Polynomial transformations

In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of polynomial. Polynomial transformations such as Tschirnhaus transformations are often used to simplify the solution of algebraic equations.

Let

be a polynomial, and

be its complex roots (not necessarily distinct).

For any constant c, the polynomial whose roots are

is

If the coefficients of P are integers and the constant ${\displaystyle c={\frac {p}{q}}}$ is a rational number, the coefficients of Q may be not integer, but the polynomial cⁿQ has integers coefficients and has the same roots as Q.

A special case is when ${\displaystyle c={\frac {a_{1}}{a_{0}}}.}$ The resulting polynomial Q does not have any term in y^{n − 1}.

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