Rational root theorem

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Topic: Rational root theorem

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	PlanetMath: proof of rational root theorem
		This is version 3 of proof of rational root theorem, born on 2002-09-25, modified 2004-12-13.
		"proof of rational root theorem" is owned by bs.
	www.planetmath.org /encyclopedia/ProofOfRationalRootTheorem.html (58 words)

	Rational root theorem - Wikipedia, the free encyclopedia
		In algebra, the rational root theorem states a constraint on solutions (also called "roots") to the polynomial equation
		The fundamental theorem of algebra states that any polynomial with integral (or real, or even complex) coefficients must have at least one root in the set of complex numbers.
		, then 0 is one of the rational roots of the equation.
	www.wikipedia.org /wiki/Rational_root_theorem (262 words)

	PlanetMath: rational root theorem
		This is version 6 of rational root theorem, born on 2001-10-15, modified 2005-03-08.
		Cross-references: fraction field, unique factorization domain, monic polynomials, theorem, root, rational, integers, polynomial
		Such theorem then states that any root in the fraction field is also in the base domain.
	planetmath.org /encyclopedia/RationalRootTheorem.html (95 words)

	EQUATION - LoveToKnow Article on EQUATION
		From the theorem that a rational symmetrical function of the roots is expressible in terms of the coefficients, it at once follows that it is possible to determine an equation (of an assignable order) having for its roots the several values of any given (unsymmetrical) function of the roots of the given equation.
		This is an easy consequence from the less general theorem, every rational and integral symmetrical function of the roots is a rational and integral function of the coefficients.
		Of course the theorem is the reverse of self-evident, and it requires proof; but provisionally assuming it as true, we derive from it the general theory of numerical equations.
	40.1911encyclopedia.org /E/EQ/EQUATION.htm (12081 words)

	Polynomial Functions - The Rational Root Theorem
		If a rational number c/d, is factored to its lowest terms, is a solution to the equation, then c is a factor of the constant term of the polynomial and d is a factor of the leading coefficient of the polynomial.
	library.thinkquest.org /10030/8trrt.htm (213 words)

	Ruler-and-compass construction - Wikipedia, the free encyclopedia
		The minimal polynomial for x is a factor of this, but if it were not irreducible, then it would have a rational root which, by the rational root theorem, must be 1 or -1, which are clearly not roots.
		It is impossible to take a square root with just a ruler, so some things cannot be constructed with a ruler that can be constructed with a compass; but (by the Poncelet-Steiner theorem) given a single circle and its center, they can be constructed.
		This requires taking the cube root of an arbitrary complex number with absolute value 1, and is likewise impossible.
	www.wikipedia.org /wiki/Ruler-and-compass_construction (1580 words)

	The factor theorem. The rational root theorem.
		If the rational number r/s is a root of a polynomial whose coefficients are integers, then the integer r is a factor of the constant term, and the integer s is a factor of the leading coefficient.
		Determine the polynomial with integer coefficients whose roots are −½, −2, −2, and sketch the graph.
		Determine the polynomial whose roots are −1, 1, 2, and sketch its graph.
	www.themathpage.com /aPreCalc/factor-theorem.htm (1496 words)

	rationalzeros
		Subject: Re: Rational Zeros Theorem Date: 6 Sep 1999 16:25:49 GMT Newsgroups: sci.math If you are referring to the theorem (I've heard it called the Rational Root Theorem or something similar) that states that for all rational solutions p/q of a polynomial equation a_nx^n + a_(n-1)x^(n-1) +...
		+ a_0, p is a factor of +/- a_0 and q is a factor of +/- a_n, the proof is as follows: Let x=p/q, where p/q is a rational root of the equation and (p,q)=0.
		Steven Sivek stevensivek@hotmail.com > Does anyone know where I could find a proof of the "Rational Zeros Theorem"?
	www.math.niu.edu /~rusin/known-math/99/rationalzeros (223 words)

	Math Forum - Ask Dr. Math
		According to the Rational Root Theorem, the rational root must be a fraction whose numerator is a factor of a(0) and whose denominator is a factor of a(n); but they can be positive or negative.
		Date: 08/28/98 at 01:34:57 From: Doctor Pat Subject: Re: Rational zeros Bill (and son), This is called the Rational Root Theorem, and I think it is usually credited to Descartes.
		So what are our choices for possible rational roots: 1/2, 2/2, 3/2, 6/2, 1/1, 2/1, 3/1, 6/1, or the negatives of any of these You will probably notice that some of these are duplicates.
	mathforum.org /library/drmath/view/56425.html (1060 words)

	Triangle From Angle Bisectors
		By the rational root theorem for polynomials with integer coefficients, the possible rational roots are r = p/q, where p is a divider of the absolute term (1 for our equation) and q is a divider of the highest power term coefficient (4 for our equation).
		Then the ratio k is always constructible (as a square root of rational number q/8).
		The possible rational roots are -1, +1, -1/2, +1/2, -1/4, +1/4 and no others.
	www.cut-the-knot.com /triangle/TriangleFromBisectors.shtml (370 words)

	Factors and Zeros
		If there are no rational roots, give up or use a calculator.
		B) The number of negative roots is either equal to the number of variations in the sign of P(x) or is less than that by an even integer.
		Every polynomial has a root in the complex numbers, moreover if the polynomial has degree n then the polynomial can be written as a product of n linear factors.
	www.ltcconline.net /greenl/courses/103a/premid3/faczero.htm (592 words)

	GraspMath College Algebra Video Series
		Topics include factoring, rational expressions, radicals, rational exponents, complex numbers, linear equations and inequalities and applications, solving quadratic equations, absolute value equations and inequalities, polynomial and rational inequalities, circles, functions and graph sketching, rational root theorem, graphing rational functions, exponential and logarithmic functions, induction, and the Binomial Theorem.
		This segment covers the use of the rational root theorem for factorization of polynomials.
		This segment covers factorial notation, binomial coefficients and the use of the binomial theorem for expansion of a binomial raised to a power as well as determination of specific terms of the expansion in case of large powers.
	www.evrmath.com /schools/colalg_VHS.html (889 words)

	Polynomial Equations
		Use the Rational Root Theorem and synthetic division to exactly determine the roots.
		Hence the rational root is -3/2 and using the calculator we see that the irrational root is 0.198.
		Since the only possible rational roots are 1, -1, 5, -5,.5, -.5, 2.5, -2.5, the possible rational roots are -5/2 and -.5.
	www.ltcc.cc.ca.us /depts/math/courses/103a/Polynomials/polyeq.htm (228 words)

	Three Problems Of Antiquity
		Then either one of its roots is rational or none of its roots is constructible.
		Among all fields that contain a constructible root of the equation choose the one with the least N. Note that, from the minimality of N, no root of the equation belongs to F
		Among the roots of this equation is z = cos(360
	www.cut-the-knot.com /arithmetic/cubic.html (769 words)

	tech.html
		A curriculum without the Rational Root Theorem need not be one that is "dumbed down." The technology approach to seeking solutions of equations does require advanced mathematical reasoning to understand how it works and to create helping tools to make it efficient.
		Instead, we proceeded to develop the Rational Root Theorem for polynomials and worked on all the associated skills needed to apply it successfully.
		Traditional text book exercises for applying the Rational Root Theorem to cubic equations have far "friendlier" coefficients, and they have been carefully rigged to yield to the technique after only a few minutes of careful work.
	home.earthlink.net /~keckcalc/tech.html (726 words)

	Chapter 8: Variation and Advanced Equations
		The power of the Remainder Theorem is explained and practiced before moving on to the Factor and Rational Root Theorems.
		The conjugate root theorem is used to solve higher degree equations which contain non-real roots.
		Then, move on to the concepts of direct and inverse variation, for which a multitude of "real life" problems exist and are practiced in the word problem section.
	www.mathmedia.com /chap8varanda.html (84 words)

	Math Forum - Ask Dr. Math
		These sixteen fractions (those with denominator 1 are actually whole numbers) are the only possible rational numbers that could be roots of the above equation.
		I do not know how to figure out a question like this: Find all possible rational roots of: 4x^3 + 3x^2 + 6x + 10 I have no idea where even to start.
		By the way, it is not very hard to see that none of the positive numbers can be roots because if you substitute them into the equation, since all the coefficients are positive, the result will be a positive number, and not zero.
	mathforum.org /library/drmath/view/56427.html (243 words)

	RATIONAL ROOTS OF POLYNOMIALS
		By the Rational Root Theorem (RRT) the possible rational solutions (roots or zeros of the polynomial) are of the form m/n where n is a factor is a factor of 6 and m is a factor of 5.
		So if we find 3 rational roots, we are done and don't have to fool with the rest.
		Now, we can do this for the non-integer rational numbers as well but sometimes this can get a little confused but is good practice if you want to go through the process.
	www.themathresource.com /Preview/CollegeAlgebra/HigherOrderPolyFcts/ratroots/index.htm (597 words)

	Mathwords: Rational Root Theorem
		A theorem that provides a complete list of possible rational roots of the polynomial equation a
		This list consists of all possible numbers of the form c/d, where c and d are integers.
	www.mathwords.com /r/rational_root_theorem.htm (60 words)

	Polynomial: Conjugate Root Theorem
		Theorem: If a polynomial f(x) with integer coefficients is irreducible in Z[x], then it is also irreducible in Q[x].
		L. The minimum polynomial of c in L is defined as the polynomial with the lowest order such that the coefficients of the polynomial are in L and c is a root of that polynomial.
		Basically, it uses “Gauss Lemma” to prove this theorem.
	www.scienceoxygen.com /mathnote/poly206.html (707 words)

	mm-163.txt
		I'd then check to see if there are any rational roots by using the rational root theorem.
		I'd then >check to see if there are any rational roots by using the rational root >theorem.
		If you know that (at least) one of the roots is a rational, then you can test all the possible roots, find a rational root a, divide p(x) by (x-a), and then use the quadratic formula on the remaining polynomial.
	www.grahamkendall.net /Mathematica/mm-163.txt (738 words)

	Rational Roots Test Lesson
		The Rational Roots (or Rational Zeroes) Test is a useful way to find your initial guesses when you are trying to find the zeroes (roots) of a polynomial.
		This is always true: If a polynomial has rational roots, then they will be fractions of the form (plus-or-minus) (factor of the constant term)/(factor of the leading coefficient).
		This gives you a list of potential rational (fractional) roots to test -- hence the name of the Test.
	www.purplemath.com /modules/rtnlroot.htm (663 words)

	SparkNotes: Polynomial Functions: Roots of Higher Degree Polynomials
		Finding the roots of higher degree polynomials is much more difficult than finding the roots of a quadratic function.
		Complex roots will not be discussed until after a thorough exploration of complex numbers and polar coordinates.
		and the denominator of the root is a factor of a
	www.sparknotes.com /math/precalc/polynomialfunctions/section5.rhtml (795 words)

	ABSTRACT ALGEBRA ON LINE: Polynomials
		If r/s is a rational root of f(x), with (r,s)=1, then r
		This process can be extended by induction until f(x)/g(x) is written as a sum of rational functions, where in each case the denominator is a power of an irreducible polynomial.
		The first step in achieving a partial fraction decomposition of f(x)/g(x) is to use Theorem 4.2.9 to write g(x) as a product of irreducible polynomials.
	www.math.niu.edu /~beachy/aaol/polynomials.html (1677 words)

	Math Forum - Ask Dr. Math
		The theorem states that if a polynomial has a rational root, then the denominator of the root must divide the coefficient of the highest power term of the polynomial, and the numerator of the root must divide the constant term of the polynomial.
		Date: 11/13/2000 at 15:57:53 From: Jacob Bucksbaum Subject: Proof of rational root theorem I have been asked to prove the Rational Root Theorem, and I am lost.
		Date: 11/13/2000 at 16:47:03 From: Doctor Rob Subject: Re: Proof of rational root theorem Thanks for writing to Ask Dr. Math, Jacob.
	www.forum.swarthmore.edu /dr.math/problems/bucksbaum.11.13.00.html (240 words)

	NCTM : Illuminations Lessons : Building Connections
		However, a students who progresses into polynomials of higher degree, normally in the second year of algebra, is taught to graph the polynomial by using the factor theorem and the rational-root theorem, both which highlight the x-intercepts of the polynomial.
		The factor theorem, used in conjunction with the fundamental theorem of algebra, is a "glue" that holds the classes of polynomial functions together.
		The absence of x-intercepts implies that no real roots exist, that is, lines cannot be drawn, because the quadratic equation cannot be factored into linear expressions over the real numbers.
	illuminations.nctm.org /lessonplans/9-12/polynomial/index.html (2617 words)

	3.3 - Real Zeros of Polynomial Functions
		After you find a possible rational root that actually works, take the quotient and continue to try to factor it until it is down to a quadratic or less.
		The actual number of negative real roots may be the maximum, or the maximum decreased by a multiple of two.
		The actual number of positive real roots may be the maximum, or the maximum decreased by a multiple of two.
	www.richland.cc.il.us /james/lecture/m116/polynomials/zeros.html (1436 words)

	Roots of Polynomials
		Roots of polynomials: rational root test, factor theorem, irrational roots.
		to see which numbers on the list could possibly be roots and test each one to find the rational roots.
		If has degree 3 or more, check for repeated rational roots by evaluating
	phoenix.goucher.edu /~kelliher/ma114/nov18.html (298 words)

	Chapter 4
		To identify all possible rational roots of a polynomial equation by using the rational root theorem
		Write the polynomial equation of least degree with roots -3,2i,and -2i.
		To use the discriminant to describe the roots of quadratic equations
	www.cedarnet.org /easthigh/staff/platte/4-notes-pcal.htm (414 words)

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