x^4 - 4x^3 + 7x^2 + 22x + 24 = 0
I have no idea how to solve this, it suddenly popped up in the quadratic section of the book. Help please!
The best we can do (if we don't use the general formula for a quartic, which is "butt ugly") is to try to find quadratic factors of the form:
(x - (a + Ib))(x - (a - Ib))
I messed with this for a while, but couldn't come up with anything intelligible.
I do not think there is nice way to solve this without the quartic formula. If instead of the 7 you have a -7 it would work.
Let me explain. Some of these polynomials can be solved by manipulating them moving factors until you bring it to the form you can solve it in.
I was able to show this polynomial is irreducible over Q. Meaning we cannot achieve this with doing nice moves, i.e. moving factors around if you know what I mean.