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The second indicates that one can remedy the divergent behavior by introducing an additional real root, at the cost of slowing down the speed of convergence. One can also in the case of odd degree polynomials first find a real root using Newton's method and/or an interval shrinking method, so that after deflation a better-behaved even-degree ...
The oldest method of finding all roots is to start by finding a single root. When a root r has been found, it can be removed from the polynomial by dividing out the binomial x – r. The resulting polynomial contains the remaining roots, which can be found by iterating on this process.
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.
Return to step 1 but use the polynomial and the initial guess . These two steps are repeated until all real zeros are found for the polynomial. If the approximated zeros are not precise enough, the obtained values can be used as initial guesses for Newton's method but using the full polynomial rather than the reduced polynomials.
Animation showing the use of synthetic division to find the quotient of + + + by .Note that there is no term in , so the fourth column from the right contains a zero.. In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division.
Ruffini's rule can be used when one needs the quotient of a polynomial P by a binomial of the form . (When one needs only the remainder, the polynomial remainder theorem provides a simpler method.) A typical example, where one needs the quotient, is the factorization of a polynomial p ( x ) {\displaystyle p(x)} for which one knows a root r :
The polynomial f is reducible by g if some monomial of f is a multiple lm(g). (So, if f is lead-reducible by g, it is also reducible, but f may be reducible without being lead-reducible.) Suppose that f is reducible by g, and let cm be a term of f such that the monomial m is a multiple of lm(g). A one-step reduction of f by g consists of ...
A polynomial f of degree n greater than one, which is irreducible over F q, defines a field extension of degree n which is isomorphic to the field with q n elements: the elements of this extension are the polynomials of degree lower than n; addition, subtraction and multiplication by an element of F q are those of the polynomials; the product ...