Math Problem Statement
Solution
The image shows a problem asking which of the given polynomials could define the function represented by the graph.
Key observations from the graph:
- Roots (x-intercepts): The graph crosses the x-axis at , , and .
- Behavior near the roots:
- At , the graph touches the x-axis and turns back up, indicating a repeated root or a double root.
- At and , the graph crosses the x-axis, indicating simple roots at these points.
Forming the polynomial based on this behavior:
- Since the graph touches the x-axis at , we have a double root at , meaning a factor of .
- The simple roots at and correspond to factors and , respectively.
Thus, the polynomial should take the form:
Checking the options:
- Option B matches this form:
Thus, the correct answer is B.
Would you like a deeper explanation of any step?
Here are five related questions:
- How do you identify simple and repeated roots from a graph?
- What is the effect of a double root on the shape of the graph?
- How would the polynomial change if the root at were a double root?
- How does factoring a polynomial help in graphing it?
- How does the leading coefficient affect the end behavior of a polynomial?
Tip: When analyzing graphs, always look at how the curve behaves near the x-intercepts to determine the multiplicity of the roots.
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Math Problem Analysis
Mathematical Concepts
Polynomial Functions
Graphing
Roots of Polynomials
Formulas
Factorized form of polynomial: P(x) = (x - r1)(x - r2)...(x - rn)
Theorems
Multiplicity of Roots Theorem
Suitable Grade Level
Grades 10-12
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