use the given zero to find all the zeros of the function.

f(x) = x^3 + x^2 + 9x + 9 r = 3i

I do not understand this!!

the zeros (roots) could be found without the given hint that one of the roots is 3i

f(x) = x^3 + x^2 + 9x + 9
= x^2(x+1) + 9(x+1)
= (x^2+9)(x+1)

for the zeros,
x+1 = 0 or x = -1
or
x^2 + 9 = 0
x^2 = -9
x = ±3√-1
= ±3i

so the roots are ±3i, -1

Using the given hint that 3i is a root, one property of irrational or complex roots is that they must come in "conjugate pairs" to end up with rational coefficients.

so 2 factors would be (x-3i) and (x+3i)

then (x+3i)(x-3i) = x^2 + 9

Using long division, divide your original function by x^2 + 9 to get the other factor of x+1

To find all the zeros of the function, we need to use the given zero and any other possible zeros.

In this case, we are given that one of the zeros, denoted as "r," is equal to 3i. It is important to note that complex zeros of a polynomial function come in conjugate pairs. This means that for every complex zero, its conjugate (with the opposite sign of the imaginary part) is also a zero.

So, if r = 3i, then its conjugate, -3i, must also be a zero of the function.

Now, to find the remaining zero, we can use polynomial long division or synthetic division to divide the given function by (x - r). In this case, (x - 3i) is the factor we'll use for division.

Let's perform the polynomial long division:

__________________________
x - 3i | x^3 + x^2 + 9x + 9

- (x^3 - 3ix^2)
______________________
(4i + 9)x^2 + 9x + 9

- ((4i + 9)x^2 - (12i + 27)x)
______________________
(21i + 9)x + 9

- ((21i + 9)x - (63i + 27))
_____________________
90i + 36

Since the result of the division, 90i + 36, does not have an 'x' term, it implies that the remaining zero is a real number, not dependent on 'x'. Therefore, the other zero is - (90i + 36) / (21i + 9). Simplifying this expression will give you the exact value of the remaining zero.