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CoefficientList.md

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CoefficientList

CoefficientList(polynomial, variable)

get the coefficient list of a polynomial.

See:

Examples

>> CoefficientList(a+b*x, x)
{a,b}

>> CoefficientList(a+b*x+c*x^2, x)
{a,b,c}

>> CoefficientList(a+c*x^2, x)
{a,0,c}

In the next line Coefficient returns the coefficient of a particular term of a polynomial. In this case (-210*c^2 * x^2*y*z^2) is a term of (c*x-2*y+z)^7 after it's expanded.

>> poly=(c*x-2*y+z)^7
(c*x-2*y+z)^7

>> Coefficient(poly, x^2*y*z^4)
-210*c^2

CoefficientList gets the same information as a list of coefficients. In the line below Part(coeff, 3,2,5) returns the coefficient of x^(3-1)*y^(2-1)*z^(5-1). In general if we say lst=CoefficientList(poly,{x1,x2,x3,...}) then Part(lst, n1, n2, ,n3, ...) will be the coefficient of x1^(n1-1)*x2^(n2-1)*x3^(n3-1)....

>> coeff=CoefficientList(poly,{x,y,z}); Part(coeff, 3,2,5)
-210*c^2

The next line gives the coefficient of x^5. As expected there is more than one term of poly with x^5 as a factor.

>> Coefficient(poly, x^5)
84*c^5*y^2-84*c^5*y*z+21*c^5*z^2

We can get the same result as the previous example if we use the next line.

>> Coefficient(poly, x, 5)
84*c^5*y^2-84*c^5*y*z+21*c^5*z^2

One can't get the result above directly from CoefficientList. Instead pieces of the above result are included in the result of CoefficientList(poly,{x,y,z}). The line below can be used to get pieces of the result above. Specifically coeff[[6]] contains all coefficients of x^5 (including those that are zero).

>> coeff[[6]]
{{0,0,21*c^5,0,0,0,0,0},{0,-84*c^5,0,0,0,0,0,0},{84*c^5,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0}}
  • Part(coeff,6,1,3) is 21*c^5 the coefficient of (x^(6 - 1) * y^(1 - 1) * z^(3 - 1)) = (x^5 * z^2) in poly.
  • Part(coeff,6,2,2) is (-84*c^5) the coefficient of (x^(6 - 1) * y^(2 - 1) * z^(2 - 1)) = (x^5 * y * z) in poly.
  • Part(coeff,6,3,1) is (84*c^5) the coeficient of (x^(6 - 1) * y^(3 - 1) * z^(1 - 1)) = (x^5 * y^2) in poly.

All other coefficients under coeff[[6]] are zero which agrees with the result of Coefficient(poly, x^5).

Related terms

Coefficient, CoefficientRules, Exponent, MonomialList

Implementation status

  • ✅ - full supported

Github