How can I solve this type of polynomial equation quickly in a college entrance exam?
In about a month, I will be sitting for my college entrance exam, and I have been practising with previous years' questions from that college. While doing so, I came across this equation:
4x^6+3x^5+2x^4=384
The question simply asks for the value of x, and there is only one answer needed. The problem is that my calculator is a Casio FX-991CW, which is the most powerful calculator permitted in the exam. It is non-programmable, but when I use its equation-solving functions for this problem, it takes roughly 55 seconds to obtain the answer, which is far too long for an entrance exam where time is critical.
The answer is x=2.
I asked some seniors how they solved it. They treated the left-hand side as a polynomial,
P(x)=4x^6+3x^5+2x^4,
and simply evaluated P(1), P(2), etc. Since P(2)=384, they immediately got x=2.
The difficulty I have with this method is that it relies heavily on guessing small integer values. What if a similar question had an answer such as
x= -1/3, x= -2, x= 1/4
or some other non-obvious value? Since this is not an MCQ, I cannot simply test the answer choices. So my question is:
What algebraic tricks or patterns would you use to solve this sort of equation quickly, with very little writing, in a college entrance exam?
I would be grateful for any efficient techniques or general patterns worth learning for this kind of problem.