coefficients of and are not real or complex numbers, "complex number" has to be replaced by "element of an algebraically closed field containing the coefficients of the input polynomials") This is exactly the defining property of the resultant This is generally difficult to compute by hand. However, as most computer algebra systems have a built-in function to compute resultants, it is straightforward to compute it with a computer. Properties If the polynomial is irreducible, then either the resulting polynomial is irreducible, or it is a power of an irreducible polynomial. Let be a root of and consider , the field extension generated by . The former case means that is a primitive element of , which has as minimal polynomial. In the latter case, belongs to a subfield of and its minimal polynomial is the irreducible polynomial that has as power. Transformation for equation-solving Polynomial transformations have been applied to the simplification of polynomial equations for solution, where possible, by radicals. Descartes introduced the transformation of a polynomial of degree which eliminates the term of degree by a translation of the roots. Such a polynomial is termed depressed. This already suffices to solve the quadratic by square roots. In the case of the cubic, Tschirnhaus transformations replace the variable by a quadratic function, thereby making it possible to eliminate two terms, and so can be used to eliminate the linear term in a depressed cubic to achieve the solution of the cubic by a combination of square and cube roots. The Bring–Jerrard transformation, which is quartic in the variable, brings a quintic into Bring-Jerrard normal form with terms of degree 5,1, and 0. References Algebra