In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation
The theorem states that each rational solution x = p⁄q, written in lowest terms so that p and q are relatively prime, satisfies:
- p is an integer factor of the constant term a0, and
- q is an integer factor of the leading coefficient an.
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 an = 1.
Application
editThe 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 x = r is found, a linear polynomial (x – r) 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
editThe general cubic equation
Proofs
editElementary proof
editLet with
Suppose P(p/q) = 0 for some coprime p, q ∈ ℤ:
To clear denominators, multiply both sides by qn:
Shifting the a0 term to the right side and factoring out p on the left side produces:
Thus, p divides a0qn. But p is coprime to q and therefore to qn, so by Euclid's lemma p must divide the remaining factor a0.
On the other hand, shifting the an term to the right side and factoring out q on the left side produces:
Reasoning as before, it follows that q divides an.[1]
Proof using Gauss's lemma
editShould 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 Q[X], then it also factors in Z[X] as a product of primitive polynomials. Now any rational root p/q corresponds to a factor of degree 1 in Q[X] of the polynomial, and its primitive representative is then qx − p, assuming that p and q are coprime. But any multiple in Z[X] of qx − p has leading term divisible by q and constant term divisible by p, which proves the statement. This argument shows that more generally, any irreducible factor of P can be supposed to have integer coefficients, and leading and constant coefficients dividing the corresponding coefficients of P.
Examples
editFirst
editIn the polynomial
Second
editIn the polynomial
Third
editEvery rational root of the polynomial
This process may be made more efficient: if P(r) ≠ 0, it can be used to shorten the list of remaining candidates.[2] For example, x = 1 does not work, as P(1) = 1. Substituting x = 1 + t yields a polynomial in t with constant term P(1) = 1, while the coefficient of t3 remains the same as the coefficient of x3. Applying the rational root theorem thus yields the possible roots , so that
True roots must occur on both lists, so list of rational root candidates has shrunk to just x = 2 and x = 2/3.
If k ≥ 1 rational roots are found, Horner's method will also yield a polynomial of degree n − k whose roots, together with the rational roots, are exactly the roots of the original polynomial. If none of the candidates is a solution, there can be no rational solution.
See also
editNotes
edit- ^ Arnold, D.; Arnold, G. (1993). Four unit mathematics. Edward Arnold. pp. 120–121. ISBN 0-340-54335-3.
- ^ King, Jeremy D. (November 2006). "Integer roots of polynomials". Mathematical Gazette. 90: 455–456. doi:10.1017/S0025557200180295.
References
edit- Miller, Charles D.; Lial, Margaret L.; Schneider, David I. (1990). Fundamentals of College Algebra (3rd ed.). Scott & Foresman/Little & Brown Higher Education. pp. 216–221. ISBN 0-673-38638-4.
- Jones, Phillip S.; Bedient, Jack D. (1998). The historical roots of elementary mathematics. Dover Courier Publications. pp. 116–117. ISBN 0-486-25563-8.
- Larson, Ron (2007). Calculus: An Applied Approach. Cengage Learning. pp. 23–24. ISBN 978-0-618-95825-2.
External links
edit- Weisstein, Eric W. "Rational Zero Theorem". MathWorld.
- RationalRootTheorem at PlanetMath
- Another proof that nth roots of integers are irrational, except for perfect nth powers by Scott E. Brodie
- The Rational Roots Test at purplemath.com