To determine the zeros of this degree 4 polynomial function and their multiplicities from the graph, let's analyze where the function intersects the x-axis.

Observations from the Graph:

1. Zero at $x = -3$: The curve touches the x-axis and bounces back, indicating a zero with even multiplicity. Since the polynomial is degree 4, this zero likely has a multiplicity of 2.

2. Zero at $x = 1$: The curve also touches the x-axis and bounces back at this point, indicating another zero with even multiplicity. This is likely also of multiplicity 2.

Conclusion:

• The zeros of the polynomial function are:
  • $x = -3$ with multiplicity 2
  • $x = 1$ with multiplicity 2

These two zeros together satisfy the degree requirement of the polynomial (degree 4).

Would you like more details or have any questions on this?

Here are five follow-up questions that might deepen your understanding:

1. How do we determine the multiplicity of a zero by observing the graph?
2. Why does the graph bounce at certain zeros instead of crossing the x-axis?
3. Can the multiplicities of zeros add up to more than the degree of the polynomial?
4. How would the graph look different if a zero had an odd multiplicity?
5. What are the implications of zero multiplicities in real-world polynomial models?

Tip: A polynomial graph touches the x-axis at zeros with even multiplicity (like 2 or 4) and crosses the x-axis at zeros with odd multiplicity (like 1 or 3).