Question

Which polynomial sequence shows that the terms of the 2nd differences are constant? (1 point) Responses {2,6,11,14} left brace 2 comma 6 comma 11 comma 14 right brace {0,4,16,34} left brace 0 comma 4 comma 16 comma 34 right brace {1,7,12,19} left brace 1 comma 7 comma 12 comma 19 right brace {−6,−1,14,39}

Answer 1

To determine which polynomial sequence shows constant second differences, we first need to calculate the first differences of the sequences and then the second differences. A polynomial sequence has constant second differences if it can be represented by a quadratic polynomial.

Let's evaluate each option one by one:

1. {2, 6, 11, 14}:

   • First differences:
     • 6 - 2 = 4
     • 11 - 6 = 5
     • 14 - 11 = 3
     • First differences: {4, 5, 3}
   • Second differences:
     • 5 - 4 = 1
     • 3 - 5 = -2
     • Second differences: {1, -2} (not constant)

2. {0, 4, 16, 34}:

   • First differences:
     • 4 - 0 = 4
     • 16 - 4 = 12
     • 34 - 16 = 18
     • First differences: {4, 12, 18}
   • Second differences:
     • 12 - 4 = 8
     • 18 - 12 = 6
     • Second differences: {8, 6} (not constant)

3. {1, 7, 12, 19}:

   • First differences:
     • 7 - 1 = 6
     • 12 - 7 = 5
     • 19 - 12 = 7
     • First differences: {6, 5, 7}
   • Second differences:
     • 5 - 6 = -1
     • 7 - 5 = 2
     • Second differences: {-1, 2} (not constant)

4. {−6, −1, 14, 39}:

   • First differences:
     • -1 - (-6) = 5
     • 14 - (-1) = 15
     • 39 - 14 = 25
     • First differences: {5, 15, 25}
   • Second differences:
     • 15 - 5 = 10
     • 25 - 15 = 10
     • Second differences: {10, 10} (constant)

After calculating the first and second differences for all options, we find that the sequence {−6, −1, 14, 39} has constant second differences.

Thus, the correct response is {−6, −1, 14, 39}.