To determine the degree of a polynomial sequence where the 6th differences are constant, we need to understand a few key points about polynomial differences and their degrees.
### Understanding Polynomial Differences:
1. Polynomial Degree and Differences:
- The degree of a polynomial indicates the highest power of the variable in the polynomial.
- Successive differences of a polynomial sequence eventually become constant. The level at which they become constant corresponds to the degree of the polynomial.
2. Relationship Between Polynomial Degree and Differences:
- For a polynomial of degree [tex]\( n \)[/tex], the [tex]\( n \)[/tex]-th differences of the sequence of values generated by this polynomial will be constant.
- For example, a linear polynomial (degree 1) has a constant first difference. A quadratic polynomial (degree 2) has a constant second difference, and so on.
### Applying the Concept:
Given that the 6th differences of a polynomial sequence are constant, the degree of the polynomial must be exactly 6. This means that the corresponding polynomial has a highest power of 6 in its expression.
Conclusion:
The degree of a polynomial sequence, where the 6th differences are constant, is 6.
So, the answer to the question is:
The degree of the polynomial sequence should be 6.
