The function below has at least one rational zero.
Use this fact to find all zeros of the function.

If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.

To find the rational zeros of the polynomial , we can use the Rational Root Theorem. This theorem states that if there are any rational zeros (given in the form of ), is a factor of the constant term (-5) and is a factor of the leading coefficient (3).

The possible rational zeros are . Testing these values:

Now, we can perform synthetic division of by :

1 | 3  -19   20   1  -5
  |        3 -16   4  5
-------------------------
  | 3  -16   4  5   0

This gives us the quotient polynomial . We now have:

Next, we can repeat the Rational Root Theorem or another method on the cubic polynomial . Testing possible rational roots again leads us to find that is also a zero:

Dividing by :

5 | 3  -16   4   5
  |         15  -5  -5
---------------------
  | 3   -1   -1   0

The result gives us . Now we factor or use the quadratic formula to solve :

Thus, the function has the zeros: