To determine which polynomial sequence has constant second differences, we must perform a step-by-step analysis of the first and second differences for each sequence. Let's analyze each sequence individually.
### Sequence 1: [tex]\(\{1, 7, 12, 19\}\)[/tex]
1. First Differences:
[tex]\[
\begin{aligned}
7 - 1 &= 6, \\
12 - 7 &= 5, \\
19 - 12 &= 7.
\end{aligned}
\][/tex]
First differences: [tex]\(\{6, 5, 7\}\)[/tex]
2. Second Differences:
[tex]\[
\begin{aligned}
5 - 6 &= -1, \\
7 - 5 &= 2.
\end{aligned}
\][/tex]
Second differences: [tex]\(\{-1, 2\}\)[/tex]
The second differences are not constant for this sequence.
### Sequence 2: [tex]\(\{0, 4, 16, 34\}\)[/tex]
1. First Differences:
[tex]\[
\begin{aligned}
4 - 0 &= 4, \\
16 - 4 &= 12, \\
34 - 16 &= 18.
\end{aligned}
\][/tex]
First differences: [tex]\(\{4, 12, 18\}\)[/tex]
2. Second Differences:
[tex]\[
\begin{aligned}
12 - 4 &= 8, \\
18 - 12 &= 6.
\end{aligned}
\][/tex]
Second differences: [tex]\(\{8, 6\}\)[/tex]
The second differences are not constant for this sequence.
### Sequence 3: [tex]\(\{2, 6, 11, 14\}\)[/tex]
1. First Differences:
[tex]\[
\begin{aligned}
6 - 2 &= 4, \\
11 - 6 &= 5, \\
14 - 11 &= 3.
\end{aligned}
\][/tex]
First differences: [tex]\(\{4, 5, 3\}\)[/tex]
2. Second Differences:
[tex]\[
\begin{aligned}
5 - 4 &= 1, \\
3 - 5 &= -2.
\end{aligned}
\][/tex]
Second differences: [tex]\(\{1, -2\}\)[/tex]
The second differences are not constant for this sequence.
### Sequence 4: [tex]\(\{-6, -1, 14, 39\}\)[/tex]
1. First Differences:
[tex]\[
\begin{aligned}
-1 - (-6) &= 5, \\
14 - (-1) &= 15, \\
39 - 14 &= 25.
\end{aligned}
\][/tex]
First differences: [tex]\(\{5, 15, 25\}\)[/tex]
2. Second Differences:
[tex]\[
\begin{aligned}
15 - 5 &= 10, \\
25 - 15 &= 10.
\end{aligned}
\][/tex]
Second differences: [tex]\(\{10, 10\}\)[/tex]
The second differences for this sequence are constant.
### Conclusion
The fourth sequence, [tex]\(\{-6, -1, 14, 39\}\)[/tex], is the polynomial sequence that has constant second differences. Thus, the answer is:
[tex]\[\{-6, -1, 14, 39\}\][/tex]
