Jääkpolünoom

Jääkpolünoom refers to the remainder when a polynomial is divided by another polynomial. When a polynomial $P(x)$ is divided by a non-zero polynomial $D(x)$, the division algorithm states that there exist unique polynomials $Q(x)$, the quotient, and $R(x)$, the remainder, such that $P(x) = D(x)Q(x) + R(x)$, where the degree of $R(x)$ is strictly less than the degree of $D(x)$, or $R(x)$ is the zero polynomial. The polynomial $R(x)$ is the jääkpolünoom.

A special case of this is the polynomial remainder theorem, which applies when the divisor is a linear polynomial of the form $(x-a)$. In this scenario, the remainder $R(x)$ is a constant, and its value is equal to $P(a)$. Thus, if $P(x)$ is divided by $(x-a)$, the jääkpolünoom is $P(a)$. This theorem is particularly useful for finding roots of polynomials.

The concept of the jääkpolünoom is fundamental in polynomial algebra and has applications in various areas of mathematics, including abstract algebra and number theory. It is used to understand the structure of polynomial rings and to perform operations like polynomial factorization and simplification. The properties of the jääkpolünoom are directly tied to the properties of the dividend and divisor polynomials.