find all rational zeros of f(x)=x^3-2x^2-5x+6

f(x)=x^3-2x^2-5x+6

Note, that if we can add the coefficients of a polynomial and we get "0," then 1 is a root (or, zero)..therefore....

1 - 2 - 5 + 6 

-1 + 1   = 0      So 1 is a root

So, using a little synthetic division, we have

1    1  -2  -5   6

           1  -1  -6

       1 -1  -6   0

So, the polynomial that remains is     x² -x -6   which can be factored as....

(x-3) (x + 2)

And the factorization is  (x-1) (x-3) (x +2) and the rational zeroes obtained by setting each factor to 0 are 1, 3 and -2

And that's it.....