SOLUTION: Find the roots of the polynomial equation x^3 + x^2 -17x +15 = 0

x³ + x² - 17x + 15 = 0

DesCartes' rule of signs:
The possibile number of positive solutions is 2 or none because
there are two sign changes going left to right.

Since the leading coefficient is 1, the feasible rational solutions
are ± the divisor of 15.  These are:

±1, ±3, ±5, ±15

We begin with trying 1 to see if it is a solution (root).  If it
is, great!  If it isn't we'll try the next one and keep trying
them until we find a rational solution or exhaust these and conclude
that it has no rational solutions.

Trying 1 as a fesible root:

1 |1  1 -17  15
  |   1   2 -15
   1  2 -15   0

Wow! the first number we tried turned out to be a solution.
That doesn't happen very often.  Usually we have to try some
others that don't give a 0 in the lower right hand corner of
the synthetic division array as a remainder.  We are lucky. 
So we have now factored the equation this way:

   x³ + x² - 17x + 15 = 0
(x - 1)(x² + 2x - 15) = 0

Now we can factor the quadratic expression in the second
parentheses:

(x - 1)(x + 5)(x - 3) = 0

Now we use the zero factor property:

x - 1 = 0;   x + 5 =  0;   x - 3 = 0
    x = 1        x = -5        x = 3

So the roots (solutions) are 1, -5, and 3

Edwin