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Rational root theorem MATHEMATICS WRITTEN BY: William L. Hosch See Article History Alternative Title: rational root test
Rational root theorem, also called rational root test, in algebra, theorem that for a polynomial equation in one variable with integer coefficients to have a solution (root) that is a rational number, the leading coefficient (the coefficient of the highest power) must be divisible by the denominator of the fraction and the constant term (the one without a variable) must be divisible by the numerator. In algebraic notation the canonical form for a polynomial equation in one variable (x) is anxn + an− 1xn − 1 + … + a1x1 + a0 = 0, where a0, a1,…, an are ordinary integers. Thus, for a polynomial equation to have a rational solution p/q, q must divide an and p must divide a0. For example, consider 3x3 − 10x2 + x + 6 = 0. The only divisors of 3 are 1 and 3, and the only divisors of 6 are 1, 2, 3, and 6. Thus, if any rational roots exist,
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modern algebra
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algebra
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binomial theorem
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elementary algebra
used to factor the equation, thereby simplifying the problem of finding further rational roots. In this
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fundamental theorem of algebra
example, the polynomial can be factored as (x − 1)(x + 2/3)(x − 3) = 0. Before computers were
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group theory
available to use the methods of numerical analysis, such calculations formed an essential part in the
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linear algebra
solution of most applications of mathematics to physical problems. The methods are still used in
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mathematics
elementary courses in analytic geometry, though the techniques are superseded once students master
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rational number
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Descartes’s rule of signs
they must have a denominator of 1 or 3 and a numerator of 1, 2, 3, or 6, which limits the choices to 1/3, 2/ , 1, 2, 3, and 6 and their corresponding negative values. Plugging the 12 candidates into the 3
equation yields the solutions − 2/3, 1, and 3. In the case of higher-order polynomials, each root can be
basic calculus. The 17th-century French philosopher and mathematician René Descartes is usually credited with devising the test, along with Descartes’s rule of signs for the number of real roots of a polynomial. The effort to find a general method of determining when an equation has a rational or real solution led to the development of group theory and modern algebra. William L. Hosch
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Algebra, branch of mathematics in which arithmetical operations and formal manipulations are applied to abstract symbols rather than specific numbers. The notion that there exists such a distinct subdiscipline of mathematics, as well as the term algebra to denote it, resulted from a
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slow historical development. This article presents that
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theorem Theorem, in mathematics and logic, a proposition or statement that is demonstrated. In geometry, a proposition is commonly
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fundamental theorem of algebra
considered as a problem (a construction to be effected) or a theorem (a statement to be proved). The statement “If two lines
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group theory
intersect, each pair of vertical angles is equal,” for example,
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Descartes’s rule of signs
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binomial theorem
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modern algebra
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elementary algebra
Arab writers usually called one of the equal factors of a number jadhr (“root”), and their medieval European translators used
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linear algebra
the Latin word radix (from which derives the adjective radical).
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linear transformation
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root Root, in mathematics, a solution to an equation, usually expressed as a number or an algebraic formula. In the 9th century,
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rational number Rational number, in arithmetic, a number that can be represented as the quotient p/q of two integers such that q ≠ 0. In
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addition to all the fractions, the set of rational numbers includes all the integers, each of which can be written as a quotient with the integer as READ MORE
Modern algebra Modern algebra, branch of mathematics concerned with the general algebraic structure of various sets (such as real numbers, complex numbers, matrices, and vector spaces),
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