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

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

So I tried -4 and it worked. So that is so far my zero of f.

so is this the right way to write the equation in order to find those zeros.

f(x)=(x+4)(x^2+0+9) ?? cuz if you do synthetic division, you would subtract 4 from 4 to get zero which is the middle term. or should it not be in the equation. and be f(x)=(x+4)(x^2+9)

Yes it is

(x+4)(x^2+9)

x = -4 you know
however
x^2+9 = 0
x^2 = -9
is x = +3i or x = -3i (not rational real)

To find the possible rational zeros of a polynomial function f(x), we can use the Rational Zero Theorem. According to the theorem, the possible rational zeros of a polynomial are all the ratios (p/q) where p is a factor of the constant term and q is a factor of the leading coefficient.

In the given function f(x) = x^3 + 4x^2 + 9x + 36, the constant term is 36 and the leading coefficient is 1.

1. Factors of the constant term (36): ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36.
2. Factors of the leading coefficient (1): ±1.

Combining the ratios of the factors, the possible rational zeros are:
±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36.

To find the zeros of the function f(x), we need to test these possible rational zeros one by one using synthetic division or any other method until we find the zeros.

Let's start with ±1:

For f(1) = 1^3 + 4(1)^2 + 9(1) + 36 = 1 + 4 + 9 + 36 = 50, f(-1) = (-1)^3 + 4(-1)^2 + 9(-1) + 36 = -1 + 4 - 9 + 36 = 30.
Since neither f(1) nor f(-1) is zero, we can conclude that 1 and -1 are not zeros of the function.

Now let's continue testing the remaining possible rational zeros one by one until we find the zeros of the function.