To determine which of the given equations could be the equation of \( m(x) \) in factored form, we need to analyze the characteristics of the graph of the polynomial function, specifically the x-intercepts (roots) and the degree of the polynomial.
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Identify the roots from each option:
- For \( m(x) = (x - 1)(x + 3)(x + 5)(x + 15) \): Roots are \( x = 1, -3, -5, -15 \) (4 roots)
- For \( m(x) = (x - 1)(x + 3)(x + 5) \): Roots are \( x = 1, -3, -5 \) (3 roots)
- For \( m(x) = (x + 1)(x - 3)(x - 5)(x - 15) \): Roots are \( x = -1, 3, 5, 15 \) (4 roots)
- For \( m(x) = (x + 1)(x - 3)(x - 5) \): Roots are \( x = -1, 3, 5 \) (3 roots)
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Characteristics of the graph:
- If the graph crosses the x-axis at each intercept, each factor corresponds to a linear factor with an odd multiplicity (1).
- If the graph touches the x-axis without crossing, the factors would have even multiplicity (2).
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Degree of the polynomial:
- A polynomial with \( n \) roots has a degree \( n \). For instance, a polynomial with 4 roots is degree 4, while one with 3 roots is degree 3.
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Conclusion:
- If the graph shows 4 x-intercepts, then you would expect a polynomial of degree 4, which could be represented by either \( (x - 1)(x + 3)(x + 5)(x + 15) \) or \( (x + 1)(x - 3)(x - 5)(x - 15) \).
- If the graph has 3 x-intercepts, then you'd align with either \( (x - 1)(x + 3)(x + 5) \) or \( (x + 1)(x - 3)(x - 5) \).
Without specific information about the behavior of the graph (like the number of x-intercepts or whether it crosses or touches the x-axis), it's challenging to definitively choose one equation. Please review the graph's characteristics to select one or provide more details regarding the graph's behavior (such as the number of roots visible).