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In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials.
Trinomial. Layers of Pascal's pyramid derived from coefficients in an upside-down ternary plot of the terms in the expansions of the powers of a trinomial. In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. [1]
Factorization of polynomials. In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of ...
Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R which is not a unit can be written as a finite product of irreducible elements pi of R: x = p1 p2 ⋅⋅⋅ pn with n ≥ 1. and this representation is unique in the following sense: If q1, ..., qm are irreducible elements of R ...
Resultant. In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients).
The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact, thus producing a lower degree polynomial whose zeros may be easier to find. Abstractly, the method is as follows: [3] Deduce the candidate of zero of the polynomial from its leading coefficient and constant term .
Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R. Then, the quotients a i / a n {\displaystyle a_{i}/a_{n}} belong to the field of fractions of R (and possibly are in R itself if a n {\displaystyle a_{n}} happens to be invertible in R ) and the roots r i {\displaystyle r_{i}} are taken in an ...
Trinomial expansion. In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by. where n is a nonnegative integer and the sum is taken over all combinations of nonnegative indices i, j, and k such that i + j + k = n. [1] The trinomial coefficients are given by.