Let be a polynomial, and be its complex roots (not necessarily distinct).
The resulting polynomial Q does not have any term in yn − 1.
A polynomial whose roots are the product by c of the roots of P is The factor cn appears here because, if c and the coefficients of P are integers or belong to some integral domain, the same is true for the coefficients of Q.
is a monic polynomial, whose coefficients belong to any integral domain containing c and the coefficients of P. This polynomial transformation is often used to reduce questions on algebraic numbers to questions on algebraic integers.
, allows to reduce any question on the roots of a polynomial, such as root-finding, to a similar question on a simpler polynomial, which is monic and does not have a term of degree n − 1.
For examples of this, see Cubic function § Reduction to a depressed cubic or Quartic function § Converting to a depressed quartic.
Let be a rational function, where g and h are coprime polynomials.
In fact, the roots of the desired polynomial Q are exactly the complex numbers y such that there is a complex number x such that one has simultaneously (if the coefficients of P, g and h 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.
is a primitive element of L, which has Q as minimal polynomial.
Descartes introduced the transformation of a polynomial of degree d which eliminates the term of degree d − 1 by a translation of the roots.
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.