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Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent.
The factor theorem states that, if r is a root of a polynomial () ... Euclidean algorithm for polynomials allows computing this greatest common factor. For example, ...
It is clear that any finite set {} of points in the complex plane has an associated polynomial = whose zeroes are precisely at the points of that set. The converse is a consequence of the fundamental theorem of algebra: any polynomial function () in the complex plane has a factorization = (), where a is a non-zero constant and {} is the set of zeroes of ().
For example, the fundamental theorem of algebra, which states that every polynomial with complex coefficients has complex roots, implies that a polynomial with integer coefficients can be factored (with root-finding algorithms) into linear factors over the complex field C.
Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem. The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial. [3]
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. [3] [4] [5] For example,
p is an integer factor of the constant term a 0, and; q is an integer factor of the leading coefficient a n. 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 ...
FWL theorem ; Factor theorem (polynomials) Fagin's theorem (computational complexity theory) Faltings's theorem (Diophantine geometry) Farrell–Markushevich theorem (complex analysis) Fáry's theorem (graph theory) Fáry–Milnor theorem (knot theory) Fatou's theorem (complex analysis) Fatou–Lebesgue theorem (real analysis)