Search results
Results from the WOW.Com Content Network
Let () be a polynomial equation, where P is a univariate polynomial of degree n.If one divides all coefficients of P by its leading coefficient, one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial.
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term.Two definitions of a monomial may be encountered: A monomial, also called a power product or primitive monomial, [1] is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. [2]
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
All possible combinations of integer factors can be tested for validity, and each valid one can be factored out using polynomial long division. If the original polynomial is the product of factors at least two of which are of degree 2 or higher, this technique only provides a partial factorization; otherwise the factorization is complete.
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).
where gcd denotes the greatest common divisor of the leading monomials of f and g. As the monomials that are reducible by both f and g are exactly the multiples of lcm, one can deal with all cases of non-uniqueness of the reduction by considering only the S-polynomials. This is a fundamental fact for Gröbner basis theory and all algorithms for ...
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer.
As the greatest common divisor of P and Q is a constant, the resultant D is not zero, and resultant theory implies that I contains all products of D by a monomial in x, y of degree m + n – 1. As D ∉ x , y , {\displaystyle D\not \in \langle x,y\rangle ,} all these monomials belong to the primary component contained in x , y . {\displaystyle ...