$$ (3)

Now, multiplying through,

$$a_1a_2\cdots a_k c_{n-k} x^n+\ldots+b_1b_2 \cdots b_k c_0=0,$$ (4)

where we have not bothered with the other terms. Since the first and last Coefficients are $d_n$ and $d_0$, all the integral roots of (1) are of the form [factors of $d_0$]/[factors of $d_n$].