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