The statement that for a polynomial p(x) over a field (or integral domain) and a scalar r, r is a root of p(x) (p(r)=0) if and only if the linear polynomial (x−r) is a factor of p(x).

Principle

Roots and linear factors are equivalent: zeros of a polynomial correspond exactly to linear factors, allowing factorization to proceed by identifying roots and extracting linear factors.

Demonstration

Example: p(x) = x^2 − 5x + 6 has p(2)=0 and p(3)=0, hence (x−2) and (x−3) are factors and p(x) = (x−2)(x−3). Extraction via polynomial division or synthetic division exhibits the linear factor explicitly.

Misapplication

Applying the theorem to functions that are not polynomials or to polynomials over rings where division by (x−r) is not well behaved; ignoring multiplicity by assuming a simple root when (x−r)^k may be the true factor.

Consequence

Translates root-finding into factorization and vice versa; enables construction of complete factorization over an algebraically closed field and underpins algorithms for solving polynomial equations numerically and symbolically.

Reversal

The reversal is the Remainder Theorem specialized: if (x−r) divides p(x) then p(r)=0. The contrast arises when no linear factor over the base field exists even though p has roots in an extension field.

Boundary

Best stated over fields or integral domains; over coefficient rings that are not integral domains or when r lies outside the coefficient ring one must treat factors in extensions. The theorem addresses linear factors only.

Semantic Tension

Tension between factoring over the base coefficient field and factoring over extensions: a polynomial may have no linear factors over Q yet factor linearly over R or C, so 'root' must be qualified by the field considered.

Synthesis

The Factor Theorem links the algebraic condition p(r)=0 to the algebraic operation of division by (x−r): identifying a root produces a linear factor and vice versa, providing the basic mechanism for stepwise polynomial factorization.