Knowing when to use factoring/how to factor/rational zeroes

Question

How do I know where I would be using factoring as opposed to rational zero theorem? Do I use Descartes rule of signs to get how many positive/negative roots are and then attempt RZT to get rational zeroes, then if the amounts do not match up, try factoring it?

On top of that, how do I know whether to complete the square or "factor it out" (is there an easy way to do this for non-$(x+a)(x+b)$-esque ones?)? IIRC, the quadratic formula is used only for $ax^2+bx+c$-type where maximum degree is $2$. Should I factor degrees greater than $2$ into some kind of $(x+a)(ax^2+bx+c)$ and then attempt the quadratic equation if its degree is higher than $2$?

Thanks, much appreciated

Answer 1

Factoring and the rational root theorem are both good when they work, and work on some problems the other doesn't. You just try both. The quadratic formula and completing the square just work for quadratic equations-when the highest power of $x$ is $x^2$. There are analogous formulas for cubic and quartic equations, but they get much messier.

Answer 2

In general the answer isn't obvious. Normally I do a quick rational zero theorem check: not doing it all the way through, just "how many options are available"? If it's $x^2+15x+1$ then I know I only have $\pm1$, neither of which work, and so I just jump to the quadratic formula without wasting my time. If there are a lot of options I just use the quadratic formula anyway. Maybe the rational zero theorem would tell me the answer, but it would take much longer.

For degrees higher than two, things can get complicated quickly. There are formulas for power $3$ and $4$ equations, but they're not pretty. Here I'll just go through the rational zero theorem. Not that great, but polynomials can't be factored in general. As they get more complicated this is one of the few things that gives a reasonable return on time for effort.

Also, note that factoring is included in the rational zero theorem already. If the rational zero theorem fails, factoring won't save you because the available factors that could work are just the integer options in RZT.