Rational root theorem

The theorem on rational zeros (also rational zeros test or lemma of Gauss ) is a statement of the rational zeros of integral polynomials. It includes a necessary criterion for the existence of a rational zero, while delivering a finite set of rational numbers, in which all rational zeros must be included.

Statement

For every rational zero of an integer polynomial that their counter the absolute term and its denominator must divide the leading coefficient of the polynomial.

So be a polynomial of degree with a rational zero point ( which are relatively prime ), then is divisible by and divisible.

Comments

When the leading coefficient of the polynomial has the value 1, then any rational zero is an integer, dividing the absolute term.

The theorem can also be used to calculate the rational zeros of rational polynomials. Because if you multiply a rational polynomial with a common multiple of the denominators of its coefficients, we obtain an integer polynomial with identical zeros, for their determination one can now apply the rational zeros test.

The theorem on rational zeros arises as a corollary to going back to Gauss general statement about polynomials over the field of fractions of a factorial ring ( see Lemma of Gauss ).

Examples

The polynomial has no rational root, since 1 and -1 are the only divisor of the absolute link and and is.

From the rational polynomial is obtained by multiplying by 30 the integer polynomial. Its rational zeros must then be contained in the set. You Verifies now all these candidates by insertion into or, one given by the zeros, and 1. Since a polynomial of degree 3 up to 3 pairs can have different zeros, there are also no other irrational zeros in this case.