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The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root x = r is found, a linear polynomial ( x – r ) can be factored out of the polynomial using polynomial long division , resulting in a polynomial of lower degree ...
The rational function defined by the quotient of two homogeneous polynomials is a homogeneous function; its degree is the difference of the degrees of the numerator and the denominator; its cone of definition is the linear cone of the points where the value of denominator is not zero.
This definition allows a limit to be defined at limit points of the domain S, if a suitable subset T which has the same limit point is chosen. Notably, the previous two-sided definition works on , which is a subset of the limit points of S.
The rational function () = is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function.
The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. [3]
If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros. Every rational function is meromorphic on the whole Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum ...
Firstly both definitions suppose that rational numbers and thus natural numbers are rigorously defined; this was done a few years later with Peano axioms. Secondly, both definitions involve infinite sets (Dedekind cuts and sets of the elements of a Cauchy sequence), and Cantor's set theory was published several years
Two numbers x, y are called algebraically dependent if there is a non-zero polynomial P in two indeterminates with integer coefficients such that P(x, y) = 0. There is a powerful theorem that two complex numbers that are algebraically dependent belong to the same Mahler class.