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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.
Abel–Ruffini theorem; Bring radical; Binomial theorem; Blossom (functional) Root of a function; nth root (radical) Surd; Square root; Methods of computing square roots; Cube root; Root of unity; Constructible number; Complex conjugate root theorem; Algebraic element; Horner scheme; Rational root theorem; Gauss's lemma (polynomial) Irreducible ...
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 a / b {\displaystyle a/b} be a root of f {\displaystyle f} in F {\displaystyle F} and assume a , b {\displaystyle a,b} are relatively prime .
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.
The rational root theorem (or integer root theorem) may be used to show that any square root of any natural number that is not a perfect square is irrational. For other proofs that the square root of any non-square natural number is irrational, see Quadratic irrational number or Infinite descent.
Sometimes one or more roots of a polynomial are known, perhaps having been found using the rational root theorem. If one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form (x − r)Q(x) where Q(x) is a polynomial of degree n − 1. Q(x) is simply the quotient obtained from the ...
Casus irreducibilis occurs when none of the roots are rational and when all three roots are distinct and real; the case of three distinct real roots occurs if and only if q 2 / 4 + p 3 / 27 < 0, in which case Cardano's formula involves first taking the square root of a negative number, which is imaginary, and then taking the ...