Restteoremen

Restteoremen, also known as the Remainder Theorem, is a fundamental concept in algebra that relates the remainder of a polynomial division to the value of the polynomial at a specific point. Specifically, it states that when a polynomial $P(x)$ is divided by a linear binomial of the form $(x-a)$, the remainder is equal to $P(a)$.

To understand why this is true, consider the division algorithm for polynomials. When $P(x)$ is divided by $(x-a)$, we can write $P(x) = Q(x)(x-a) + R$, where $Q(x)$ is the quotient polynomial and $R$ is the remainder. Since the divisor $(x-a)$ is of degree 1, the remainder $R$ must be a constant (a polynomial of degree 0). If we substitute $x=a$ into this equation, we get $P(a) = Q(a)(a-a) + R$. This simplifies to $P(a) = Q(a)(0) + R$, which further simplifies to $P(a) = R$. Thus, the remainder $R$ is indeed equal to $P(a)$.

The Remainder Theorem has several practical applications. It provides a quick way to check if a specific value is a root of a polynomial without performing long division. If $P(a) = 0$, then according to the Remainder Theorem, the remainder when $P(x)$ is divided by $(x-a)$ is 0. This means $(x-a)$ is a factor of $P(x)$, which is the essence of the Factor Theorem, a direct consequence of the Remainder Theorem. This theorem is a cornerstone for understanding polynomial factorization and solving polynomial equations.