Suppose we have the polynomial
$f(x) = a x^4 + bx^3 + cx^2+dx+e $
where a-e are real.
Can someone please help explain why the following is definitely not true:
$f(x)= 1$ has 1 distinct real solution
$f(x)= 2$ has 3 distinct real solutions
$f(x)= 3$ has 2 distinct real solutions
and $f(x)= 4$ has 4 distinct real solutions
yet the following could be true
$f(x)= 1$ has 1 distinct real solution
$f(x)= 2$ has 2 distinct real solutions
$f(x)= 3$ has 4 distinct real solutions
and $f(x)= 4$ has 3 distinct real solutions
I thought that by the fundamental theorem of algebra that f(x) has 4 roots. Therefore $f(x) - \alpha$ for some constant $\alpha$ should have four roots. And because complex roots come in pairs how could for example $f(x)= 4$ have 3 distinct real solutions?