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In these examples, the (negative) least absolute remainder is obtained from the least positive remainder by subtracting 5, which is d. This holds in general. When dividing by d, either both remainders are positive and therefore equal, or they have opposite signs. If the positive remainder is r 1, and the negative one is r 2, then r 1 = r 2 + d.
In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) [1] is an application of Euclidean division of polynomials.It states that, for every number , any polynomial is the sum of () and the product of and a polynomial in of degree one less than the degree of .
Using Euclidean division, 9 divided by 4 is 2 with remainder 1. In other words, each person receives 2 slices of pie, and there is 1 slice left over. This can be confirmed using multiplication, the inverse of division: if each of the 4 people received 2 slices, then 4 × 2 = 8 slices were given out in total. Adding the 1 slice remaining, the ...
In each step k of the Euclidean algorithm, the quotient q k and remainder r k are computed for a given pair of integers r k−2 and r k−1. r k−2 = q k r k−1 + r k. The computational expense per step is associated chiefly with finding q k, since the remainder r k can be calculated quickly from r k−2, r k−1, and q k. r k = r k−2 − q ...
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.
Assuming that [a − r, a + r] ⊂ I and r < R, all these series converge uniformly on (a − r, a + r). Naturally, in the case of analytic functions one can estimate the remainder term R k ( x ) {\textstyle R_{k}(x)} by the tail of the sequence of the derivatives f′ ( a ) at the center of the expansion, but using complex analysis also ...
Polynomial long division can be used to find the equation of the line that is tangent to the graph of the function defined by the polynomial P(x) at a particular point x = r. [3] If R(x) is the remainder of the division of P(x) by (x – r) 2, then the equation of the tangent line at x = r to the graph of the function y = P(x) is y = R(x ...
The b values are the coefficients of the result (R(x)) polynomial, the degree of which is one less than that of P(x). The final value obtained, s, is the remainder. The polynomial remainder theorem asserts that the remainder is equal to P(r), the value of the polynomial at r.