Math Problem Statement
Solution
The graph provided shows a polynomial function, and the problem asks to identify its equation from the given options. Based on the graph:
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Roots of the Polynomial:
The graph crosses or touches the x-axis at three points: -
Behavior at Each Root:
- At , the graph touches the x-axis and turns around, indicating a double root (multiplicity 2).
- At , the graph crosses the x-axis with a linear (straight-through) behavior, indicating a single root (multiplicity 1).
- At , the graph touches the x-axis and turns around, indicating another double root (multiplicity 2).
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Equation of the Polynomial:
Based on the roots and their multiplicities, the equation can be written as:
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Answer:
The correct equation from the options is:
Would you like further details about the reasoning? Here are some related questions to deepen your understanding:
5 Related Questions:
- How can the multiplicity of a root be determined by looking at a graph?
- Why do double roots cause the graph to touch but not cross the x-axis?
- How can we write the equation of a polynomial if the graph shows a triple root at a certain point?
- What is the general form of a polynomial function with distinct roots and their multiplicities?
- How does the leading coefficient affect the end behavior of a polynomial graph?
Tip:
When analyzing polynomial graphs, always check the behavior at the x-intercepts to determine the root multiplicities.
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Math Problem Analysis
Mathematical Concepts
Polynomial Functions
Roots and Multiplicity
Graphing Polynomials
Formulas
y = (x - r1)^m1 (x - r2)^m2 ... (x - rn)^mn, where r is the root and m is the multiplicity
Theorems
Root-Multiplicity Behavior Theorem
Suitable Grade Level
Grades 10-12
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