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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.
[note 5] If =, then it says a rational root of a monic polynomial over integers is an integer (cf. the rational root theorem). To see the statement, let / be a root of in and assume , are relatively prime.
Degree: The maximum exponents among the monomials.; Factor: An expression being multiplied.; Linear factor: A factor of degree one.; Coefficient: An expression multiplying one of the monomials of the polynomial.
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 ...
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 ...
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The rational root test allows finding q and p by examining a finite number of cases (because q must be a divisor of a, and p must be a divisor of d). Thus, one root is =, and the other roots are the roots of the other factor, which can be found by polynomial long division.