In the provided graph, we can analyze the zeros and their multiplicities by observing how the graph behaves at each zero.

Identifying Zeros and Their Multiplicities

• D = (-2, 0):

  • The graph crosses the x-axis at $(-2, 0)$ and changes direction, indicating this is a simple zero with a multiplicity of 1.

• E = (-1, 0):

  • The graph touches the x-axis at $(-1, 0)$ but does not cross it, indicating this is a zero with even multiplicity. In this case, it appears to be a double root (multiplicity 2) since it just touches the axis and turns back.

• C = (2, 0):

  • The graph crosses the x-axis at $(2, 0)$ and changes direction, indicating this is a simple zero with a multiplicity of 1.

Explanation of Multiplicity

• A zero with multiplicity 1 means the graph crosses the x-axis at this point.
• A zero with even multiplicity means the graph touches the x-axis but does not cross it, and the direction of the curve changes at this point.

So, the zero $(-1, 0)$ has a multiplicity of 2, as the graph only touches the x-axis at this point without crossing it.

Would you like further details or have any questions?

Here are 5 questions you might want to ask:

1. How do you determine the multiplicity of a root from the polynomial equation?
2. What does it mean for a root to have an odd multiplicity?
3. Can you explain the impact of root multiplicity on the shape of the graph?
4. How can you factorize a polynomial given its roots and their multiplicities?
5. What is the significance of the derivative in understanding multiplicities of roots?

Tip: Observing how the graph behaves at each x-intercept can give you insights into the multiplicity of each zero. A graph that touches but does not cross the x-axis at a point indicates an even multiplicity at that zero.