Math Problem Statement

Starting from least to​ greatest, the first zero of the function is of multiplicity    enter your response here​, so the graph of f ▼ crosses touches the​ x-axis at xequals    enter your response here. The second zero of the function is of multiplicity    enter your response here​, so the graph of f ▼ touches crosses the​ x-axis at xequals    enter your response here. The third zero of the function is of multiplicity    enter your response here​, so the graph of f ▼ crosses touches the​ x-axis at xequals    enter your response here. The fourth zero of the function is of multiplicity    enter your response here​, so the graph of f ▼ touches crosses the​ x-axis at xequals    enter your response here.

Solution

To complete the statement, you need to identify the zeros of the function and their multiplicities. Multiplicity affects how the graph behaves at each zero. Here's how it works:

  • Multiplicity of 1 (odd multiplicity): The graph crosses the x-axis at the zero.
  • Multiplicity of 2 (even multiplicity): The graph touches the x-axis but doesn't cross.

Let's assume you know the zeros and their multiplicities. You would fill in the blanks based on that information.

For example:

  • Starting from least to greatest, the first zero of the function is of multiplicity 2, so the graph of ff touches the x-axis at x=3x = -3.
  • The second zero of the function is of multiplicity 1, so the graph of ff crosses the x-axis at x=0x = 0.
  • The third zero of the function is of multiplicity 3, so the graph of ff crosses the x-axis at x=2x = 2.
  • The fourth zero of the function is of multiplicity 2, so the graph of ff touches the x-axis at x=4x = 4.

Relative Questions:

  1. How does the graph behave when a zero has odd multiplicity?
  2. What happens to the graph at a zero with even multiplicity?
  3. Can a zero have a multiplicity higher than 2, and how does it affect the graph?
  4. How do you determine the zeros of a polynomial function?
  5. Why is the order of zeros important when sketching the graph of a polynomial?

Tip:

Multiplicity tells you not only where the graph touches or crosses the x-axis but also how "flat" or "sharp" the behavior of the graph is at those points.

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Math Problem Analysis

Mathematical Concepts

Polynomials
Zeros of Functions
Multiplicity

Formulas

Multiplicity impacts whether a graph crosses or touches the x-axis

Theorems

Multiplicity and Behavior of Polynomial Graphs

Suitable Grade Level

Grades 10-12