Question
The sum of a sequence of consecutive numbers from 1 to n is given by the formula an = x3 + 2x2−3x + 7 . Which level of differences of this polynomial is a constant value?(1 point) Responses 3rd differences 3rd differences 1st differences 1st differences 7th differences 7th differences 2nd differences
Answers
GPT-4o mini
To determine which level of differences of the polynomial \( a_n = x^3 + 2x^2 - 3x + 7 \) is constant, we need to analyze the polynomial's degree.
Since the polynomial is of degree 3 (the highest power of \( x \) is 3), we know that:
- The 1st differences (the differences between consecutive terms) will be a polynomial of degree \( 2 \).
- The 2nd differences (the differences of the 1st differences) will be a polynomial of degree \( 1 \).
- The 3rd differences (the differences of the 2nd differences) will be a constant value (i.e., a polynomial of degree \( 0 \)).
Therefore, the answer is:
**3rd differences**
Since the polynomial is of degree 3 (the highest power of \( x \) is 3), we know that:
- The 1st differences (the differences between consecutive terms) will be a polynomial of degree \( 2 \).
- The 2nd differences (the differences of the 1st differences) will be a polynomial of degree \( 1 \).
- The 3rd differences (the differences of the 2nd differences) will be a constant value (i.e., a polynomial of degree \( 0 \)).
Therefore, the answer is:
**3rd differences**