Math Problem Statement
Solution
Let's analyze the graph and the corresponding statements about the degree 4 polynomial function.
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Statement A: The leading coefficient is positive.
- The end behavior of the graph shows that as , , and as , . This indicates that the leading coefficient is positive for a degree 4 polynomial.
- This statement is true.
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Statement B: The constant term is negative.
- The constant term is the -intercept of the polynomial, which occurs when . From the graph, the -intercept is positive (above the x-axis).
- This statement is false.
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Statement C: It has 2 relative maximums.
- Looking at the graph, there are two points where the function has relative maximums: one near and another around .
- This statement is true.
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Statement D: It has 4 linear factors.
- A degree 4 polynomial typically has four factors if all the roots are real. This graph crosses the x-axis four times, suggesting four real roots.
- This statement is true.
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Statement E: One of the factors is .
- Looking at the x-intercepts, the function crosses the x-axis at around , so could be one of the factors.
- This statement is true.
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Statement F: One of the zeros is .
- From the graph, it does not seem that the function crosses the x-axis at ; it crosses at and around .
- This statement is false.
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Statement G: There is a relative minimum between and .
- Indeed, there is a relative minimum around .
- This statement is true.
Correct Statements:
- A, C, D, E, G
Would you like any further clarifications or details on these? Here are 5 related questions for deeper understanding:
- What does the behavior of a degree 4 polynomial tell us about its graph?
- How can we determine the number of real roots from a polynomial graph?
- What is the significance of the leading coefficient in determining end behavior?
- How do relative maximum and minimum points help us analyze a graph?
- Why does the constant term correspond to the -intercept?
Tip: Always check the end behavior of a polynomial by looking at the degree and sign of the leading coefficient—it gives important clues about the function's general shape.
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Math Problem Analysis
Mathematical Concepts
Polynomial Functions
Degree of Polynomials
End Behavior
Relative Maximum and Minimum
Roots of Polynomial Equations
Formulas
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Theorems
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Suitable Grade Level
Grades 9-12
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