### Instruction: Find all the integer roots of
\[x^3 - 3x^2 - 13x + 15 = 0.\]Enter all the integer roots, separated by commas.### Response: I notice that this is a cubic equation, so it might have up to three roots.
To find them, I could try to factor the cubic expression into a product of linear and quadratic factors, or use the rational root theorem.
I think the latter might be faster, since it gives me a list of possible roots to test.
The rational root theorem says that if there is a rational root of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p and q must be integers.
In this case, the constant term is 15 and the leading coefficient is 1, so the possible rational roots are p/q, where p is a factor of 15 and q is a factor of 1.
The factors of 15 are 1, -1, 3, -3, 5, -5, 15, and -15, and the factors of 1 are 1 and -1.
So the possible rational roots are 1, -1, 3, -3, 5, -5, 15, and -15.
I can test each of these by plugging them into the cubic equation and seeing if they make it zero.
If I find one root, I can use synthetic division to reduce the cubic to a quadratic, and then find the other roots by factoring or using the quadratic formula.
I start with 1.
If I plug in 1, I get 1^3 - 3(1)^2 - 13(1) + 15, which is 0.
So 1 is a root.
I use synthetic division to divide the cubic by x-1 and get a quotient of x^2 - 2x - 15.
This quotient can be factored as (x-5)(x+3), so the other roots are 5 and -3.
Therefore, the integer roots of the cubic equation are 1, 5, and -3.