termorder = forward, reverse

By default, the first entry of the output will be the constant coefficient of the polynomial.  If reverse is specified, the last entry of the output will be the constant coefficient.

form = monomial, horner

The polynomial is constructed in standard monomial form by default, but can be created in Horner form also known as nested multiplication form.

The FromCoefficientList(L, x) calling sequence returns a polynomial in x from a list of coefficients.

The FromCoefficientVector(L, x) calling sequence returns a polynomial in x from a vector of coefficients.

These are inverse commands to CoefficientList and CoefficientVector.

FromCoefficientVector is written to handle sparse Vector inputs efficiently.

with(PolynomialTools):

FromCoefficientList([1,-1,2,3],x);

$3x^3+2x^2-x+1$

FromCoefficientList([1,-1,2,3],x,form=horner);

$x\left(-1+x\left(2+3x\right)\right)+1$

FromCoefficientList([1,0,0,2,0],x,termorder=reverse);

$x^4+2x$

FromCoefficientVector(<y^3,-y^2,y-2,1>,x);

$y^3-y^{2}x+\left(y-2\right)x^2+x^3$

FromCoefficientList([],x);

$0$

FromCoefficientVector(CoefficientVector(0,x),x);

$0$

v1 := Vector(500000001,shape=unit[300]):

FromCoefficientVector(v1,x);

$x^{300}$

v2 := Vector(200000001,shape=scalar[50000,a]):

FromCoefficientVector(v2,x);

$a{x}^{50000}$

v3 := Vector(10, shape=constant[b^2-1]):

FromCoefficientVector(v3,x);

$\left(b^2-1\right)\left(x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1\right)$

v4 := Vector(1000000001,{(1) = 5, (1000000001) = 1},storage = sparse):

FromCoefficientVector(v4,x);

$x^{1000000000}+5$