Problem - 2184
Let $z$ be a complex number and $k$ be a known real number. Find the maximum value of $|z^2 +kz+1|$ if $|z|=1$.
First, because $|z|=1$, we have $$|z^2 +kz+1| = |z(z+k+\overline{z})| = |z+k+\overline{z}|$$
Now, let $z=a+bi$, where $a^2 + b^2 = 1$. Then
$$|z+k+\overline{z}|= |2a + k| \le |2a| + |k| \le \boxed{2 + |k|}$$