I know that $2\cos^2 x +\sin x-1$ can't be factored, but I don't really know how to explain it. The best explanation I can come up with is "it would be like factoring $2x^2 + y -1$; it just doesn't work", but that hardly explains exactly why it wouldn't work. Does anyone know a better way of putting it?
I'm not asking if it's impossible to factor at all, I know the equation can be changed to only having sines and from there factored like a regular quadratic. That's not the question. The question is how do I put into words that the equation, exactly as written above, cannot be factored unless you change the equation into $-2 \sin^2x+\sinx+1$ and from there factor it as $(-2 \sinx-1)(\sinx-1)=0$. I've already done that latter part, I just need a way to explain why I have to do it, why it's necessary in the first place to change $2\cos^2 x +\sin x-1$ in order to factor it. I'm sorry for being unclear.
To clear up some confusion, here's what the assignment asks for:
Given the equation $2 \cos^2x+\sinx-1=0$,
Explain why this equation cannot be factored.
Use a trigonometric identity to change the equation into one that can be factored.
Factor the equation.
Determine all solutions in the interval 0≤x≤2π.
I've had no problems with steps 2, 3 and 4, so I did those first. However, I'm not sure how to word the answer to 1 right, because I can't think of anything else than "it just doesn't work".