Rational root theorem

Rational root theorem

In algebra, the rational root theorem (or 'rational root test' to find the zeros) states a constraint on solutions (or roots) to the polynomial equation

:a_nx^n+a_{n-1}x^{n-1}+cdots+a_0 = 0,!

with integer coefficients.

Let "a""0" and "a""n" be nonzero.Then each rational solution "x"can be written in the form "x" = "p"/"q" for "p" and "q" satisfying two properties:
* "p" is an integer factor of the constant term "a"0, and
* "q" is an integer factor of the leading coefficient "a""n".

Thus, a list of possible rational roots of the equation can be derived using the formula x = pm frac{p}{q}.

For example, every rational solution of the equation

:3x^3 - 5x^2 + 5x - 2 = 0,!must be among the numbers symbolically indicated by

frac{1,2}{1,3},,

which gives the list of possible answers:

:1, -1, 2, -2, frac{1}{3}, -frac{1}{3}, frac{2}{3}, -frac{2}{3},.

These root candidates can be tested, for example using the Horner scheme. In this particular case there is exactly one rational root.

If a root "r""1"is found, the Horner scheme will also yield a polynomial of degree "n" − 1 whose roots, together with "r""1", are exactly the roots of the original polynomial.

It may also be the case that none of the candidates is a solution; in this case the equation has no rational solution. The fundamental theorem of algebrastates that any polynomial with integer (or real, or even complex)coefficients must have at least one root in the set of complex numbers.Any polynomial of odd degree (degree being "n" in the example above) with real coefficients must have a root in the set of real numbers.

If the equation lacks a constant term "a"0, then 0 is one of the rational roots of the equation.

The theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials.

The integral root theorem is a special case of the rational root theorem if the leading coefficient "a""n"=1.

External links

* [http://www.cut-the-knot.org/Generalization/RationalRootTheorem.shtml Another proof that nth roots of integers are irrational, except for perfect nth powers] by Scott E. Brodie


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Rational trigonometry — is a recently introduced approach to trigonometry that eschews all transcendental functions (such as sine and cosine) and all proportional measurements of angles. In place of angles, it characterizes the separation between lines by a quantity… …   Wikipedia

  • Root of unity — The 5th roots of unity in the complex plane In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially …   Wikipedia

  • Root-finding algorithm — A root finding algorithm is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. Such an x is called a root of the function f. This article is concerned with finding scalar, real or complex roots,… …   Wikipedia

  • Square root of 2 — The square root of 2, also known as Pythagoras constant, often denoted by:sqrt{2} or √2but can also be written as:2^{1/2},,is the positive real number that, when multiplied by itself, gives the number 2. Its numerical value approximated to 65… …   Wikipedia

  • Fundamental theorem of algebra — In mathematics, the fundamental theorem of algebra states that every non constant single variable polynomial with complex coefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed.Sometimes,… …   Wikipedia

  • Square root of 5 — The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. This number appears in the formula for the golden ratio. It can be denoted in surd form as::sqrt{5}.It is an irrational algebraic number.… …   Wikipedia

  • Square root — Measured fall time of a small steel sphere falling from various heights. The data is in good agreement with the predicted fall time of , where h is the height and g is the acceleration of gravity. In mathematics, a square root of a number x is a… …   Wikipedia

  • Pythagorean theorem — See also: Pythagorean trigonometric identity The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c) …   Wikipedia

  • nth root — In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals x rn = x, where n is the degree of the root. A root of degree 2 is usually called a square root and a root of degree 3, a cube root. For example …   Wikipedia

  • Square root of a matrix — In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product B · B is equal to A.[1] Contents 1 Properties 2 Computation methods …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”