##  [Factor Theorem](/factor-theorem-0) 

 Definition

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

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

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

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

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

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

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

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

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.