(also called Rational Zeros Theorem)

Theorem
Let there be a polynomial P(x) = c0x0 + c1x1 + ... + cnxn 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|c0 and q|cn.

Proof
c0x0 + c1x1 + ... + cnxn = 0
c0(p/q)0 + c1(p/q)1 + ... + cn(p/q)n = 0
c0p0qn + c1p1qn-1 + ... + cnpnq0 = 0 * qn
c0p0qn + c1p1qn-1 + ... + cn-1pn-1q1 = -cnpn
q divides left hand side, therefore q|cnpn.
By a corollary of Euclid's First Theorem, q|cnpn ⇒ q|cn.
Similarly it can be shown that p|c0.
QED

This also proves the square root of any positive integer is either integral or irrational as a special case.

Log in or registerto write something here or to contact authors.