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Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem. The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial. [3]
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The rings for which such a theorem exists are called Euclidean domains, but in this generality, uniqueness of the quotient and remainder is not guaranteed. [8] Polynomial division leads to a result known as the polynomial remainder theorem: If a polynomial f(x) is divided by x − k, the remainder is the constant r = f(k). [9] [10]
The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division. [19] When the denominator b(x) is monic and linear, that is, b(x) = x − c for some constant c, then the polynomial remainder theorem asserts that the remainder of the division of a(x) by b(x) is the evaluation a ...
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.
This is the form of the remainder term mentioned after the actual statement of Taylor's theorem with remainder in the mean value form. The Lagrange form of the remainder is found by choosing G ( t ) = ( x − t ) k + 1 {\displaystyle G(t)=(x-t)^{k+1}} and the Cauchy form by choosing G ( t ) = t − a {\displaystyle G(t)=t-a} .
Chevalley–Warning theorem (field theory) Chinese remainder theorem (number theory) Choi's theorem on completely positive maps (operator theory) Chomsky–Schützenberger enumeration theorem (formal language theory) Chomsky–Schützenberger representation theorem (formal language theory) Choquet–Bishop–de Leeuw theorem (functional analysis)
Divide the highest term of the remainder by the highest term of the divisor (x 2 ÷ x = x). Place the result (+x) below the bar. x 2 has been divided leaving no remainder, and can therefore be marked as used. The result x is then multiplied by the second term in the divisor −3 = −3x. Determine the partial remainder by subtracting 0x − ...