Any polynomial f(y) divided by a number in a form (y-d) yields a remainder "r" that can also be represented as another polynomial f(d), where "d" and "r" are integers and y is a variable constituting the dividend polynomial. This statement presents the basic idea that the remainder obtained after division of f(y) can be also obtained by simply calculating the polynomial value of f(d), given that the values of "y" and "d" are known.
The implementation of the remainder theorem is usually carried out over polynomials of different degrees in order to obtain their remainder values. The "degree" of a polynomial refers to the highest power of its variables, and there is no evident relationship between this power and the value of remainder obtained. An exemplary implementation of remainder theorem over a sample polynomial can be explained by considering the sample polynomial f(y) = 2y-4 divided by (y-3); since y=3, therefore, putting "y" in f(y) results in 2(3) - 4 which gives 2 as the remainder of this division process. In this way, the remainder theorem makes it possible to obtain the value of the remainder without carrying out the entire long division process.
The remainder theorem is used extensively by mathematics students when manipulating polynomials of higher degrees, the division of which is a difficult and time-consuming operation. Furthermore, this generalized theorem is also employed in engineering software and electronic mathematical applications, through which polynomials of higher degrees and longer arithmetic structures are divided without any complexity.
The most relevant association of remainder theorem is with the process of "polynomial long division" (another method of polynomial division), which employs a more detailed procedure to divide polynomials of different degrees. Additionally, this theorem finds close association with "Little Bezout's theorem," which in fact is the original form of remainder theorem. Another association of remainder theorem is with factor theorem, which is simultaneously used with the former to obtain the roots of a polynomial. Finally, a theorem known as "Chinese remainder theorem" is a modification of remainder theorem, but its application is entirely in the domain of advanced number theory (a branch of pure mathematics) instead of in common algebraic equations.