Question

List possible rational zeros of f using the rational zero theorem. Then find all thezeros of the function.

Answer 1

The first term is x^3 with coefficient 1. Let's call this p = 1
The last term is 36, so we'll call this q = 36

To find all of the possible rational roots or zeros, we divide each factor of q over each factor of p

Factors of p = 1:
-1 and 1

Factors of q = 36: 
-1, 1,
-2, 2
-3, 3
-4, 4
-6, 6
-9,9
-12, 12
-18,18
-36,36

As you can see there's a lot of possible roots. Luckily, p = 1 so that means we simply need to list the factors of q (the plus and minus versions)

The possible rational roots are listed above when I listed all the possible factors of q. Those are the possible x values that make f(x) equal to 0. You need to plug each of those values into f(x) to see which result in 0. It turns out that only x = -4 leads to f(x) = 0 being true. None of the other possible rational roots are actual rational roots.