f(x) = x^3 + 10x^2 + 27x - 38
has one integer root. What is it?
The integer root is 19.
Look at the coefficients of the powers of x. These are 1, 10 and 27. Their sum is 1+10+27 = 38. Hence, if x = 1 then f(x) = 0, so the integer root is x = 1.