In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations such as Tschirnhaus transformations are often used to simplify the solution of algebraic equations. Simple examples Translating the roots Let be a polynomial, and be its complex roots (not necessarily distinct). For any constant , the polynomial whose roots are is If the coefficients of are integers and the constant is a rational number, the coefficients of may be not integers, but the polynomial has integer coefficients and has the same roots as . A special case is when The resulting polynomial does not have any term in . Reciprocals of the roots Let be a polynomial. The polynomial whose roots are the reciprocals of the roots of as roots is its reciprocal polynomial Scaling the roots Let be a polynomial, and be a non-zero constant. A polynomial whose roots are the product by of the roots of is The factor appears here because, if and the coefficients of are integers or belong to some integral domain, the same is true for the coefficients of . In the special case where , all coefficients of are multiple of , and is a monic polynomial, whose coefficients belong to any integral domain containing and the coefficients of . This polynomial transformation is often used to reduce questions on algebraic numbers to questions on algebraic integers. Combining this with a translation of the roots by , 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 . For examples of this, see Cubic function § Reduction to a depressed cubic or Quartic function § Converting to a depressed quartic. Transformation by a rational function All preceding examples are polynomial transformations by a rational function, also called Tschirnhaus transformations. Let be a rational function, where and are coprime polynomials. The polynomial transformation of a polynomial by is the polynomial (defined up to the product by a non-zero constant) whose roots are the images by of the roots of . Such a polynomial transformation may be computed as a resultant. In fact, the roots of the desired polynomial are exactly the complex numbers such that there is a complex number such that one has simultaneously (if the