Linear transformations of roots

A linear transformation changes the roots of a polynomial by substitively (p x+ q) for x in the polynomial.
The linear transformation of roots of f(x) by the relation x -> px + q is denoted by f(px + q).
If p > 0 in the transformation x -> px + q, the roots of the polynomial are compressed towards the origin by a factor of 1/p.
If p < 0, the roots are not only compressed towards the origin by a factor of 1/p , but also reflected over the y-axis.
Adding q to each root of the polynomial (i.e., transforming x -> x + q) translates the roots to the left if q > 0 and to the right if q < 0.
The linear transformation does not change the number of roots, but their values and locations along the x-axis.
If a polynomial has a root at x = c, the transformed polynomial has a root at x = (c-q)/p
The sum and product of the roots of the transformed polynomial can also be obtained using transformations. For example, if α and β are the roots of x^2 + bx + c = 0 then the roots of (px+q)^2, p(px+q) and p^2c are α/p - q/p and β/p - q/p.
These properties can be used to solve polynomial equations when roots are transformed. It is often easier to solve for the transformed roots then convert back to the original roots.
Working practice of such problems often involve substitution, factorising, or deploying Vieta’s formulas.
It’s crucial to understand the effects of transformations on roots visually, including translations, dilations, rotations, and reflections.