To determine which of the given equations could be the equation of \( m(x) \) in factored form based on its graph, we should consider the following:
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Roots/Zeros: The factors of the polynomial indicate the x-values where the graph crosses the x-axis. For example, if you have a factor like \( (x - r) \), then \( x = r \) is a root.
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Multiplicity: The way the graph behaves at each root (crossing or touching the x-axis) can inform us about the multiplicity of the roots.
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Number of Factors: The number of factors indicates the degree of the polynomial. For example, three factors yield a polynomial of degree 3, and four factors result in a polynomial of degree 4.
Now, let's analyze the provided options:
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\( m(x) = (x + 1)(x - 3)(x - 5)(x - 15) \)
- This polynomial has 4 distinct roots: \( -1, 3, 5, 15 \). It's a degree 4 polynomial.
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\( m(x) = (x - 1)(x + 3)(x + 5) \)
- This polynomial has 3 distinct roots: \( 1, -3, -5 \). It's a degree 3 polynomial.
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\( m(x) = (x + 1)(x - 3)(x - 5) \)
- This polynomial has 3 distinct roots: \( -1, 3, 5 \). It's a degree 3 polynomial.
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\( m(x) = (x - 1)(x + 3)(x + 5)(x + 15) \)
- This polynomial has 4 distinct roots: \( 1, -3, -5, -15 \). It's a degree 4 polynomial.
To choose the correct equation, you need to consider the number of root crossings shown in the graph you have, along with the nature of those crossings (whether they cross or touch the x-axis).
- If the graph has 4 distinct crossings at different x-values (including \( -1 \)), the first option would be valid.
- If the graph has 3 crossings at the roots provided in options 2 or 3 (and the behavior at each root is different), then you might consider one of those.
- If it shows 4 roots but does not cross at all or is only touching at certain ones like \( 1, -3, -5, -15 \), then the fourth option could be the answer.
Without seeing the graph, I can't definitively tell you which one is correct, but you should choose based on the number of distinct roots and how the graph behaves at those roots.