Let f(x) be a polynomial of degree 4. Under what condition f(x) has no roots?

Answer 1

A polynomial that has no roots means it has no x intercepts. If you start with a 4th degree parent function like f(x)=x^4 and then shift it up vertically there will be no x intercepts.

f(x)=x^4+5 has no x intercepts and therefore no roots.