Math Problem Statement
Solution
The graph provided shows a polynomial function, and the task is to write its formula for the polynomial function of least degree.
Here’s how we can approach it:
Step 1: Identify the Zeros (Roots)
From the graph:
- The function crosses the x-axis at approximately , , and . Therefore, the zeros of the function are , , and .
Step 2: Determine the Multiplicity of Each Root
- At , the graph passes through the axis, indicating it has a root of multiplicity 1.
- At , the graph also passes through the axis, which suggests this root also has a multiplicity of 1.
- At , the graph touches the axis and bounces back up, which indicates a root with an even multiplicity. A multiplicity of 2 is the simplest assumption for the least degree.
Step 3: Construct the Polynomial
Given the roots, the polynomial should include factors that correspond to each root:
- For , the factor is .
- For , the factor is .
- For , the factor is .
Thus, the polynomial is of the form: where is a constant to be determined.
Step 4: Estimate the Leading Coefficient
By looking at the behavior of the graph as , the graph opens upwards, meaning is positive. To further refine , we can use the y-intercept (which appears to be at around ) to determine the precise value of , but for now, assume for simplicity.
Thus, the polynomial can be written as:
Would you like further details or explanations about any step?
Here are some additional questions to deepen understanding:
- How does the multiplicity of a root affect the shape of the graph at that root?
- What happens to the degree of a polynomial when a root has higher multiplicity?
- How can you verify if the leading coefficient is correct using another point on the graph?
- What are some methods for identifying the degree of a polynomial based on its graph?
- How would the graph behave differently if were negative?
Tip: Always double-check for symmetry in the graph when determining the form of the polynomial—this can give clues about the degree and behavior of the function.
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Math Problem Analysis
Mathematical Concepts
Polynomial Functions
Roots and Zeros
Multiplicity
Formulas
f(x) = a * (x + 2)(x)(x - 3)^2
Theorems
Root Multiplicity Theorem
Suitable Grade Level
Grades 11-12 (Precalculus/Calculus)
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