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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 remainder is multiplied by 3 to get feet and carried up to the feet column. Long division of the feet gives 1 remainder 29 which is then multiplied by twelve to get 348 inches. Long division continues with the final remainder of 15 inches being shown on the result line.
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 remainder R such that A = BQ + R, and either R = 0 or the degree of R is lower than the degree of B.
The polynomial remainder theorem may be used to evaluate () by calculating the remainder, . Although polynomial long division is more difficult than evaluating the function itself, synthetic division is computationally easier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem.
But by the polynomial remainder theorem, we know that the remainder is (). Thus, () =. In this example, if =, =, =, = we can see that =, =, =, =, the entries in the third row. 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 ...
For six months or so in 2021, as vaccines paved an economic reopening from the COVID-19 pandemic and fresh waves of federal benefits flowed to household bank accounts, President Joe Biden's ...
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
Jessica Capshaw and Camilla Luddington talk about their new podcast and reflect on their time on Grey's Anatomy. (Corbis via Getty Images) (Stephane Cardinale - Corbis via Getty Images)
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