### Instruction: Find all the integer roots of
\[x^3 - 3x^2 - 13x + 15 = 0.\]Enter all the integer roots, separated by commas.### Response: I know that if a polynomial has integer roots, they must be factors of the constant term.
So the possible integer roots are 1, -1, 3, -3, 5, -5, 15, and -15.
I could try plugging them all in and see which ones make the polynomial zero, but that would take too long.
A faster way is to use synthetic division and the remainder theorem, which says that if I divide the polynomial by x - r, where r is a root, the remainder will be zero.
So I start with 1 and write the coefficients of the polynomial in a row, leaving a space below them.
Then I bring down the first coefficient, which is 1, and write it below the line.
Next, I multiply 1 by 1 and add it to the second coefficient, which is -3, and write the result below the line, which is -2.
Then I multiply 1 by -2 and add it to the third coefficient, which is -13, and write the result below the line, which is -15.
Finally, I multiply 1 by -15 and add it to the last coefficient, which is 15, and write the result below the line, which is 0.
This means that 1 is a root of the polynomial, and I can write the polynomial as (x - 1) times the quotient, which is x^2 - 2x - 15.
Now I can factor the quotient further, since it is a quadratic.
I look for two numbers that multiply to -15 and add to -2.
Those numbers are -5 and 3.
So I can write the quotient as (x - 5)(x + 3).
This means that the polynomial can be written as (x - 1)(x - 5)(x + 3), and the other two integer roots are 5 and -3.
Therefore, the integer roots of the polynomial are 1, 5, and -3.