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In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by linear monic polynomials (known as Ruffini's rule ), but the method can be generalized to division by any polynomial .
Ruffini's rule. In mathematics, Ruffini's rule is a method for computation of the Euclidean division of a polynomial by a binomial of the form x – r. It was described by Paolo Ruffini in 1809. [1] The rule is a special case of synthetic division in which the divisor is a linear factor.
So, synthetic division (which was actually invented and published by Ruffini 10 years before Horner's publication) is easier to use; it can be shown to be equivalent to Horner's method. As a consequence of the polynomial remainder theorem, the entries in the third row are the coefficients of the second-degree polynomial, the quotient of f ( x ...
Polynomial long division is an algorithm that implements the Euclidean division of polynomials, which starting from two polynomials A (the dividend) and B (the divisor) produces, if B is not zero, a quotient Q and a remainderR such that. and either R = 0 or the degree of R is lower than the degree of B. These conditions uniquely define Q and R ...
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 .
The polynomial remainder theorem follows from the theorem of Euclidean division, which, given two polynomials f(x) (the dividend) and g(x) (the divisor), asserts the existence (and the uniqueness) of a quotient Q(x) and a remainder R(x) such that. If the divisor is where r is a constant, then either R(x) = 0 or its degree is zero; in both cases ...
In mathematics and computer science, polynomial evaluation refers to computation of the value of a polynomial when its indeterminates are substituted for some values. In other words, evaluating the polynomial at consists of computing See also Polynomial ring § Polynomial evaluation. For evaluating the univariate polynomial the most naive ...
Long division. In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals ( positional notation) that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another ...