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...

perfect square) or irrational. The rational root theorem (or integer root theorem) may be used to show that any square root of any natural number that is not...

Waerden cites the polynomial f(x) = x5 − x − 1. By the rational root theorem, this has no rational zeroes. Neither does it have linear factors modulo 2...

The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial...

the resulting sextic polynomial has a rational root, which can be easily tested for using the rational root theorem. Around 1770, Joseph Louis Lagrange...

p_{i}} do not have any common root. An efficient method to compute this factorization is Yun's algorithm. Rational root theorem Pan, Victor Y. (January 1997)...

and constant term a 0 {\displaystyle a_{0}} . (See Rational Root Theorem.) Use the factor theorem to conclude that ( x − a ) {\displaystyle (x-a)} is...

square roots Cube root Root of unity Constructible number Complex conjugate root theorem Algebraic element Horner scheme Rational root theorem Gauss's lemma...

polynomial can be obtained. For example, if the rational root theorem produces a single (rational) root of a quintic polynomial, it can be factored out...

rule of signs Newton's identities Gauss–Lucas theorem Properties of polynomial roots Rational root theorem Symmetric polynomial and elementary symmetric...

analysis) – Limit of roots of sequence of functions Rational root theorem – Relationship between the rational roots of a polynomial and its extreme coefficients...

some ai, then a root of it will generate the asserted N0.) Construction of elliptic curves with large rank. Hilbert's irreducibility theorem is used as a...

(polynomials) Polynomial remainder theorem (polynomials) Primitive element theorem (field theory) Rational root theorem (algebra, polynomials) Solutions...

k\in \mathbb {Z} } , and so k must be a root of p(x); but also k must divide 2 (by the rational root theorem); that is, k = 1, 2, −1 or −2, and none of...

result of the rational root theorem for the case of a monic polynomial. Gaussian integer Eisenstein integer Root of unity Dirichlet's unit theorem Fundamental...

it has a rational root. By the rational root theorem, this root must be ±1, ±⁠1/2⁠, ±⁠1/4⁠ or ±⁠1/8⁠, but none of these is a root. Therefore, p(t) is...

exactly the rational numbers that are also algebraic integers. This results from the rational root theorem, which asserts that, if the rational number p...

unknown, it is possible to solve the equation for rational-valued unknowns (see Rational root theorem), and then find solutions to the Diophantine equation...

{\displaystyle (x^{6}-2x^{3}-1)=0} . This polynomial has no rational roots, since the rational root theorem shows that the only possibilities are ±1, but x0 is...

{\displaystyle {\tfrac {3}{2}}} is a root, so there can be no other rational root. Applying the factor theorem leads finally to the factorization 2 x...

square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. (See square root of 2 for...

fundamental theorem of algebra, C is algebraically closed, i.e., any polynomial equation with complex coefficients has a complex solution. The rational and the...

of the rational numbers Q, since Q has characteristic 0 and therefore every finite extension over Q is separable. Using the fundamental theorem of Galois...

neither a nor b is a multiple root of p, then V(a) − V(b) is the number of distinct real roots of P. The proof of the theorem is as follows: when the value...

Pfaffian functions. Sturm's theorem – Counting polynomial roots in an interval Rational root theorem – Relationship between the rational roots of a polynomial...

polynomial whose arguments are all conjugates of one another gives a rational number. The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann...

immediately from the theorem: Gelfond–Schneider constant 2 2 = 2.665144142 … {\displaystyle 2^{\sqrt {2}}=2.665144142\ldots } and its square root 2 2 = 2 2 = 1...

algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the...

extension of the rational numbers. The Kronecker–Weber theorem therefore guarantees that the square root of five can be written as a rational linear combination...

In geometry, Descartes' theorem states that for every four kissing, or mutually tangent circles, the radii of the circles satisfy a certain quadratic...