To find the zeros of the polynomial function [tex]\( f(x) = x^3 + 6x^2 + 9x \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
Step-by-Step Solution:
1. Factor the polynomial:
The given polynomial is [tex]\( f(x) = x^3 + 6x^2 + 9x \)[/tex]. Notice that each term in the polynomial has a common factor of [tex]\( x \)[/tex].
2. Extract the common factor:
Factor out [tex]\( x \)[/tex] from each term:
[tex]\[
f(x) = x(x^2 + 6x + 9).
\][/tex]
3. Factor the quadratic expression:
Next, we need to factor [tex]\( x^2 + 6x + 9 \)[/tex]. We notice that this quadratic expression is a perfect square trinomial:
[tex]\[
x^2 + 6x + 9 = (x + 3)^2.
\][/tex]
4. Write the polynomial in its factored form:
Incorporating the factorization, the polynomial can be written as:
[tex]\[
f(x) = x(x + 3)^2.
\][/tex]
5. Find the zeros of each factor:
Now, set each factor equal to zero and solve for [tex]\( x \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
x = 0 \quad \text{(This is one of the zeros)}.
\][/tex]
- For [tex]\( (x + 3)^2 = 0 \)[/tex]:
[tex]\[
x + 3 = 0 \implies x = -3 \quad \text{(This is another zero)}.
\][/tex]
6. List all the zeros:
The zeros of the polynomial function are:
[tex]\[
x = 0 \quad \text{and} \quad x = -3.
\][/tex]
Therefore, the correct answer is [tex]\( \boxed{x = 0, x = -3} \)[/tex], which corresponds to option C.
