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The number of complex roots equals 6 minus the number of real roots. In algebra, a sextic (or hexic) polynomial is a polynomial of degree six. A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form:
For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...
The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or "name". It was derived from the term binomial by replacing the Latin root bi-with the Greek poly-. That is, it means a sum of many terms (many monomials). The word polynomial was first used in the 17th century. [6]
The square-free factorization of a polynomial p is a factorization = where each is either 1 or a polynomial without multiple roots, and two different do not have any common root. An efficient method to compute this factorization is Yun's algorithm .
In mathematics, an algebraic equation or polynomial equation is an equation of the form =, where P is a polynomial with coefficients in some field, often the field of the rational numbers. For example, x 5 − 3 x + 1 = 0 {\displaystyle x^{5}-3x+1=0} is an algebraic equation with integer coefficients and
If one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form (x − r)Q(x) where Q(x) is a polynomial of degree n − 1. Q ( x ) is simply the quotient obtained from the division process; since r is known to be a root of P ( x ), it is known that the remainder must be zero.
To work with a polynomial system whose coefficients belong to a number field, it suffices to consider this generator as a new variable and to add the equation of the generator to the equations of the system. Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers.
In the polynomial + the only possible rational roots would have a numerator that divides 6 and a denominator that divides 1, limiting the possibilities to ±1, ±2, ±3, and ±6. Of these, 1, 2, and –3 equate the polynomial to zero, and hence are its rational roots (in fact these are its only roots since a cubic polynomial has only three roots).
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