Transforming the roots of a polynomial is a technique for constructing a polynomial whose roots are related to (or transformed from) the roots of another polynomial. For example, if then one polynomial whose roots are the reciprocals of the roots of is Similarly, a polynomial whose roots are one more than the roots of is
The general question considered in this wiki is the following:
Let be a monic polynomial with roots Suppose are obtained from by some (simple) function, e.g. or What is the monic polynomial whose roots are the ?
In general, it is understood that the roots are counted with multiplicity, and that the respective multiplicities of each root and degrees of and will be equal.
Here are two easy examples.
Translation. Let be a monic polynomial with roots What is the monic polynomial whose roots are ?
Clearly will do the trick: if and only if which happens if and only if or This is an informal argument, since it does not formally address repeated roots, so here is a more rigorous proof:
Write and Then by simple substitution put in for in the expression for .
For instance, let , then The roots of are and the roots of are , as desired.
Scaling. Let be a monic polynomial with roots What is the monic polynomial whose roots are
This is similar: will work.
If then which is The last step eliminates the by multiplying each of the factors by
For instance, let Then The roots of are and the roots of are as desired. The formula works just as well if the roots of are complex.
Inversion. Let be a monic polynomial with roots Suppose for all What is the monic polynomial whose roots are
Here the natural choice is but this is not a polynomial: if However, it becomes a polynomial after multiplying by So Note that this polynomial is precisely with coefficients reversed. Note that because it is times the product of the So the monic polynomial we want is
For example, if the polynomial with reversed coefficients is and Its roots are which are the reciprocals of the roots of
Slightly more difficult is the problem of finding polynomials whose roots are squares of the roots of the original polynomial.
Squaring. Let be a monic polynomial with roots What is the monic polynomial whose roots are
First note that has roots and has roots Then is a monic polynomial of degree whose roots are But note that is an even polynomial, so it is a polynomial in : for some monic of degree Then will be the polynomial that the problem asks for.
An explicit formula for is This formula doesn't make it clear why is a polynomial, but the previous paragraph ensures that it does.
Here is an explicit example: if then so will be the polynomial with roots As before, the roots of are and it is easy to check that the roots of are as desired.
These techniques can be combined to produce polynomials with roots which are interesting functions of the original roots.
Let Call its roots
What is the monic cubic polynomial whose roots are
The monic polynomial whose roots are is The monic polynomial whose roots are is The monic polynomial whose roots are is
In fact, the roots of are so should have roots This is easy to check directly.
More generally, suppose is a rational function, with and polynomials. Given a polynomial with roots the way to construct the polynomial with roots is as follows:
The roots of are so they are the solutions to the two-variable system of equations The monic polynomial with these roots is (up to a scaling factor) the resultant
Let Suppose its roots are Find the polynomial whose roots are
We want the resultant which is So
The resultant is not easy to compute by hand, but is well-suited to computer computations. This is how computer algebra packages solve the problem of finding polynomials whose roots are rational functions of the roots of existing polynomials.