To determine the degree of the polynomial that produces the given successive results \(24, 0, -24, 0, 144, 504\), we will analyze the differences between the values.
Let's denote the polynomial as \(P(x)\) evaluated at consecutive integers starting from \(x=0\):
- \(P(0) = 24\)
- \(P(1) = 0\)
- \(P(2) = -24\)
- \(P(3) = 0\)
- \(P(4) = 144\)
- \(P(5) = 504\)
Now, let's compute the first differences: \[ \begin{align*} P(1) - P(0) & = 0 - 24 = -24 \ P(2) - P(1) & = -24 - 0 = -24 \ P(3) - P(2) & = 0 - (-24) = 24 \ P(4) - P(3) & = 144 - 0 = 144 \ P(5) - P(4) & = 504 - 144 = 360 \ \end{align*} \] First differences: \(-24, -24, 24, 144, 360\)
Next, we compute the second differences: \[ \begin{align*} -24 - (-24) & = 0 \ 24 - (-24) & = 48 \ 144 - 24 & = 120 \ 360 - 144 & = 216 \ \end{align*} \] Second differences: \(0, 48, 120, 216\)
Next, we compute the third differences: \[ \begin{align*} 48 - 0 & = 48 \ 120 - 48 & = 72 \ 216 - 120 & = 96 \ \end{align*} \] Third differences: \(48, 72, 96\)
Next, we compute the fourth differences: \[ \begin{align*} 72 - 48 & = 24 \ 96 - 72 & = 24 \ \end{align*} \] Fourth differences: \(24, 24\)
Next, we compute the fifth differences: \[ 96 - 72 = 24 \] Fifth differences: \(24\)
The fifth differences are constant, which means the polynomial is of degree 5. Therefore, the degree of the polynomial that produces the given successive results is \(\boxed{5}\).