CoefficientList

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`CoefficientList`

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CoefficientList[poly,var]

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

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CoefficientList[poly,{var₁,var₂,…}]

gives an array of coefficients of the var_i.

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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,var_i].
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)Summary of the most common use cases

Find the coefficients in a polynomial:

In[1]:=1

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https://wolfram.com/xid/0rsvpuzzu-dci

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0rsvpuzzu-pxu

Out[1]=1

Matrix of coefficients for a quadratic function:

In[1]:=1

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https://wolfram.com/xid/0rsvpuzzu-llp

Out[1]=1

Scope (2)Survey of the scope of standard use cases

Univariate polynomial coefficient lists:

In[1]:=1

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https://wolfram.com/xid/0rsvpuzzu-fgzndi

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rsvpuzzu-dz6sqx

Out[2]=2

Multivariate polynomial coefficient lists:

In[1]:=1

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https://wolfram.com/xid/0rsvpuzzu-c2jgv

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rsvpuzzu-p0i44

Out[2]=2

Options (1)Common values & functionality for each option

Modulus (1)

Coefficient list over the integers modulo 2:

In[1]:=1

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https://wolfram.com/xid/0rsvpuzzu-b160gh

Out[1]=1

Properties & Relations (4)Properties of the function, and connections to other functions

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

In[1]:=1

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https://wolfram.com/xid/0rsvpuzzu-gf0zzl

In[2]:=2

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https://wolfram.com/xid/0rsvpuzzu-ihkoto

Out[2]=2

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

In[3]:=3

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https://wolfram.com/xid/0rsvpuzzu-dg0m56

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0rsvpuzzu-e399e1

Out[4]=4

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

In[1]:=1

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https://wolfram.com/xid/0rsvpuzzu-oxg

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rsvpuzzu-l7o

Out[2]=2

Fold the operation for multivariate polynomials:

In[3]:=3

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https://wolfram.com/xid/0rsvpuzzu-b6zbac

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0rsvpuzzu-mat6i

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0rsvpuzzu-jrp64a

Out[5]=5

Polynomial multiplication is convolution as performed by ListConvolve:

In[1]:=1

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https://wolfram.com/xid/0rsvpuzzu-mcf

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rsvpuzzu-s7b

Out[2]=2

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

In[1]:=1

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https://wolfram.com/xid/0rsvpuzzu-d999g9

In[2]:=2

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https://wolfram.com/xid/0rsvpuzzu-bb5pjo

Out[2]=2

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

In[3]:=3

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https://wolfram.com/xid/0rsvpuzzu-raoe4

Out[3]=3

The coefficient of :

In[4]:=4

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https://wolfram.com/xid/0rsvpuzzu-ivgm7a

Out[4]=4

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

In[5]:=5

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https://wolfram.com/xid/0rsvpuzzu-wpcpc

Out[5]=5

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):

In[6]:=6

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https://wolfram.com/xid/0rsvpuzzu-gbmlhf

Out[6]=6

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