jakojäännöksen

Jako jäännöksen, known in English as the remainder theorem, is a concept in abstract algebra, specifically concerning polynomial rings. It describes a property of the division of a polynomial by a linear factor.

The theorem states that for a polynomial P(x) and a linear factor (x - a), the remainder of the division of P(x) by (x - a) is equal to P(a). In simpler terms, if you substitute the root of the linear factor into the polynomial, you get the remainder you would obtain if you performed polynomial long division.

This theorem is a fundamental result that simplifies the process of finding remainders. Instead of carrying out the full division algorithm, one can simply evaluate the polynomial at a specific value. This is particularly useful in polynomial factorization and in determining if a specific value is a root of a polynomial.

A direct consequence of the remainder theorem is the factor theorem. The factor theorem states that (x - a) is a factor of a polynomial P(x) if and only if P(a) = 0. This means that if evaluating the polynomial at 'a' results in zero, then 'a' is a root of the polynomial and (x - a) is a factor. This theorem is a cornerstone for finding roots and factoring polynomials.