Statement is mathematically incorrect and misleading: polynomial equations of degree > 4 are unsolvable by radicals.
Teacher (I assume, author as well) ought to be ashamed!

You are mistaken - polinimials of degree > 4 can't be solved by CLOSED ANALYTICAL formula (Abel's theorem) but there exist numerical methods!

There is no mistake: equations of degrees higher than FOUR cannot be solved in radicals in the general form, so equations of degrees higher than FIVE cannot be solved in radicals too.