In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, 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
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
:
with
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
coefficients
and
. Solutions of the equation are also called
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
or zeroes of the
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
on the left side.
The theorem states that each
rational solution , written in lowest terms so that and are
relatively prime, satisfies:
* is an integer
factor of the
constant term
In mathematics, a constant term is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial
:x^2 + 2x + 3,\
the 3 is a constant term.
After like terms are com ...
, and
* is an integer factor of the leading
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
.
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
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, bec ...
, 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
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. 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
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...
, are the remaining two roots of the cubic, avoiding cube roots.
Proofs
Elementary proof
Let
with
Suppose for some
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
:
:
To clear denominators, multiply both sides by :
:
Shifting the term to the right side and factoring out on the left side produces:
:
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 on the left side produces:
:
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
In the polynomial
:
any rational root fully reduced would have to have a numerator that divides evenly into 1 and a denominator that divides evenly into 2. Hence the only possible rational roots are ±1/2 and ±1; since neither of these equates the polynomial to zero, it has no rational roots.
Second
In the polynomial
:
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 has only three roots; in general, a polynomial could have some rational and some
irrational roots.)
Third
Every rational root of the polynomial
:
must be among the numbers symbolically indicated by:
:
These 8 root candidates can be tested by evaluating , for example using
Horner's method. It turns out there is exactly one with .
This process may be made more efficient: if , it can be used to shorten the list of remaining candidates.
For example, does not work, as . Substituting yields a polynomial in with constant term , while the coefficient of remains the same as the coefficient of . 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 and .
If rational roots are found, Horner's method will also yield a polynomial of degree 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
*
Fundamental theorem of algebra
*
Integrally closed domain
*
Descartes' rule of signs
*
Gauss–Lucas theorem
*
Properties of polynomial roots
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Mathematics
* Property (mathematics)
Philosophy and science
* Property (philosophy), in philosophy an ...
*
Content (algebra)
*
Eisenstein's criterion
Notes
References
*Charles D. Miller, Margaret L. Lial, David I. Schneider: ''Fundamentals of College Algebra''. Scott & Foresman/Little & Brown Higher Education, 3rd edition 1990, , pp. 216–221
*Phillip S. Jones, Jack D. Bedient: ''The historical roots of elementary mathematics''. Dover Courier Publications 1998, , pp. 116–117 ()
*Ron Larson: ''Calculus: An Applied Approach''. Cengage Learning 2007, , pp. 23–24 ()
External links
*{{MathWorld, urlname=RationalZeroTheorem, title=Rational Zero Theorem
''RationalRootTheorem''at
PlanetMath
Another proof that nth roots of integers are irrational, except for perfect nth powersby Scott E. Brodie
at purplemath.com
Theorems about polynomials
Root-finding algorithms