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 Math Help > Basic Algebra > Polynomial > Rational Root

Theorem: Rational roots of a monic polynomial with integer coefficients are inetegers.

Let f(x) be a monic polynomial given by

f(x) = a0 + a1x + a2x2 + a3x3 + ... + an-1xn-1 + xn.

If ξ is a rational root of the equation f(x)=0, then ξ is an integer.

Proof: If a0=0, then 0 is a root of f, which satisfies this theorem.  In that case, f(x) = x·g(x), where g(x), given by

g(x) = a1 + a2x + a3x2 + ... + an-1xn- + xn-1,

has all the same roots as f(x) except possibly 0.  If a1=0, then the process can be repeated, stripping off zero-roots, until we find a constant term not equal to zero.

Now we assume a0 ≠ 0.  The root, ξ, is rational, so we can let ξ=r/s, where GCD(r,s)=1.  Substituting this root in place of x, and multiplying through by sn,

a0sn + a1rsn-1 + a2r2sn-2 + ... + an-1rn-1s + rn = 0

For an arbitrary prime, p, if s were divisible by p, then every term except possibly rn would be divisible by p, so rn would have to be divisible by p as well.  Hence p|r, contradicting GCD(r,s)=1, and so the theorem is proved.

### Application of Rational Root Theorem

Theorem: If n and N are natural numbers, and if N is not the nth power of a natural number, then the nth root of N is irrational.

Proof: The nth root of N is a root of the polynomial,

f(x) = xn − N

Suppose for contradiction that ξ is a rational root of f.  The rational root theorem tells us that ξ is a an integer, which means N = ξn , a contradiction.

### Related pages in this website

Synthetic Division

Polynomial Division

Polynomial Remainder Theorem

Formal Power Series

Generating Function

The webmaster and author of this Math Help site is Graeme McRae.