How do you use the rational roots theorem to find all possible zeros of #f(x)=-2x^4+5x^3+2x^2+18#?

The rational roots theorem (or test) is more simple than you might think. Here I have the explanation: you take the coefficient( or number without any unknown value such as x,y,z,.. near it) which in this case is 18 then you factorize it which will be:

1,2,3,6,9,18 (factors of 18)

you divide each one of these by the coefficient of the x with the largest exponent (in this case= #-2x^4# but you need only -2).
Remember: you do not need to take into account the negative sign, because you need to write near each value the plus or minus sign.Another way to think this is plus plus=plus, Plus minus=minus (the same vice versa), minus*minus=plus ( these are the same for division)

The answers are: #+- 0.5#, #+- 1#, #+-3/2#, #+- 3#, #+- 4.5 (or 9/2)#, #+- 9#
Now, you need to use these values and test them whether they are zeros with synthetic division.

Realize that when having found one zero, your answer of the synthetic division is equal to the coefficients of the result, each one multiplied by the #x^(Y-1)#, where Y= original largest x exponent - 1
If there are any further questions, write a comment