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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.
This application also invokes the integer root theorem, a stronger version of the rational root theorem for the case when () is a monic polynomial with integer coefficients; for such a polynomial, all roots are necessarily integers (which is not, as 2 is not a perfect square) or irrational.
Fundamental theorem of algebra – Every polynomial has a real or complex root; Hurwitz's theorem (complex analysis) – Limit of roots of sequence of functions; Rational root theorem – Relationship between the rational roots of a polynomial and its extreme coefficients
By the rational root theorem, this has no rational zeroes. Neither does it have linear factors modulo 2 or 3. The Galois group of f(x) modulo 2 is cyclic of order 6, because f(x) modulo 2 factors into polynomials of orders 2 and 3, (x 2 + x + 1)(x 3 + x 2 + 1). f(x) modulo 3 has no linear or quadratic factor, and hence is irreducible. Thus its ...
(±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 first complete root-isolation procedure results of Sturm's theorem (1829), which expresses the number of real roots in an interval in terms of the number of sign variations of the values of a sequence of polynomials, called Sturm's sequence, at the ends of the interval.
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.