Which Polynomial Sequence Shows That The Terms Of The 2nd Differences Are Constant? -6,-1,14,39 2,6,11,14 0,4,16,34 1,7,12,19

To determine which polynomial sequence shows that the terms of the second differences are constant, we need to calculate the first and second differences for each sequence. If the second differences are constant, the sequence is generated by a quadratic polynomial.

Let's analyze each sequence:

### Sequence 1: \(-6, -1, 14, 39\)
- **First Differences:** 
  \[
  -1 - (-6) = 5, \quad 14 - (-1) = 15, \quad 39 - 14 = 25
  \]
  First differences: \(5, 15, 25\)
  
- **Second Differences:** 
  \[
  15 - 5 = 10, \quad 25 - 15 = 10
  \]
  Second differences: \(10, 10\) (constant)

### Sequence 2: \(2, 6, 11, 14\)
- **First Differences:** 
  \[
  6 - 2 = 4, \quad 11 - 6 = 5, \quad 14 - 11 = 3
  \]
  First differences: \(4, 5, 3\)
  
- **Second Differences:** 
  \[
  5 - 4 = 1, \quad 3 - 5 = -2
  \]
  Second differences: \(1, -2\) (not constant)

### Sequence 3: \(0, 4, 16, 34\)
- **First Differences:** 
  \[
  4 - 0 = 4, \quad 16 - 4 = 12, \quad 34 - 16 = 18
  \]
  First differences: \(4, 12, 18\)
  
- **Second Differences:** 
  \[
  12 - 4 = 8, \quad 18 - 12 = 6
  \]
  Second differences: \(8, 6\) (not constant)

### Sequence 4: \(1, 7, 12, 19\)
- **First Differences:** 
  \[
  7 - 1 = 6, \quad 12 - 7 = 5, \quad 19 - 12 = 7
  \]
  First differences: \(6, 5, 7\)
  
- **Second Differences:** 
  \[
  5 - 6 = -1, \quad 7 - 5 = 2
  \]
  Second differences: \(-1, 2\) (not constant)

### Conclusion:
The first sequence \(-6, -1, 14, 39\) is the only sequence where the second differences are constant, indicating that it is generated by a quadratic polynomial.

Would you like further details or have any questions? Here are some related questions:

1. How do you determine the degree of a polynomial from a sequence of numbers?
2. What is the relationship between a polynomial's degree and the constancy of differences?
3. Can you give an example of a cubic sequence and explain the differences?
4. How can you find the quadratic polynomial given a sequence?
5. What are some real-world applications of identifying polynomial sequences?

**Tip:** If the second differences are constant, the sequence is quadratic; if the third differences are constant, the sequence is cubic, and so on.

Discover how to identify polynomial sequences where the terms of the second differences are constant. Analyze specific sequences like -6, -1, 14, 39 and understand the mathematical process behind determining if a sequence is generated by a quadratic polynomial. Suitable for high school students and those interested in advanced algebraic concepts.

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