(also called Rational Zeros Theorem)
Theorem
Let there be a
polynomial P(x) = c
0x
0 + c
1x
1 + ... + c
nx
n with
integer coefficients (in other words, the polynomial is an
integer polynomial, or belongs to
Z[x]). If there exists a
rational root p/q in
lowest terms, then p|c
0 and q
|c
n.
Proof
c
0x
0 + c
1x
1 + ... + c
nx
n = 0
c
0(p/q)
0 + c
1(p/q)
1 + ... + c
n(p/q)
n = 0
c
0p
0q
n + c
1p
1q
n-1 + ... + c
np
nq
0 = 0 * q
n
c
0p
0q
n + c
1p
1q
n-1 + ... + c
n-1p
n-1q
1 = -c
np
n
q divides left hand side, therefore q|c
np
n.
By a corollary of
Euclid's First Theorem, q|c
np
n ⇒ q|c
n.
Similarly it can be shown that p|c
0.
QED
This also proves
the square root of any positive integer is either integral or irrational as a special case.