Let $|z|=1, $ prove that $|z^2-3z+1|\leq 5$ ....

Question

Problem :

Let $|z|=1, $ prove that $|z^2-3z+1|\leq 5$

My approach :

Let $z = x +iy$

$ \Rightarrow (x^2+y^2)=1$

$\Rightarrow |z| =1 $ represent a circle with centre at (0,0) and radius 1

Now how to use this in the given problem please guide further thanks.

Answer 1

By triangle inequality we have $$|z^2-3z+1|\leq|z^2|+3|z|+1=5$$