### Instruction: Find all the integer roots of
\[x^3 - 3x^2 - 13x + 15 = 0.\]Enter all the integer roots, separated by commas.### Response: To find the integer roots of a cubic polynomial, I can use the rational root theorem, which says that any rational root must be of the form p/q, where p is a factor of the constant term (15) and q is a factor of the leading coefficient (1).
So the possible rational roots are the factors of 15, which are ±1, ±3, ±5, and ±15.
I can use synthetic division to test each one until I find a root, or I can graph the function and estimate the roots visually.
I'll try synthetic division first, starting with the easiest ones.
If I divide by 1, I get
\[\begin{array}{c|cccc}
1 & 1 & -3 & -13 & 15\\
  &   & 1  & -2  & -15\\
\hline
  & 1 & -2 & -15 & 0
\end{array}\]
So 1 is a root, and the quotient is x^2 - 2x - 15, which I can factor as (x - 5)(x + 3).
That means the other two roots are 5 and -3, which are also factors of 15.
So I found all the integer roots without having to test the others.
I can also check my answer by plugging in the roots into the original equation and making sure they make it zero.