Question

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

Answer 1

To find the degree of the polynomial that produces the given successive results, we can observe the differences between the successive outputs.

Let's denote the outputs as \( f(1) = 24 \), \( f(2) = 0 \), \( f(3) = -24 \), \( f(4) = 0 \), \( f(5) = 144 \), and \( f(6) = 504 \).

Now, let's compute the first differences:

\[
\begin{align*}
f(2) - f(1) & = 0 - 24 = -24 \\
f(3) - f(2) & = -24 - 0 = -24 \\
f(4) - f(3) & = 0 - (-24) = 24 \\
f(5) - f(4) & = 144 - 0 = 144 \\
f(6) - f(5) & = 504 - 144 = 360 \\
\end{align*}
\]

The first differences are:
\(-24, -24, 24, 144, 360\).

Next, we compute the second differences:

\[
\begin{align*}
-24 - (-24) & = 0 \\
24 - (-24) & = 48 \\
144 - 24 & = 120 \\
360 - 144 & = 216 \\
\end{align*}
\]

The second differences are:
\(0, 48, 120, 216\).

Next, we compute the third differences:

\[
\begin{align*}
48 - 0 & = 48 \\
120 - 48 & = 72 \\
216 - 120 & = 96 \\
\end{align*}
\]

The third differences are:
\(48, 72, 96\).

Next, we compute the fourth differences:

\[
\begin{align*}
72 - 48 & = 24 \\
96 - 72 & = 24 \\
\end{align*}
\]

The fourth differences are:
\(24, 24\).

The differences stabilized (the same value) at the fourth difference level, which indicates that the original polynomial is of degree 4.

Thus, the degree of the polynomial is \( \boxed{4} \).