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Wikipedia / Rational root theorem

In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or theorem) states a constraint on rational solutions of a polynomial equation with integer. Solutions of the equation are also called roots or zeros of the polynomial on the left side.

The theorem states that each rational solution written in lowest terms (that is, and are relatively prime), satisfies:

The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is.

Application

The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root is found, a linear polynomial can be factored out of the polynomial using polynomial long division, resulting in a polynomial of lower degree whose roots are also roots of the original polynomial.

Cubic equation

The general cubic equation with integer coefficients has three solutions in the complex plane. If the rational root test finds no rational solutions, then the only way to express the solutions algebraically uses cube roots. But if the test finds a rational solution , then factoring out leaves a quadratic polynomial whose two roots, found with the quadratic formula, are the remaining two roots of the cubic, avoiding cube roots.

Proofs

Elementary proof

Suppose for some coprime

To clear denominators, multiply both sides by

Shifting the term to the right side and factoring out

Thus, divides . But is coprime to and therefore to , so by Euclid's lemma must divide the remaining factor .

On the other hand, shifting the term to the right side and factoring out

Reasoning as before, it follows that divides .

Proof using Gauss's lemma

Should there be a nontrivial factor dividing all the coefficients of the polynomial, then one can divide by the greatest common divisor of the coefficients so as to obtain a primitive polynomial in the sense of Gauss's lemma; this does not alter the set of rational roots and only strengthens the divisibility conditions. That lemma says that if the polynomial factors in , then it also factors in as a product of primitive polynomials. Now any rational root corresponds to a factor of degree 1 in of the polynomial, and its primitive representative is then , assuming that and are coprime. But any multiple in of has leading term divisible by and constant term divisible by , which proves the statement. This argument shows that more generally, any irreducible factor of can be supposed to have integer coefficients, and leading and constant coefficients dividing the corresponding coefficients of.

Examples

First

; since neither of these equates the polynomial to zero, it has no rational roots.

Second

the only possible rational roots would have a numerator that divides 6 and a denominator that divides 1, limiting the possibilities to ±1, ±2, ±3, and ±6. Of these, 1, 2, and –3 equate the polynomial to zero, and hence are its rational roots (in fact these are its only roots since a cubic polynomial has only three roots).

Third

These 8 possible values for

However, these eight computations may be rather tedious, and some tricks allow avoiding some of them.

all terms of

So, is not a root. Moreover, if one sets is a polynomial in with the same first coefficient and constant term . The rational root theorem implies thus that a rational root of and thus that the rational roots of This shows again that any rational root of is positive, and the only remaining candidates are and .

To show that are multiples of is not. So, their sum cannot be zero.

needs to be computed to verify that it is a root of the polynomial.

Fourth

must be integer.

Consider the quadratic equation:

Simplify the coefficients:

where:

  • , obviously integer, as negation of an integer,

  • , also integer, as the product of two integers.

Apply the rational root theorem:

are integers.

See also

References

Further reading