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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 a n = 1.
Theorem — The number of strictly positive roots (counting multiplicity) of is equal to the number of sign changes in the coefficients of , minus a nonnegative even number. If b 0 > 0 {\displaystyle b_{0}>0} , then we can divide the polynomial by x b 0 {\displaystyle x^{b_{0}}} , which would not change its number of strictly positive roots.
An element a of F is integral over R if it is a root of a monic polynomial with coefficients in R. A complex number that is integral over the integers is called an algebraic integer. This terminology is motivated by the fact that the integers are exactly the rational numbers that are also algebraic integers.
Rational root theorem; Gauss's lemma (polynomial) Irreducible polynomial. Eisenstein's criterion; Primitive polynomial; Fundamental theorem of algebra; Hurwitz polynomial; Polynomial transformation; Tschirnhaus transformation; Galois theory; Discriminant of a polynomial. Resultant; Elimination theory. Gröbner basis; Regular chain; Triangular ...
In some cases a brute force approach can be used, as mentioned above. In some other cases, in particular if the equation is in one unknown, it is possible to solve the equation for rational-valued unknowns (see Rational root theorem), and then find solutions to the Diophantine equation by restricting the solution set to integer-valued solutions ...
(±1 cannot be a root because a 0 /a n is not integral.) It so happens that neither 1 nor -1 is a root, but this is not implied by the rational root theorem. There is also no theorem saying that ±1 can only be a root if a 0 /a n is integral. For example, 3x 3-x 2-x-1 = 0 has a root x=1, even though -1/3 is not integral.
The polynomial P(x) has a rational root (this can be determined using the rational root theorem). The resolvent cubic R 3 (y) has a root of the form α 2, for some non-null rational number α (again, this can be determined using the rational root theorem). The number a 2 2 − 4a 0 is the square of a rational number and a 1 = 0. Indeed:
The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real roots appear in pairs of the form (a + ib, a – ib). It follows that the roots of a polynomial with real coefficients are mirror-symmetric with respect to the real axis.
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