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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.
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.
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 .
Triangles whose side lengths and areas are all rational numbers (positive rational solutions of the above equation) are sometimes also called Heronian triangles or rational triangles; [5] in this article, these more general triangles will be called rational Heronian triangles. Every (integral) Heronian triangle is a rational Heronian triangle.
Since x = cos 20° is a root of p(t), the minimal polynomial for cos 20° is a factor of p(t). Because p(t) has degree 3, if it is reducible over by Q then 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.
Triangles based on Pythagorean triples are Heronian, meaning they have integer area as well as integer sides. The possible use of the 3 : 4 : 5 triangle in Ancient Egypt, with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras' theorem was known at that time, have been much debated. [3]