Certainly! Let's go through the problem step by step to determine which zeroes of [tex]\( f(x) \)[/tex] we should choose.
1. Review the given zeroes and their multiplicities:
- [tex]\(-3\)[/tex] with multiplicity 1
- [tex]\(3\)[/tex] with multiplicity 1
- [tex]\(0\)[/tex] with multiplicity 1
- [tex]\(0\)[/tex] with multiplicity 3
- [tex]\(3\)[/tex] with multiplicity 0
2. Identify which zeroes have a multiplicity of at least 1:
- If a zero has a multiplicity of at least 1, it means it is indeed a zero of the function.
- Zeroes with multiplicity [tex]\(\geq 1\)[/tex] from the list are:
[tex]\[
\begin{align*}
&-3 \text{ (multiplicity 1)}, \\
&3 \text{ (multiplicity 1)}, \\
&0 \text{ (multiplicity 1)}, \\
&0 \text{ (multiplicity 3)}.
\end{align*}
\][/tex]
3. Compile the list of selected zeroes:
- From the above step, the zeroes [tex]\(-3\)[/tex], [tex]\(3\)[/tex], and [tex]\(0\)[/tex] meet the condition of having a multiplicity of at least 1.
4. Notice the repetitions:
- The zero [tex]\(0\)[/tex] appears with two different multiplicities (1 and 3). While constructing our final list, we should take care not to repeat zeroes unnecessarily. However, since [tex]\(0\)[/tex] meets the criteria multiple times, we acknowledge each instance it meets the condition of multiplicity [tex]\(\geq 1\)[/tex], which explains multiple occurrences in our final list.
5. Final List of Zeroes:
- Hence, the final selection of zeroes of [tex]\(f(x)\)[/tex] includes [tex]\(-3\)[/tex], [tex]\(3\)[/tex], and two occurrences of [tex]\(0\)[/tex].
Thus, the zeroes of [tex]\(f(x)\)[/tex] we should choose are:
[tex]\[
[-3, 3, 0, 0]
\][/tex]
This concludes the detailed step-by-step solution to determine the zeroes of [tex]\(f(x)\)[/tex].
