I wonder if you could graph it? Just in case you can, I'll include that.
You could try 1 in the original equation
x^3 + x^2 - 17x + 15 = y If it equals 0 when you put 1 in, then that is a solution.
1 + 1 - 17 + 15 and it does = 0.
One factor is (x - 1) <<<< answer 1
Notice that the sign changes. The reason for that is you want to put 1 in for the x and have the result of x - 1 = 0.
Now we could try - 1 to see if it will turn the original equation into 0. It might.
x^3 + x^2 - 17x + 15 = y
(-1)^3 (-1)^2 - 17(-1) + 15 which not going to equal 0 because 17 becomes a plus.
At this point, we could do a division
x - 1 || x^3 + x^2 - 17x + 15 || x^2 + 2x - 15
x^3 - x^2
========
2x^2 - 17x
2x^2 - 2x
========
- 15x + 15
- 15x + 15
=======
0
we now need to factor x^2 + 2x - 15 which factors easily into
(x + 5)(x - 3)=y1
x + 5 = 0
x = - 5
x - 3 = 0
x = 3
So the three roots are 1,3,-5 <<< answer. Look ma, no calculator and as promised, a graph to confirm.
The factors are y = (x - 1)(x - 3)(x + 5)