--- title: "FactorList" language: "en" type: "Symbol" summary: "FactorList[poly] gives a list of the factors of a polynomial, together with their exponents." keywords: - factors list - irreducible factors - list of factors in polynomials - prime factors - unique factorization domain - factors canonical_url: "https://reference.wolfram.com/language/ref/FactorList.html" source: "Wolfram Language Documentation" related_guides: - title: "Polynomial Factoring & Decomposition" link: "https://reference.wolfram.com/language/guide/PolynomialFactoring.en.md" - title: "Polynomial Algebra" link: "https://reference.wolfram.com/language/guide/PolynomialAlgebra.en.md" - title: "Finite Fields" link: "https://reference.wolfram.com/language/guide/FiniteFields.en.md" related_functions: - title: "FactorTermsList" link: "https://reference.wolfram.com/language/ref/FactorTermsList.en.md" - title: "TrigFactorList" link: "https://reference.wolfram.com/language/ref/TrigFactorList.en.md" - title: "CoefficientList" link: "https://reference.wolfram.com/language/ref/CoefficientList.en.md" - title: "Factor" link: "https://reference.wolfram.com/language/ref/Factor.en.md" - title: "Decompose" link: "https://reference.wolfram.com/language/ref/Decompose.en.md" - title: "FiniteField" link: "https://reference.wolfram.com/language/ref/FiniteField.en.md" related_tutorials: - title: "Algebraic Operations on Polynomials" link: "https://reference.wolfram.com/language/tutorial/AlgebraicManipulation.en.md#13694" --- # FactorList FactorList[poly] gives a list of the factors of a polynomial, together with their exponents. ## Details and Options * The first element of the list is always the overall numerical factor. It is ``{1, 1}`` if there is no overall numerical factor. * ``FactorList`` takes the following options: | | | | | ----------------- | ----- | ---------------------------------------------------------------- | | Extension | None | generators for the algebraic number field to be used | | GaussianIntegers | False | whether to allow Gaussian integer coefficients | | Modulus | 0 | modulus to assume for integers | | Trig | False | whether to do trigonometric as well as algebraic transformations | * ``FactorList[poly, Modulus -> p]`` requires that ``p`` be prime. * ``FactorList[poly, Extension -> {a1, a2, …}]`` allows coefficients that are arbitrary rational combinations of the ``ai``. --- ## Examples (25) ### Basic Examples (2) List the irreducible factors of polynomials: ```wl In[1]:= FactorList[x ^ 2 - 1] Out[1]= {{1, 1}, {-1 + x, 1}, {1 + x, 1}} In[2]:= FactorList[2x ^ 3 + 2x ^ 2 - 2x - 2] Out[2]= {{2, 1}, {-1 + x, 1}, {1 + x, 2}} ``` --- List the irreducible factors of multivariate polynomials: ```wl In[1]:= FactorList[x ^ 3 + 2x ^ 2y + x y ^ 2 + x ^ 2z - y ^ 2z - x z ^ 2 - 2y z ^ 2 - z ^ 3] Out[1]= {{1, 1}, {x - z, 1}, {x + y + z, 2}} ``` ### Scope (11) #### Basic Uses (4) A univariate polynomial: ```wl In[1]:= FactorList[x ^ 3 - 6x ^ 2 + 11x - 6] Out[1]= {{1, 1}, {-3 + x, 1}, {-2 + x, 1}, {-1 + x, 1}} ``` --- A multivariate polynomial: ```wl In[1]:= FactorList[2x ^ 3 y - 2a ^ 2x y - 3a ^ 2x ^ 2 + 3a ^ 4] Out[1]= {{1, 1}, {a - x, 1}, {a + x, 1}, {3 a^2 - 2 x y, 1}} ``` --- A polynomial with multiple factors: ```wl In[1]:= FactorList[x ^ 8 + 11x ^ 7 + 43x ^ 6 + 59x ^ 5 - 35x ^ 4 - 151x ^ 3 - 63x ^ 2 + 81x + 54] Out[1]= {{1, 1}, {-1 + x, 2}, {1 + x, 2}, {2 + x, 1}, {3 + x, 3}} ``` --- A rational function: ```wl In[1]:= FactorList[(x ^ 3 + 2x ^ 2) / (x ^ 2 - 4y ^ 2) - (x + 2) / (x ^ 2 - 4y ^ 2)] Out[1]= {{1, 1}, {-1 + x, 1}, {1 + x, 1}, {2 + x, 1}, {x - 2 y, -1}, {x + 2 y, -1}} ``` #### Advanced Uses (7) The irreducible factors over the Gaussian integers: ```wl In[1]:= FactorList[x ^ 2 + 1, GaussianIntegers -> True] Out[1]= {{1, 1}, {-I + x, 1}, {I + x, 1}} ``` --- The irreducible factors over an algebraic extension: ```wl In[1]:= FactorList[x ^ 4 - 2, Extension -> Sqrt[2]] Out[1]= {{-1, 1}, {Sqrt[2] - x^2, 1}, {Sqrt[2] + x^2, 1}} ``` --- The irreducible factors over the integers modulo 2: ```wl In[1]:= FactorList[x ^ 4 + 1, Modulus -> 2] Out[1]= {{1, 1}, {1 + x, 4}} ``` --- The irreducible factors over a finite field: ```wl In[1]:= ℱ = FiniteField[17, 3]; In[2]:= FactorList[ℱ[3]x ^ 2 + ℱ[1771]] Out[2]= [image] In[3]:= FactorList[x ^ 3 + 5x + 19, Extension -> ℱ] Out[3]= [image] ``` --- The irreducible factors over an extension of a finite field: ```wl In[1]:= ℱ = FiniteField[29, 3]; \[ScriptCapitalG] = FiniteField[29, 6]; ℰ = FiniteFieldEmbedding[ℱ, \[ScriptCapitalG]]; ``` A polynomial irreducible over $\mathcal{F}$ factors after embedding $\mathcal{F}$ in a larger field $\mathcal{G}$ : ```wl In[2]:= FactorList[ℱ[12]x ^ 2 + ℱ[34]x + ℱ[56]] Out[2]= [image] In[3]:= FactorList[ℱ[12]x ^ 2 + ℱ[34]x + ℱ[56], Extension -> ℰ] Out[3]= [image] ``` --- List factors of non-polynomial expressions: ```wl In[1]:= FactorList[E ^ (2 x) - 2E ^ x + 1] Out[1]= {{1, 1}, {-1 + E^x, 2}} In[2]:= FactorList[x ^ 2 - Sqrt[x]] Out[2]= {{1, 1}, {-1 + Sqrt[x], 1}, {Sqrt[x], 1}, {1 + Sqrt[x] + x, 1}} ``` --- List factors of a polynomial of degree $2000$ : ```wl In[1]:= rpoly[n_] := RandomInteger[{-2 ^ 10, 2 ^ 10}, {n + 1}].x ^ Range[0, n] SeedRandom[1234]; p = rpoly[1000]; q = rpoly[1000]; r = Expand[p q]; In[2]:= (facs = FactorList[r]);//AbsoluteTiming Out[2]= {1.3239, Null} In[3]:= {Exponent[#[[1]], x], #[[2]]}& /@ facs Out[3]= {{0, 1}, {1000, 1}, {1000, 1}} ``` ### Options (7) #### Extension (4) Factor over algebraic number fields: ```wl In[1]:= FactorList[1 + x ^ 4, Extension -> Sqrt[2]] Out[1]= {{-1, 1}, {-1 + Sqrt[2] x - x^2, 1}, {1 + Sqrt[2] x + x^2, 1}} In[2]:= FactorList[1 + x ^ 4, Extension -> {Sqrt[2], I}] Out[2]= {{1, 1}, {4, -1}, {Sqrt[2] + (1 + I) x, 1}, {Sqrt[2] + (1 - I) x, 1}, {Sqrt[2] - (1 - I) x, 1}, {Sqrt[2] - (1 + I) x, 1}} ``` --- ``Extension -> Automatic`` automatically extends to a field that covers the coefficients: ```wl In[1]:= FactorList[2 + 2Sqrt[2]x + x ^ 2] Out[1]= {{1, 1}, {2 + 2 Sqrt[2] x + x^2, 1}} In[2]:= FactorList[2 + 2Sqrt[2]x + x ^ 2, Extension -> Automatic] Out[2]= {{1, 1}, {Sqrt[2] + x, 2}} ``` --- Factor a polynomial with integer coefficients over a finite field: ```wl In[1]:= ℱ = FiniteField[73, 3]; In[2]:= FactorList[x ^ 3 + 21x + 3, Extension -> ℱ] Out[2]= [image] ``` --- Factor a polynomial with coefficients in a finite field: ```wl In[1]:= ℱ = FiniteField[7, 2]; In[2]:= FactorList[ℱ[1]x ^ 4 + ℱ[234]x + ℱ[567]] Out[2]= [image] ``` Embedding $\mathcal{F}$ in a larger field $\mathcal{G}$ allows further factorization: ```wl In[3]:= \[ScriptCapitalG] = FiniteField[7, 4]; ℰ = FiniteFieldEmbedding[ℱ, \[ScriptCapitalG]]; In[4]:= FactorList[ℱ[1]x ^ 4 + ℱ[234]x + ℱ[567], Extension -> ℰ] Out[4]= [image] ``` #### GaussianIntegers (1) Factor over Gaussian integers: ```wl In[1]:= FactorList[1 + x ^ 2, GaussianIntegers -> True] Out[1]= {{1, 1}, {-I + x, 1}, {I + x, 1}} In[2]:= FactorList[1 + x ^ 2, Extension -> I] Out[2]= {{1, 1}, {-I + x, 1}, {I + x, 1}} ``` #### Modulus (1) Factor over finite fields: ```wl In[1]:= FactorList[5 + x ^ 12] Out[1]= {{1, 1}, {5 + x^12, 1}} In[2]:= FactorList[5 + x ^ 12, Modulus -> 7] Out[2]= {{1, 1}, {2 + x^3, 1}, {5 + x^3, 1}, {4 + x^6, 1}} ``` #### Trig (1) Factor a trigonometric expression: ```wl In[1]:= FactorList[Sin[x] + Sin[y], Trig -> True] Out[1]= {{2, 1}, {Cos[(x/2) - (y/2)], 1}, {Sin[(x/2) + (y/2)], 1}} ``` ### Applications (2) ``FactorList`` can be useful in determining the behavior of functions: ```wl In[1]:= f = x ^ 3 - x ^ 2 - 16x - 20; In[2]:= FactorList[f] Out[2]= {{1, 1}, {-5 + x, 1}, {2 + x, 2}} ``` $f$ has roots at $-2$ and $5$, at $-2$ it does not cross the $x$ axis, and at $5$ it crosses the $x$ axis: ```wl In[3]:= Plot[-20 - 16 x - x^2 + x^3, {x, -4, 6}] Out[3]= [image] ``` --- Show that a polynomial is a perfect square: ```wl In[1]:= f = 62500 - 112500 x + 15625 x^2 + 47500 x^3 - 3500 x^4 - 9380 x^5 - 1106 x^6 + 628 x^7 + 208 x^8 + 24 x^9 + x^10; ``` Compute the factors and note that the constant factor is positive: ```wl In[2]:= fac = FactorList[f] Out[2]= {{1, 1}, {-2 + x, 2}, {-1 + x, 2}, {5 + x, 6}} ``` Extract the exponents of nonconstant factors: ```wl In[3]:= exp = Transpose[Rest[fac]][[2]] Out[3]= {2, 2, 6} ``` Check that every factor has even exponent and thus $f$ is a square: ```wl In[4]:= AllTrue[exp, EvenQ] Out[4]= True ``` ### Properties & Relations (3) ``FactorList`` gives a list of irreducible factors: ```wl In[1]:= FactorList[x ^ 8 + 11x ^ 7 + 43x ^ 6 + 59x ^ 5 - 35x ^ 4 - 151x ^ 3 - 63x ^ 2 + 81x + 54] Out[1]= {{1, 1}, {-1 + x, 2}, {1 + x, 2}, {2 + x, 1}, {3 + x, 3}} ``` This multiplies the factors together: ```wl In[2]:= Times@@Power@@@% Out[2]= (-1 + x)^2 (1 + x)^2 (2 + x) (3 + x)^3 ``` ``Factor`` gives a product of factors: ```wl In[3]:= Factor[x ^ 8 + 11x ^ 7 + 43x ^ 6 + 59x ^ 5 - 35x ^ 4 - 151x ^ 3 - 63x ^ 2 + 81x + 54] Out[3]= (-1 + x)^2 (1 + x)^2 (2 + x) (3 + x)^3 ``` ``Expand`` combines all the factors back together: ```wl In[4]:= Expand[%] Out[4]= 54 + 81 x - 63 x^2 - 151 x^3 - 35 x^4 + 59 x^5 + 43 x^6 + 11 x^7 + x^8 ``` --- ``FactorSquareFreeList`` gives a list of square-free factors: ```wl In[1]:= FactorSquareFreeList[x ^ 8 + 11x ^ 7 + 43x ^ 6 + 59x ^ 5 - 35x ^ 4 - 151x ^ 3 - 63x ^ 2 + 81x + 54] Out[1]= {{1, 1}, {2 + x, 1}, {3 + x, 3}, {-1 + x^2, 2}} ``` --- ``FactorInteger`` gives a list of prime factors of an integer: ```wl In[1]:= FactorInteger[1234567] Out[1]= {{127, 1}, {9721, 1}} In[2]:= Times@@Power@@@% Out[2]= 1234567 ``` ## See Also * [`FactorTermsList`](https://reference.wolfram.com/language/ref/FactorTermsList.en.md) * [`TrigFactorList`](https://reference.wolfram.com/language/ref/TrigFactorList.en.md) * [`CoefficientList`](https://reference.wolfram.com/language/ref/CoefficientList.en.md) * [`Factor`](https://reference.wolfram.com/language/ref/Factor.en.md) * [`Decompose`](https://reference.wolfram.com/language/ref/Decompose.en.md) * [`FiniteField`](https://reference.wolfram.com/language/ref/FiniteField.en.md) ## Tech Notes * [Algebraic Operations on Polynomials](https://reference.wolfram.com/language/tutorial/AlgebraicManipulation.en.md#13694) ## Related Guides * [Polynomial Factoring & Decomposition](https://reference.wolfram.com/language/guide/PolynomialFactoring.en.md) * [Polynomial Algebra](https://reference.wolfram.com/language/guide/PolynomialAlgebra.en.md) * [Finite Fields](https://reference.wolfram.com/language/guide/FiniteFields.en.md) ## History * Introduced in 1988 (1.0) \| Updated in 1996 (3.0) ▪ [2022 (13.2)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn132.en.md) ▪ [2023 (13.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn133.en.md)