# Factor theorem

The factor theorem is a statement in algebra that a number \( a \) is a zero of a polynomial \( f(x) \) if and only if \( x - a \) is a factor of \( f(x) \); equivalently, \( f(a) = 0 \) exactly when \( f(x) \) is divisible by \( x - a \).<sup>[1](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/16:_An_Introduction_to_Rings_and_Fields/16.03:_Polynomial_Rings)</sup> It is a direct consequence of the remainder theorem, which identifies the remainder of division by a linear polynomial with the value of the polynomial at a point.<sup>[2](https://mathcenter.oxford.emory.edu/site/math108/rationalRoots/)</sup>

| Key fact | Detail |
|---|---|
| Statement | For a field \( F \), \( a \in F \) is a zero of \( f(x) \in F[x] \) if and only if \( x - a \) is a factor of \( f(x) \) in \( F[x] \)<sup>[1](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/16:_An_Introduction_to_Rings_and_Fields/16.03:_Polynomial_Rings)</sup> |
| Companion result | Remainder theorem: the remainder of \( f(x) \) divided by \( x - c \) is \( f(c) \)<sup>[2](https://mathcenter.oxford.emory.edu/site/math108/rationalRoots/)</sup> |
| Key hypothesis | The divisor \( x - a \) is monic (leading coefficient 1), so division works even over a general commutative ring<sup>[3](https://alphabetamath.cc/item/cor-factor-theorem-over-a-commutative-ring)</sup> |
| Main corollary | A nonzero polynomial of degree \( n \) over a field has at most \( n \) zeros<sup>[1](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/16:_An_Introduction_to_Rings_and_Fields/16.03:_Polynomial_Rings)</sup> |
| Practical use | Testing candidate roots by substitution, then dividing out known factors to reduce degree<sup>[4](https://amsi.org.au/ESA_Senior_Years/SeniorTopic2/2e/2e_2content_3.html)</sup> |
| Ring generality | Over any commutative ring \( R \), \( f(a) = 0 \) if and only if \( x - a \) divides \( f \) in \( R[x] \)<sup>[3](https://alphabetamath.cc/item/cor-factor-theorem-over-a-commutative-ring)</sup> |

## Statement and relation to the remainder theorem

The theorem is usually presented alongside the remainder theorem. Dividing \( f(x) \) by \( x - c \) gives a quotient \( q(x) \) and a remainder that must be a constant \( r \), because the remainder's degree is strictly less than the divisor's degree of 1:<sup>[2](https://mathcenter.oxford.emory.edu/site/math108/rationalRoots/)</sup>

\[ f(x) = q(x) \cdot (x - c) + r. \]

Substituting \( x = c \) collapses the product term, so \( f(c) = r \). This is the remainder theorem: evaluating \( f(c) \) reveals the remainder without carrying out the division.<sup>[2](https://mathcenter.oxford.emory.edu/site/math108/rationalRoots/)</sup> The factor theorem follows immediately as the special case \( r = 0 \): the remainder vanishes precisely when \( f(c) = 0 \), and zero remainder means \( x - c \) divides \( f(x) \) evenly.<sup>[2](https://mathcenter.oxford.emory.edu/site/math108/rationalRoots/)</sup>

## Proof

The standard proof over a field \( F \) runs as follows.<sup>[1](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/16:_An_Introduction_to_Rings_and_Fields/16.03:_Polynomial_Rings)</sup>

1. Apply the division property for polynomials to \( f(x) \) and \( g(x) = x - a \), obtaining unique \( q(x), r(x) \in F[x] \) with \( f(x) = (x - a) \cdot q(x) + r(x) \) and \( \deg r(x) < \deg(x - a) = 1 \). The remainder is therefore a constant \( c \in F \).<sup>[1](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/16:_An_Introduction_to_Rings_and_Fields/16.03:_Polynomial_Rings)</sup>
2. If \( a \) is a zero, then \( 0 = f(a) = (a - a) \, q(a) + c = c \), so \( c = 0 \) and \( f(x) = (x - a) \cdot q(x) \).<sup>[1](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/16:_An_Introduction_to_Rings_and_Fields/16.03:_Polynomial_Rings)</sup>
3. Conversely, if \( f(x) = (x - a) \, q(x) \), evaluating at \( a \) gives \( f(a) = q(a) \cdot (a - a) = 0 \).<sup>[3](https://alphabetamath.cc/item/cor-factor-theorem-over-a-commutative-ring)</sup>

The critical ingredient is the division property's guarantee that \( \deg r(x) < \deg g(x) \), which forces the remainder to be a constant.<sup>[1](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/16:_An_Introduction_to_Rings_and_Fields/16.03:_Polynomial_Rings)</sup>

## General form over commutative rings

The theorem extends beyond fields. Over any commutative ring \( R \) and any \( a \in R \), there is a unique \( q \in R[x] \) such that

\[ f = q \cdot (x - a) + f(a), \]

so \( f(a) = 0 \) if and only if \( x - a \) divides \( f \) in \( R[x] \).<sup>[3](https://alphabetamath.cc/item/cor-factor-theorem-over-a-commutative-ring)</sup> The proof relies on division by the monic polynomial \( x - a \), which needs no invertible leading coefficient and therefore works over a ring; the zero ring is handled separately because there \( x - a = 0 \) is not monic.<sup>[3](https://alphabetamath.cc/item/cor-factor-theorem-over-a-commutative-ring)</sup>

In the language of ring theory, evaluation at \( a \) is a ring homomorphism \( \operatorname{ev}_a : R[x] \to R \), and the factor theorem says its kernel is exactly the ideal generated by \( x - a \): a polynomial evaluates to zero at \( a \) precisely when it is a multiple of \( x - a \).<sup>[5](https://www.math.purdue.edu/~walther/teach/453/2025/453notes.pdf)</sup><sup> • </sup><sup>[6](https://androma.org/theorems/17387)</sup>

## Consequences

**Bound on the number of zeros.** Over a field, a nonzero polynomial of degree \( n \) has at most \( n \) zeros. Each zero \( a \) yields a factor \( x - a \) by the factor theorem, and peeling off factors reduces the degree by one each time, so the process terminates in at most \( n \) steps.<sup>[1](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/16:_An_Introduction_to_Rings_and_Fields/16.03:_Polynomial_Rings)</sup> The same argument shows that if \( r_1, r_2, \ldots, r_k \) are distinct zeros of \( f \), then the product \( (x - r_1)(x - r_2) \cdots (x - r_k) \) is a factor of \( f(x) \).<sup>[4](https://amsi.org.au/ESA_Senior_Years/SeniorTopic2/2e/2e_2content_3.html)</sup> Applied to a polynomial of degree \( n \) with \( n + 1 \) distinct real solutions, this would force a factor of degree \( n + 1 \), a contradiction; hence a degree-\( n \) equation has at most \( n \) distinct real solutions, and the graph \( y = f(x) \) has at most \( n \) x-intercepts.<sup>[4](https://amsi.org.au/ESA_Senior_Years/SeniorTopic2/2e/2e_2content_3.html)</sup>

**Where the bound can fail.** The field (or integral domain) hypothesis matters. In the ring \( \mathbb{Z}/12\mathbb{Z} \), the polynomial \( x^2 - 1 \) has four roots, namely \( 1, 5, 7, \) and \( 11 \) modulo 12, and factorizations such as \( x^2 - 1 = (x-1)(x+1) = (x-5)(x-7) \) coexist; the failure comes from [zero divisors](https://www.edgepedia.org/zero-divisor), since \( \mathbb{Z}/12\mathbb{Z} \) is not a domain.<sup>[5](https://www.math.purdue.edu/~walther/teach/453/2025/453notes.pdf)</sup> Over a domain the at-most-\( n \) bound holds even counting roots with multiplicity.<sup>[5](https://www.math.purdue.edu/~walther/teach/453/2025/453notes.pdf)</sup>

**Dependence on the ground field.** A zero must belong to the coefficient field for the factor theorem to apply over that field. The polynomial \( x^2 - 2 \) has zeros \( \pm\sqrt{2} \) and factors as \( (x - \sqrt{2})(x + \sqrt{2}) \) over the reals, but neither factor lies in \( \mathbb{Q}[x] \), so over the rationals the polynomial is irreducible despite having real roots.<sup>[1](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/16:_An_Introduction_to_Rings_and_Fields/16.03:_Polynomial_Rings)</sup> Similarly, \( x^2 + 1 \) has no roots in \( \mathbb{Q} \).<sup>[5](https://www.math.purdue.edu/~walther/teach/453/2025/453notes.pdf)</sup> Allowing complex numbers removes this obstruction: by the fundamental theorem of algebra, every polynomial equation of degree \( n \) with complex coefficients has \( n \) roots in the complex numbers, counted with multiplicity.<sup>[7](https://mathshistory.st-andrews.ac.uk//HistTopics/Fund_theorem_of_algebra/)</sup><sup> • </sup><sup>[8](https://arxiv.org/abs/2510.10301)</sup>

## Use in factoring polynomials

The theorem converts root-finding into factoring and back. To factor a polynomial of degree greater than 2, one hunts for a value \( c \) with \( f(c) = 0 \), then divides by \( x - c \) to obtain a quotient of lower degree and repeats.<sup>[4](https://amsi.org.au/ESA_Senior_Years/SeniorTopic2/2e/2e_2content_3.html)</sup><sup> • </sup><sup>[2](https://mathcenter.oxford.emory.edu/site/math108/rationalRoots/)</sup> Conversely, once a linear factor is known, division by it produces the remaining factorization.

For polynomials with integer coefficients, the [rational root theorem](https://www.edgepedia.org/rational-root-theorem) narrows the search: if \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \) has a rational root \( c = p/q \) in lowest terms, then \( p \) divides \( a_0 \) and \( q \) divides \( a_n \), leaving only a small number of candidate values to test by substitution.<sup>[2](https://mathcenter.oxford.emory.edu/site/math108/rationalRoots/)</sup> The theorem also generalizes to non-monic linear divisors: since \( P(\tfrac{1}{2}) = 0 \) implies \( (2x - 1) \) is a factor of \( P(x) \), a root of \( P \) at \( a/b \) corresponds to a factor \( (bx - a) \).<sup>[9](https://learningbro.com/courses/edexcel-alevel-maths-algebra-functions/lessons/factor-theorem)</sup>

Computationally, the remainder theorem justifies evaluating \( f(c) \) by synthetic division, which produces the quotient coefficients and the remainder \( f(c) \) directly and more cheaply than full polynomial division.<sup>[10](https://mathresearch.utsa.edu/wiki/index.php?title=Remainder_and_Factor_Theorem)</sup>

## History

The underlying observation, that a polynomial vanishing at \( t \) has a factor \( x - t \), was known to [Thomas Harriot](https://www.edgepedia.org/thomas-harriot), but it did not become well known until [René Descartes](https://www.edgepedia.org/rene-descartes) stated it in *La géométrie* in 1637.<sup>[7](https://mathshistory.st-andrews.ac.uk//HistTopics/Fund_theorem_of_algebra/)</sup> The factor theorem supplies the algebraic step in the standard proof that a degree-\( n \) polynomial over a field has at most \( n \) roots, a count that the fundamental theorem of algebra raises to exactly \( n \) once complex roots are admitted.<sup>[1](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/16:_An_Introduction_to_Rings_and_Fields/16.03:_Polynomial_Rings)</sup><sup> • </sup><sup>[7](https://mathshistory.st-andrews.ac.uk//HistTopics/Fund_theorem_of_algebra/)</sup>

## References

1. [Polynomial Rings (Division Property, Factor Theorem, zeros bound), Mathematics LibreTexts, Applied Discrete Structures](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/16:_An_Introduction_to_Rings_and_Fields/16.03:_Polynomial_Rings)
2. [The Remainder, Factor, and Rational Roots Theorems, Emory University Math Center](https://mathcenter.oxford.emory.edu/site/math108/rationalRoots/)
3. [Factor theorem over a commutative ring, Alphabeta Math](https://alphabetamath.cc/item/cor-factor-theorem-over-a-commutative-ring)
4. [Solving polynomial equations, AMSI Senior Years](https://amsi.org.au/ESA_Senior_Years/SeniorTopic2/2e/2e_2content_3.html)
5. [Divisibility, Field Extensions (Week 11/12 notes), Purdue University Math 453](https://www.math.purdue.edu/~walther/teach/453/2025/453notes.pdf)
6. [Kernel of Evaluation at a Coefficient-Ring Element, Androma](https://androma.org/theorems/17387)
7. [The fundamental theorem of algebra, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//HistTopics/Fund_theorem_of_algebra/)
8. [Around the "Fundamental Theorem of Algebra", arXiv](https://arxiv.org/abs/2510.10301)
9. [The Factor Theorem and Remainder Theorem, Edexcel A Level Maths (Learning Bro)](https://learningbro.com/courses/edexcel-alevel-maths-algebra-functions/lessons/factor-theorem)
10. [Remainder and Factor Theorem, UTSA Department of Mathematics](https://mathresearch.utsa.edu/wiki/index.php?title=Remainder_and_Factor_Theorem)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Ring theory › Factorization and orders › Factorization of polynomials over rings and fields*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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