Diminishing roots of a equations

Question

I have $f(x)=x^3-6x^2+12x-8$ and need to diminish each root by $3$ without actaully calculating each root. I can achive this by by setting $$f(x)=(x+3)^3-6(x+3)^2+12(x+3)-8$$

This can be easily expanded using binomial theorem. But what if $f(x)$ is of degree $7?, 10?$
How would I proceed in those cases without expading, say, power $9$ ?

Answer 1

Hint

Taylor's formula is an exact formula for polynomials.