(also called Rational Zeros Theorem)

__Theorem__
Let there be a

polynomial P(x) = c

_{0}x

^{0} + c

_{1}x

^{1} + ... + c

_{n}x

^{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

_{0}x

^{0} + c

_{1}x

^{1} + ... + c

_{n}x

^{n} = 0

c

_{0}(p/q)

^{0} + c

_{1}(p/q)

^{1} + ... + c

_{n}(p/q)

^{n} = 0

c

_{0}p

^{0}q

^{n} + c

_{1}p

^{1}q

^{n-1} + ... + c

_{n}p

^{n}q

^{0} = 0 * q

^{n}
c

_{0}p

^{0}q

^{n} + c

_{1}p

^{1}q

^{n-1} + ... + c

_{n-1}p

^{n-1}q

^{1} = -c

_{n}p

^{n}
q divides left hand side, therefore q|c

_{n}p

^{n}.

By a corollary of

Euclid's First Theorem, q|c

_{n}p

^{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.