### Instruction: Find all the integer roots of
\[2x^3 + 13x^2 - 23x + 8 = 0.\]Enter all the integer roots, separated by commas.### Response: I recognize this as a cubic equation, which might have up to three roots.
One way to find them is to factor the equation into a product of linear and quadratic factors.
To do that, I need to find at least one root first.
I can try some small integers and see if they make the equation zero.
For example, if I plug in x = 1, I get 2 + 13 - 23 + 8 = 0, so x = 1 is a root.
That means I can divide the equation by x - 1 using polynomial long division or synthetic division.
I get \[2x^3 + 13x^2 - 23x + 8 = (x - 1)(2x^2 + 15x - 8).\]
Now I can look for roots of the quadratic factor.
I can use the quadratic formula or try to factor it by finding two numbers that multiply to -16 and add to 15.
I see that 16 and -1 work, so \[2x^2 + 15x - 8 = (2x - 1)(x + 8).\]
That means the other roots are x = 1/2 and x = -8.
However, the question asks for integer roots, so I only include x = 1 and x = -8 in my answer.