If the Coefficients of the Polynomial
![\begin{displaymath}
d_nx^n+d_{n-1}x^{n-1}+\ldots +d_0=0
\end{displaymath}](p2_1513.gif) |
(1) |
are specified to be Integers, then integral Roots must have a Numerator which is a
factor of
and a Denominator which is a factor of
(with either sign possible). This follows since a
Polynomial of Order
with
integral Roots can be expressed as
![\begin{displaymath}
(a_1x+b_1)(a_2x+b_2)\cdots(a_kx+b_k)(c_{n-k}x^{n-k}+\ldots+c_0) = 0,
\end{displaymath}](p2_1516.gif) |
(2) |
where the Roots are
,
, ..., and
. Factoring out the
s,
![\begin{displaymath}
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.
\end{displaymath}](p2_1520.gif) |
(3) |
Now, multiplying through,
![\begin{displaymath}
a_1a_2\cdots a_k c_{n-k} x^n+\ldots+b_1b_2 \cdots b_k c_0=0,
\end{displaymath}](p2_1521.gif) |
(4) |
where we have not bothered with the other terms. Since the first and last Coefficients are
and
, all the integral roots of (1) are of the form [factors of
]/[factors of
].
© 1996-9 Eric W. Weisstein
1999-05-25