Math Problem Statement
Solution
The graph shows a degree 5 polynomial, and you're asked to complete the table of values based on the x-values of the zeros and their corresponding multiplicities.
Analyzing the graph:
-
The polynomial crosses the x-axis at , indicating a zero at .
- Since the graph crosses the x-axis at this point, the multiplicity of this root is odd. We already see in the table that its multiplicity is missing.
-
The polynomial also crosses the x-axis at , and since it crosses straight through, the multiplicity is 1 (this value is already given in the table).
-
The graph touches the x-axis at approximately , meaning it doesn't cross but touches the x-axis and turns around, indicating a multiplicity greater than 1 (typically 2).
Table Completion:
- The zero at has multiplicity 1 because it crosses the x-axis.
- The root at should be added to the table with multiplicity 2.
Here's the completed table:
Root with | Multiplicity |
---|---|
1 | |
2 | |
1 |
Would you like a more detailed explanation of how multiplicities work, or have any questions?
Here are 5 related questions:
- How can you tell if a root has an odd or even multiplicity just by looking at the graph?
- What would happen if a root has a multiplicity of 3?
- Can a polynomial of degree 5 have more than 5 roots?
- How does the degree of the polynomial affect the shape of the graph near a root with multiplicity greater than 1?
- How can you factor this polynomial based on the roots identified?
Tip: The degree of a polynomial dictates the maximum number of roots (real or complex) it can have.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Polynomials
Zeros of a Polynomial
Multiplicity of Roots
Formulas
-
Theorems
Multiplicity Theorem for Polynomials
Suitable Grade Level
Grades 10-12
Related Recommendation
Analyzing Degree 5 Polynomial Graph to Determine Roots and Their Multiplicities
Finding Roots and Multiplicities from a Degree 5 Polynomial Graph
Finding Zeros and Multiplicities of a Degree 4 Polynomial Function
Polynomial Analysis: Zeros, Multiplicity, Degree, and Leading Coefficient
Finding Zeros and Multiplicities of a Degree 4 Polynomial from a Graph