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

f (x) = x^3 + 4x^2 + 9x+36

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.