Which polynomial sequence shows that the terms of the 2nd differences are constant?

A. [tex]$\{1, 7, 12, 19\}$[/tex]

B. [tex]$\{0, 4, 16, 34\}$[/tex]

C. [tex]$\{2, 6, 11, 14\}$[/tex]

D. [tex]$\{-6, -1, 14, 39\}$[/tex]



Answer :

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]