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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 :
Solving for , = = = = = Thus, the power rule applies for rational exponents of the form /, where is a nonzero natural number. This can be generalized to rational exponents of the form p / q {\displaystyle p/q} by applying the power rule for integer exponents using the chain rule, as shown in the next step.
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 two polynomials q(x) (the quotient) and r(x) (the remainder) which satisfy: [7]
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 .
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.
So, for example, the () = words using 0s and 1s are ,,,,,. To obtain the Gaussian binomial coefficient ( m r ) q {\displaystyle {\tbinom {m}{r}}_{q}} , each word is associated with a factor q d , where d is the number of inversions of the word, where, in this case, an inversion is a pair of positions where the left of the pair holds the letter ...
In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers) variables, and the basic algebraic operations: addition (+), subtraction (-), multiplication (×), division (÷), whole number powers, and roots (fractional powers).
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and () The quotient rule states that the derivative of h(x) is