To find the degree of the polynomial that produces the given successive results \(24, 0, -24, 0, 144, 504\), we can analyze the differences between the values.
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List the values: \[ f(0) = 24, \quad f(1) = 0, \quad f(2) = -24, \quad f(3) = 0, \quad f(4) = 144, \quad f(5) = 504 \]
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Calculate the first differences: \[ \begin{align*} f(1) - f(0) &= 0 - 24 = -24 \ f(2) - f(1) &= -24 - 0 = -24 \ f(3) - f(2) &= 0 - (-24) = 24 \ f(4) - f(3) &= 144 - 0 = 144 \ f(5) - f(4) &= 504 - 144 = 360 \ \end{align*} \] First differences: \(-24, -24, 24, 144, 360\).
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Calculate the second differences: \[ \begin{align*} -24 - (-24) &= 0 \ 24 - (-24) &= 48 \ 144 - 24 &= 120 \ 360 - 144 &= 216 \ \end{align*} \] Second differences: \(0, 48, 120, 216\).
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Calculate the third differences: \[ \begin{align*} 48 - 0 &= 48 \ 120 - 48 &= 72 \ 216 - 120 &= 96 \ \end{align*} \] Third differences: \(48, 72, 96\).
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Calculate the fourth differences: \[ \begin{align*} 72 - 48 &= 24 \ 96 - 72 &= 24 \ \end{align*} \] Fourth differences: \(24, 24\).
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Calculate the fifth differences: \[ 24 - 24 = 0 \] Fifth differences: \(0\).
Since the fourth differences are constant, this indicates that the polynomial is of degree 4.
Therefore, the degree of the polynomial that produces the given successive results is \(\boxed{4}\).
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