To find the zeros of the polynomial function
Step 1: Identify the Zeros
To find the zeros of the function, set
→ → →
Step 2: Find the Multiplicities
- The zero
comes from the factor which has a multiplicity of 1 (since it's linear). - The zero
comes from the factor which has a multiplicity of 2 (since it's a quadratic).
Summary of Zeros and Multiplicities
- Zeros:
and - Multiplicity of
: 1 - Multiplicity of
: 2
Determining the Behavior of the Graph at Each Zero
- For a zero with odd multiplicity, the graph crosses the x-axis.
- For a zero with even multiplicity, the graph touches the x-axis and turns around.
At the Zeros:
- At
: The multiplicity is 1 (odd), so the graph crosses the x-axis. - At
: The multiplicity is 2 (even), so the graph touches the x-axis and turns around.
Final Answers
-
The zero(s) is/are:
-
The multiplicity of the largest zero (
): -
The multiplicity of the other zero (
): -
The graph touches the x-axis and turns around at:
Selection Summary:
- Choose option A: There are two zeros. The multiplicity of the largest zero is 1, and the multiplicity of the other zero is 2.
- The function crosses the x-axis at
and touches the x-axis and turns around at .