--- title: "FactorTermsList" language: "en" type: "Symbol" summary: "FactorTermsList[poly] gives a list in which the first element is the overall numerical factor in poly, and the second element is the polynomial with the overall factor removed. FactorTermsList[poly, {x1, x2, ...}] gives a list of factors of poly. The first element in the list is the overall numerical factor. The second element is a factor that does not depend on any of the xi. Subsequent elements are factors which depend on progressively more of the xi." keywords: - content of a polynomial - polynomial factors - icontent canonical_url: "https://reference.wolfram.com/language/ref/FactorTermsList.html" source: "Wolfram Language Documentation" related_guides: - title: "Polynomial Factoring & Decomposition" link: "https://reference.wolfram.com/language/guide/PolynomialFactoring.en.md" related_functions: - title: "FactorTerms" link: "https://reference.wolfram.com/language/ref/FactorTerms.en.md" - title: "FactorList" link: "https://reference.wolfram.com/language/ref/FactorList.en.md" - title: "FactorSquareFreeList" link: "https://reference.wolfram.com/language/ref/FactorSquareFreeList.en.md" related_tutorials: - title: "Algebraic Operations on Polynomials" link: "https://reference.wolfram.com/language/tutorial/AlgebraicManipulation.en.md#13694" --- # FactorTermsList FactorTermsList[poly] gives a list in which the first element is the overall numerical factor in poly, and the second element is the polynomial with the overall factor removed. FactorTermsList[poly, {x1, x2, …}] gives a list of factors of poly. The first element in the list is the overall numerical factor. The second element is a factor that does not depend on any of the xi. Subsequent elements are factors which depend on progressively more of the xi. ## Details and Options * ``FactorTermsList`` takes the following options: | | | | | -------- | ----- | ---------------------------------------------------------------- | | Modulus | 0 | modulus to assume for integers | | Trig | False | whether to do trigonometric as well as algebraic transformations | --- ## Examples (14) ### Basic Examples (2) Pull out an overall numerical factor, but do no further factoring: ```wl In[1]:= FactorTermsList[2x ^ 2 - 2] Out[1]= {2, -1 + x^2} ``` --- Pull out factors that do not depend on $x$ : ```wl In[1]:= FactorTermsList[3 + 3a + 6a x + 6x + 12a x ^ 2 + 12x ^ 2, x] Out[1]= {3, 1 + a, 1 + 2 x + 4 x^2} ``` ### Scope (8) #### Basic Uses (5) A univariate polynomial: ```wl In[1]:= FactorTermsList[8x ^ 3 - 6x ^ 2 + 22x - 6] Out[1]= {2, -3 + 11 x - 3 x^2 + 4 x^3} ``` --- A multivariate polynomial: ```wl In[1]:= FactorTermsList[6a ^ 2 + 9x ^ 2 + 12b ^ 2] Out[1]= {3, 2 a^2 + 4 b^2 + 3 x^2} ``` --- A rational function: ```wl In[1]:= FactorTermsList[7 x + (14y + 21) / z] Out[1]= {7, x + (3/z) + (2 y/z)} ``` --- A polynomial with complex coefficients: ```wl In[1]:= FactorTermsList[5I x ^ 2 + 20x I + 10] Out[1]= {5 I, -2 I + 4 x + x^2} ``` --- A non-polynomial expression: ```wl In[1]:= FactorTermsList[15Sin[x] ^ 2 + 100Log[x]f[x] + 50E ^ x] Out[1]= {5, 10 E^x + 20 f[x] Log[x] + 3 Sin[x]^2} ``` #### Advanced Uses (3) List the overall numerical factor, and then factors that do not depend on $x$ : ```wl In[1]:= FactorTermsList[-6 y - 6 a y + 2 x^2 y + 2 a x^2 y + 4 a y^2 + 4 a^2 y^2, x] Out[1]= {2, y + a y, -3 + x^2 + 2 a y} ``` --- List the overall numerical factor, then factors that do not depend on $x$ and $y$, and then factors that do not depend on $x$ : ```wl In[1]:= FactorTermsList[-6 y - 6 a y + 2 x^2 y + 2 a x^2 y + 4 a y^2 + 4 a^2 y^2, {x, y}] Out[1]= {2, 1 + a, y, -3 + x^2 + 2 a y} ``` --- Pull out overall numerical factor over the integers modulo 3: ```wl In[1]:= FactorTermsList[8x ^ 2 + 5, Modulus -> 3] Out[1]= {2, 1 + x^2} ``` ### Options (1) #### Modulus (1) Pull out overall numerical factor over integers modulo 7: ```wl In[1]:= FactorTermsList[3x + 10, Modulus -> 7] Out[1]= {3, 1 + x} ``` ### Applications (1) Define a large polynomial: ```wl In[1]:= f = 2 x ^ 2 y z + 2 x ^ 2 y + 4 x ^ 2z + 4 x ^ 2 + 4 y ^ 2z ^ 2 + 4 z y ^ 2 + 8 z ^ 2 y + 2 z y - 6 y - 12 z - 12; ``` Pull out an overall numerical factor: ```wl In[2]:= FactorTermsList[f] Out[2]= {2, -6 + 2 x^2 - 3 y + x^2 y - 6 z + 2 x^2 z + y z + x^2 y z + 2 y^2 z + 4 y z^2 + 2 y^2 z^2} ``` Pull out factors that do not depend on $x$ : ```wl In[3]:= FactorTermsList[f, x] Out[3]= {2, 2 + y + 2 z + y z, -3 + x^2 + 2 y z} ``` Pull out factors that do not depend on $x$ and $y$ and then factors that do not depend on $x$ : ```wl In[4]:= FactorTermsList[f, {x, y}] Out[4]= {2, 1 + z, 2 + y, -3 + x^2 + 2 y z} ``` ### Properties & Relations (2) ``FactorTermsList`` gives a list of factors: ```wl In[1]:= FactorTermsList[14x + 21y + 35x y + 63] Out[1]= {7, 9 + 2 x + 3 y + 5 x y} ``` This multiplies the factors together: ```wl In[2]:= Times@@% Out[2]= 7 (9 + 2 x + 3 y + 5 x y) ``` ``FactorTerms`` gives a product of factors: ```wl In[3]:= FactorTerms[14x + 21y + 35x y + 63] Out[3]= 7 (9 + 2 x + 3 y + 5 x y) ``` ``Expand`` combines the factors back together: ```wl In[4]:= Expand[%] Out[4]= 63 + 14 x + 21 y + 35 x y ``` --- ``FactorList`` gives a list of all irreducible factors: ```wl In[1]:= FactorTermsList[4x ^ 3 - 4] Out[1]= {4, -1 + x^3} In[2]:= FactorList[4x ^ 3 - 4] Out[2]= {{4, 1}, {-1 + x, 1}, {1 + x + x^2, 1}} ``` ## See Also * [`FactorTerms`](https://reference.wolfram.com/language/ref/FactorTerms.en.md) * [`FactorList`](https://reference.wolfram.com/language/ref/FactorList.en.md) * [`FactorSquareFreeList`](https://reference.wolfram.com/language/ref/FactorSquareFreeList.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) ## History * Introduced in 1988 (1.0)