When finding all the roots of polynomials, how do you find the possible rational zeros to know which numbers you plug back into the function to get the number you synthetically divide by?

You would have to break the polynomial down into smaller binomials or monomials. Take x^2-6x-9 for instance. It would break down into (x+3)(x-3) through the a^2+2ab+b^2 formula. After that set the two nomials equal to zero and solve for x. After that, you would plug back in x to see whatever x works. You may more than two nomials after breaking it apart so keep that in mind.

-6 is the correct option from the above given options