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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.
If necessary, simplify the long division problem by moving the decimals of the divisor and dividend by the same number of decimal places, to the right (or to the left), so that the decimal of the divisor is to the right of the last digit. When doing long division, keep the numbers lined up straight from top to bottom under the tableau.
After three steps, the middle card (*) is the one in all chosen piles. The Twenty-One Card Trick, also known as the 11th card trick or three column trick, is a simple self-working card trick that uses basic mathematics to reveal the user's selected card. The game uses a selection of 21 cards out of a standard deck. These are shuffled and the ...
Because matrix multiplication is not commutative, one can also define a left division or so-called backslash-division as A \ B = A −1 B. For this to be well defined, B −1 need not exist, however A −1 does need to exist. To avoid confusion, division as defined by A / B = AB −1 is sometimes called right division or slash-division in this ...
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 :
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.
Euclidean division of polynomials is very similar to Euclidean division of integers and leads to polynomial remainders. Its existence is based on the following theorem: Given two univariate polynomials a ( x ) and b ( x ) (where b ( x ) is a non-zero polynomial) defined over a field (in particular, the reals or complex numbers ), there exist ...
Indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. While Euclid took the first step on the way to the existence of prime factorization, Kamāl al-Dīn al-Fārisī took the final step [ 8 ] and stated for the first time the fundamental theorem of arithmetic.