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CoefficientList[poly,var]

gives a list of coefficients of powers of var in poly, starting with power 0.

CoefficientList[poly,{var1,var2,…}]

gives an array of coefficients of the vari.

CoefficientList[poly,{var1,var2,…},{dim1,dim2,…}]

gives an array of dimensions {dim1,dim2,…}, truncating or padding with zeros as needed.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Modulus  
Properties & Relations  
See Also
Tech Notes
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CoefficientList

CoefficientList[poly,var]

gives a list of coefficients of powers of var in poly, starting with power 0.

CoefficientList[poly,{var1,var2,…}]

gives an array of coefficients of the vari.

CoefficientList[poly,{var1,var2,…},{dim1,dim2,…}]

gives an array of dimensions {dim1,dim2,…}, truncating or padding with zeros as needed.

Details and Options

  • The dimensions of the array returned by CoefficientList are determined by the values of the Exponent[poly,vari].
  • Terms that do not contain positive integer powers of a particular variable are included in the first element of the list for that variable.
  • CoefficientList always returns a full rectangular array. Combinations of powers that do not appear in poly give zeros in the array.
  • CoefficientList[0,var] gives {}.
  • CoefficientList works whether or not poly is explicitly given in expanded form.

Examples

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Basic Examples  (3)

Find the coefficients in a polynomial:

Wolfram Language code: CoefficientList[1 + 6x - x ^ 4, x]

CoefficientList works even when the polynomial has not been expanded out:

Wolfram Language code: CoefficientList[(1 + x) ^ 10 , x]

Matrix of coefficients for a quadratic function:

Wolfram Language code: CoefficientList[1 + a x ^ 2 + b x y + c y ^ 2, {x, y}]

Scope  (2)

Univariate polynomial coefficient lists:

Wolfram Language code: CoefficientList[(2x + 3) ^ 5, x]
Wolfram Language code: CoefficientList[a x ^ 4 + b x ^ 3 + c x ^ 2 + d x + e, x]

Multivariate polynomial coefficient lists:

Wolfram Language code: CoefficientList[(3x + 4 y + 1) ^ 3, {x, y}]
Wolfram Language code: CoefficientList[(x + y + z + 1)(2x + 3y ^ 2 + 4z ^ 3 + 5), {x, y, z}]

Options  (1)

Modulus  (1)

Coefficient list over the integers modulo 2:

Wolfram Language code: CoefficientList[(x + 1) ^ 5, x, Modulus -> 2]

Properties & Relations  (4)

Use Coefficient to get a coefficient at a specified power of the variable:

Wolfram Language code: f = (2x + 3) ^ 3;
Wolfram Language code: Coefficient[f, x, 2]

The list of coefficients can be obtained using Coefficient and Exponent:

Wolfram Language code: Coefficient[f, x, #]& /@ Range[0, Exponent[f, x]]
Wolfram Language code: CoefficientList[f, x]

FromDigits can reconstruct a univariate polynomial from the list of its coefficients:

Wolfram Language code: CoefficientList[a + b x + c x ^ 2, x]
Wolfram Language code: FromDigits[Reverse[%], x]

Fold the operation for multivariate polynomials:

Wolfram Language code: CoefficientList[(x + 2y) ^ 3, {x, y}]
Wolfram Language code: Fold[FromDigits[Reverse[#1], #2]&, %, {x, y}]
Wolfram Language code: Expand[(x + 2y) ^ 3 - %]

Polynomial multiplication is convolution as performed by ListConvolve:

Wolfram Language code: CoefficientList[(a + b x + c x ^ 2)(1 + 2x + 3x ^ 2 + 4x ^ 3), x]
Wolfram Language code: ListConvolve[{a, b, c}, {1, 2, 3, 4}, {1, -1}, 0]

For multivariate polynomials, CoefficientList gives a tensor of the coefficients:

Wolfram Language code: f = (3x + 5y) ^ 4;
Wolfram Language code: cl = CoefficientList[f, {x, y}]

CoefficientArrays gives the list of arrays of polynomial coefficients ordered by total degrees:

Wolfram Language code: ca = CoefficientArrays[f, {x, y}]

The coefficient of :

Wolfram Language code: Coefficient[f, x y ^ 3]

In cl, the coefficient of x^a y^b is the element at position {a+1,b+1}:

Wolfram Language code: cl[[1 + 1, 1 + 3]]

In ca, the position of this coefficient is a+b+1 followed by a 1s and b 2s (1 and 2 indicate the first and second variables):

Wolfram Language code: ca[[5, 1, 2, 2, 2]]
Wolfram Research (1988), CoefficientList, Wolfram Language function, https://reference.wolfram.com/language/ref/CoefficientList.html (updated 2015).

Text

Wolfram Research (1988), CoefficientList, Wolfram Language function, https://reference.wolfram.com/language/ref/CoefficientList.html (updated 2015).

CMS

Wolfram Language. 1988. "CoefficientList." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/CoefficientList.html.

APA

Wolfram Language. (1988). CoefficientList. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CoefficientList.html

BibTeX

@misc{reference.wolfram_2026_coefficientlist, author="Wolfram Research", title="{CoefficientList}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/CoefficientList.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_coefficientlist, organization={Wolfram Research}, title={CoefficientList}, year={2015}, url={https://reference.wolfram.com/language/ref/CoefficientList.html}, note=[Accessed: 12-June-2026]}