Bot GPT 4-o mini To determine whether \( x + 2 \) is a factor of \( x^3 - 3x + 2 \), we can use the Factor Theorem, which states that \( (x - c) \) is a factor of a polynomial \( P(x) \) if and only if \( P(c) = 0 \). In our case, we can rewrite \( x + 2 \) as \( x - (-2) \), indicating that we should check \( P(-2) \).
Let's evaluate \( P(-2) = (-2)^3 - 3(-2) + 2 \).
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Calculate \( (-2)^3 \): \[ (-2)^3 = -8 \]
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Calculate \( -3(-2) \): \[ -3(-2) = 6 \]
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Now substitute these values back into \( P(-2) \): \[ P(-2) = -8 + 6 + 2 \]
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Combine the terms: \[ P(-2) = -8 + 6 + 2 = -8 + 8 = 0 \]
Since \( P(-2) = 0 \), by the Factor Theorem, \( x + 2 \) is indeed a factor of \( x^3 - 3x + 2 \).
Conclusion
James is correct; \( x + 2 \) is a factor of \( x^3 - 3x + 2 \). We confirmed this by evaluating the polynomial at \( x = -2 \) and showing that it equals 0.
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