Final answer:
The multiplicity of a zero affects the behavior of a polynomial function's graph around that zero, influencing the steepness and whether the graph crosses or touches the x-axis at the zero.
Step-by-step explanation:
The multiplicity of a zero in a polynomial function is a critical factor that affects how the graph behaves around that zero. Specifically, the multiplicity influences whether the graph touches or passes through the x-axis at the zero, as well as the steepness and the concavity of the graph locally around the zero. For example:
- If a zero has an odd multiplicity, the graph will cross the x-axis at that point.
- If a zero has an even multiplicity, the graph will merely touch the x-axis and turn back around, remaining on the same side of the x-axis.
- Higher multiplicities can cause the curve to appear flatter as it approaches the zero, whereas a multiplicity of one will result in a sharper intersection with the x-axis.
In sum, the correct answer to the question is: B) The multiplicity influences the steepness of the curve around the zero. It determines how the graph behaves as it passes through the x-axis at that point. A higher multiplicity will often lead to a more leveled off appearance at the zero, while a lower multiplicity will typically imply a sharper crossing through the zero.