 Factor theorem

For a polynomial $f$ over a field, $f(a)=0$ exactly when $X-a$ divides $f(X)$. Polynomial division by the monic linear factor givesThe constant polynomial remainder is thus the value obtained by evaluation. For $p(X)=X^3-3X^2+4$, the root of a polynomial $2$ gives factor $X-2$ and polynomial quotient $X^2-X-2$. The latter also vanishes at $2$, explaining the repeated factor. This provides a practical way to move between evaluation and factorization.

The Factor Theorem is a fundamental principle in algebra that relates to polynomials. It provides a way to determine whether a given polynomial has a particular linear factor. Specifically, the theorem states: If \( f(x) \) is a polynomial and \( c \) is a constant, then \( (x - c) \) is a factor of \( f(x) \) if and only if \( f(c) = 0 \).