7x³ - 6x² + 2x - 1 = 0

Every candidate for a rational solution is ± a fraction whose
numerator is a divisor of the absolute value of the constant term,
-1, and whose denominator is a divisor of the absolute value of the
leading coefficient, 7.

The only divisor of |-1| is 1
The only divisors of |7| are 1 and 7

Therefore the only candidates for rational solutions are
 and .  The only ones of those which
are integers are 1 and -1, so we see if either of those is a 
solution:

We see if 1 is a solution using synthetic division:

1| 7  -6  2  -1
 |     7  1   3
   7   1  3   2

That left a remainder of 2, not 0, so 1 is not a solution


We also see if -1 is a solution

-1| 7  -6  2  -1
  |    -7 13 -15
    7 -13 15 -16

That left a remainder of -16, not 0, so -1 is not a solution
either.

Those two, 1 and -1,  were the only possible candidates for integer
solutions.  So there are none.

Edwin