Question

Find the degree of the polynomial that produces these succesive results:
24,0,-24,0,144,504

Answer 1

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 \(\boxed{5}\).