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Chapter 2: Problem 83

Explain the relationship between the multiplicity of a zero and whether or not the graph crosses or touches the \(x\) -axis and turns around at that zero.

Short Answer

Expert verified
The multiplicity of a zero dictates how the graph behaves at that zero. If the multiplicity is even, the graph touches the x-axis at that zero and turns around. If it is odd, the graph crosses the x-axis at that zero.

Step by step solution

01

Understanding Zeros and their Multiplicities

In a polynomial function, zeros, or the roots of the function, are the x-values that make the function equal to zero. Each zero has a multiplicity, which refers to the number of times the factor appears in the function. Essentially, it indicates how many times that particular root occurs.
02

Touching x-axis and Turning Around

When the multiplicity of a zero is even, the graph touches the x-axis at that point and then turns around. In simpler terms, the graph does not cross the x-axis; it merely bounces back on the same side of the x-axis. This is because even multiplicities lead to positive values whether the x is positive or negative, creating a curve that touches the axis but doesn't cross it.
03

Crossing the x-axis

When the multiplicity of a zero is odd, the behavior of the graph is different. Here, the graph will cross over the x-axis at the point representing the zero. This happens because an odd multiplicity can yield both positive and negative values for different x's, thus leading to a crossing over the x-axis. Odd multiplicities create a curve that goes from one side of the x-axis to the other.

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